Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.3% → 83.6%

Time: 27.2s

Alternatives: 16

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% 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: 83.6% accurate, 0.6× speedup

Math

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

FPCore

NOTE: D 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 -5e-310)
   (*
    (/ (sqrt (- d)) (sqrt (- h)))
    (*
     (sqrt (/ d l))
     (- 1.0 (* 0.5 (* h (/ (pow (* M (* 0.5 (/ D d))) 2.0) l))))))
   (*
    (/ (sqrt d) (sqrt h))
    (*
     (/ (sqrt d) (sqrt l))
     (- 1.0 (* 0.5 (/ h (/ l (pow (/ (* M D) (* d 2.0)) 2.0)))))))))

C

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

Fortran

NOTE: D 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 <= (-5d-310)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * (h * (((m * (0.5d0 * (d_1 / d))) ** 2.0d0) / l)))))
    else
        tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0d0 - (0.5d0 * (h / (l / (((m * d_1) / (d * 2.0d0)) ** 2.0d0))))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: D 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, -5e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h / N[(l / N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 62.4%

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

   3. Simplified 63.2%

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

      1. expm1-log1p-u 62.7%

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

      2. expm1-udef 62.7%

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

      3. div-inv 62.7%

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

      4. associate-*l* 62.7%

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

      5. metadata-eval 62.7%

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

   5. Applied egg-rr 62.7%

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

      1. expm1-def 62.7%

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

      2. expm1-log1p 63.2%

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

      3. associate-*r/ 68.1%

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

      4. associate-*l/ 68.8%

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

      5. *-commutative 68.8%

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

   7. Simplified 68.8%

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

      1. frac-2neg 68.8%

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

      2. sqrt-div 82.3%

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

   9. Applied egg-rr 82.3%

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

   if -4.999999999999985e-310 < d

   1. Initial program 61.4%

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

   3. Simplified 60.5%

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

      1. expm1-log1p-u 60.3%

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

      2. expm1-udef 60.3%

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

      3. div-inv 60.3%

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

      4. associate-*l* 60.3%

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

      5. metadata-eval 60.3%

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

   5. Applied egg-rr 60.3%

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

      1. expm1-def 60.3%

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

      2. expm1-log1p 60.5%

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

      3. associate-*r/ 60.9%

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

      4. associate-*l/ 62.5%

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

      5. *-commutative 62.5%

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

   7. Simplified 62.5%

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

      1. clear-num 62.5%

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

      2. un-div-inv 62.5%

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

      3. associate-*r* 62.5%

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

      4. metadata-eval 62.5%

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

      5. div-inv 62.5%

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

      6. *-commutative 62.5%

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

      7. frac-times 63.4%

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

   9. Applied egg-rr 63.4%

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

       1. sqrt-div 74.0%

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

       2. div-inv 74.0%

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

   11. Applied egg-rr 74.0%

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

       1. associate-*r/ 74.0%

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

       2. *-rgt-identity 74.0%

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

   13. Simplified 74.0%

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

       1. sqrt-div 89.1%

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

       2. div-inv 89.0%

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

   15. Applied egg-rr 89.0%

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

       1. associate-*r/ 89.1%

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

       2. *-rgt-identity 89.1%

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

   17. Simplified 89.1%

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

4. Final simplification 85.4%

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

Alternative 2: 79.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot t_1\right) \cdot \frac{h}{\ell}\right)\\ t_3 := D \cdot \left(M \cdot \frac{0.5}{d}\right)\\ \mathbf{if}\;t_2 \leq -4 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \left(t_3 \cdot \left(\frac{h}{\ell} \cdot t_3\right)\right)\right)\right)\\ \mathbf{elif}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 5 \cdot 10^{+221}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{t_1}}\right)\right) \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

NOTE: D 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)))
        (t_1 (pow (/ (* M D) (* d 2.0)) 2.0))
        (t_2
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (* 0.5 t_1) (/ h l)))))
        (t_3 (* D (* M (/ 0.5 d)))))
   (if (<= t_2 -4e-276)
     (* (sqrt (/ d h)) (* t_0 (- 1.0 (* 0.5 (* t_3 (* (/ h l) t_3))))))
     (if (or (<= t_2 0.0) (not (<= t_2 5e+221)))
       (fabs (/ d (sqrt (* h l))))
       (* (* t_0 (- 1.0 (* 0.5 (/ h (/ l t_1))))) (/ 1.0 (sqrt (/ h d))))))))

C

D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = pow(((M * D) / (d * 2.0)), 2.0);
	double t_2 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((0.5 * t_1) * (h / l)));
	double t_3 = D * (M * (0.5 / d));
	double tmp;
	if (t_2 <= -4e-276) {
		tmp = sqrt((d / h)) * (t_0 * (1.0 - (0.5 * (t_3 * ((h / l) * t_3)))));
	} else if ((t_2 <= 0.0) || !(t_2 <= 5e+221)) {
		tmp = fabs((d / sqrt((h * l))));
	} else {
		tmp = (t_0 * (1.0 - (0.5 * (h / (l / t_1))))) * (1.0 / sqrt((h / d)));
	}
	return tmp;
}

Fortran

NOTE: D 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 / l))
    t_1 = ((m * d_1) / (d * 2.0d0)) ** 2.0d0
    t_2 = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((0.5d0 * t_1) * (h / l)))
    t_3 = d_1 * (m * (0.5d0 / d))
    if (t_2 <= (-4d-276)) then
        tmp = sqrt((d / h)) * (t_0 * (1.0d0 - (0.5d0 * (t_3 * ((h / l) * t_3)))))
    else if ((t_2 <= 0.0d0) .or. (.not. (t_2 <= 5d+221))) then
        tmp = abs((d / sqrt((h * l))))
    else
        tmp = (t_0 * (1.0d0 - (0.5d0 * (h / (l / t_1))))) * (1.0d0 / sqrt((h / d)))
    end if
    code = tmp
end function

Java

D = Math.abs(D);
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 t_1 = Math.pow(((M * D) / (d * 2.0)), 2.0);
	double t_2 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((0.5 * t_1) * (h / l)));
	double t_3 = D * (M * (0.5 / d));
	double tmp;
	if (t_2 <= -4e-276) {
		tmp = Math.sqrt((d / h)) * (t_0 * (1.0 - (0.5 * (t_3 * ((h / l) * t_3)))));
	} else if ((t_2 <= 0.0) || !(t_2 <= 5e+221)) {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	} else {
		tmp = (t_0 * (1.0 - (0.5 * (h / (l / t_1))))) * (1.0 / Math.sqrt((h / d)));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = math.pow(((M * D) / (d * 2.0)), 2.0)
	t_2 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((0.5 * t_1) * (h / l)))
	t_3 = D * (M * (0.5 / d))
	tmp = 0
	if t_2 <= -4e-276:
		tmp = math.sqrt((d / h)) * (t_0 * (1.0 - (0.5 * (t_3 * ((h / l) * t_3)))))
	elif (t_2 <= 0.0) or not (t_2 <= 5e+221):
		tmp = math.fabs((d / math.sqrt((h * l))))
	else:
		tmp = (t_0 * (1.0 - (0.5 * (h / (l / t_1))))) * (1.0 / math.sqrt((h / d)))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0
	t_2 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(0.5 * t_1) * Float64(h / l))))
	t_3 = Float64(D * Float64(M * Float64(0.5 / d)))
	tmp = 0.0
	if (t_2 <= -4e-276)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_0 * Float64(1.0 - Float64(0.5 * Float64(t_3 * Float64(Float64(h / l) * t_3))))));
	elseif ((t_2 <= 0.0) || !(t_2 <= 5e+221))
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(Float64(t_0 * Float64(1.0 - Float64(0.5 * Float64(h / Float64(l / t_1))))) * Float64(1.0 / sqrt(Float64(h / d))));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = ((M * D) / (d * 2.0)) ^ 2.0;
	t_2 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((0.5 * t_1) * (h / l)));
	t_3 = D * (M * (0.5 / d));
	tmp = 0.0;
	if (t_2 <= -4e-276)
		tmp = sqrt((d / h)) * (t_0 * (1.0 - (0.5 * (t_3 * ((h / l) * t_3)))));
	elseif ((t_2 <= 0.0) || ~((t_2 <= 5e+221)))
		tmp = abs((d / sqrt((h * l))));
	else
		tmp = (t_0 * (1.0 - (0.5 * (h / (l / t_1))))) * (1.0 / sqrt((h / d)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D 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]}, Block[{t$95$1 = N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = 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 * t$95$1), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-276], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(1.0 - N[(0.5 * N[(t$95$3 * N[(N[(h / l), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$2, 0.0], N[Not[LessEqual[t$95$2, 5e+221]], $MachinePrecision]], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(t$95$0 * N[(1.0 - N[(0.5 * N[(h / N[(l / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $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)))) < -4e-276

   1. Initial program 86.4%

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

   3. Simplified 86.4%

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

      1. expm1-log1p-u 85.2%

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

      2. expm1-udef 85.2%

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

      3. div-inv 85.2%

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

      4. associate-*l* 85.2%

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

      5. metadata-eval 85.2%

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

   5. Applied egg-rr 85.2%

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

      1. expm1-def 85.2%

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

      2. expm1-log1p 86.4%

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

      3. associate-*r/ 83.8%

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

      4. associate-*l/ 86.5%

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

      5. *-commutative 86.5%

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

   7. Simplified 86.5%

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

      1. clear-num 86.4%

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

      2. un-div-inv 86.4%

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

      3. associate-*r* 86.4%

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

      4. metadata-eval 86.4%

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

      5. div-inv 86.4%

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

      6. *-commutative 86.4%

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

      7. frac-times 86.4%

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

   9. Applied egg-rr 86.4%

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

       1. associate-/r/ 86.4%

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

       2. unpow2 86.4%

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

       3. associate-*r* 88.9%

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

       4. div-inv 88.9%

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

       5. associate-*l* 87.6%

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

       6. *-commutative 87.6%

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

       7. associate-/r* 87.6%

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

       8. metadata-eval 87.6%

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

       9. div-inv 87.6%

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

       10. associate-*l* 89.0%

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

       11. *-commutative 89.0%

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

       12. associate-/r* 89.0%

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

       13. metadata-eval 89.0%

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

   11. Applied egg-rr 89.0%

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

   if -4e-276 < (*.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 5.0000000000000002e221 < (*.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 20.9%

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

   2. Taylor expanded in d around inf 29.6%

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

      1. add-sqr-sqrt 28.8%

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

      2. sqrt-unprod 32.8%

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

      3. pow2 32.8%

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

      4. sqrt-div 32.8%

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

      5. metadata-eval 32.8%

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

      6. un-div-inv 32.8%

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

   4. Applied egg-rr 32.8%

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

      1. unpow2 32.8%

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

      2. rem-sqrt-square 55.5%

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

   6. Simplified 55.5%

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

   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)))) < 5.0000000000000002e221

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-udef 99.3%

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

      3. div-inv 99.3%

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

      4. associate-*l* 99.3%

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

      5. metadata-eval 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. expm1-def 99.3%

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

      2. expm1-log1p 99.3%

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

      3. associate-*r/ 99.3%

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

      4. associate-*l/ 99.3%

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

      5. *-commutative 99.3%

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

   7. Simplified 99.3%

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

      1. clear-num 99.3%

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

      2. un-div-inv 99.3%

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

      3. associate-*r* 99.3%

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

      4. metadata-eval 99.3%

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

      5. div-inv 99.3%

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

      6. *-commutative 99.3%

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

      7. frac-times 99.3%

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

   9. Applied egg-rr 99.3%

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

       1. clear-num 99.3%

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

       2. sqrt-div 99.5%

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

       3. metadata-eval 99.5%

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

   11. Applied egg-rr 99.5%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -4 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0 \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+221}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}}}\right)\right) \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \end{array} \]

Alternative 3: 78.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot t_1\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t_2 \leq -4 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 5 \cdot 10^{+221}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{t_1}}\right)\right) \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \end{array} \end{array} \]

FPCore

NOTE: D 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)))
        (t_1 (pow (/ (* M D) (* d 2.0)) 2.0))
        (t_2
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (* 0.5 t_1) (/ h l))))))
   (if (<= t_2 -4e-276)
     (*
      (sqrt (/ d h))
      (*
       t_0
       (- 1.0 (* 0.5 (pow (* (* M (* 0.5 (/ D d))) (sqrt (/ h l))) 2.0)))))
     (if (or (<= t_2 0.0) (not (<= t_2 5e+221)))
       (fabs (/ d (sqrt (* h l))))
       (* (* t_0 (- 1.0 (* 0.5 (/ h (/ l t_1))))) (/ 1.0 (sqrt (/ h d))))))))

C

D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = pow(((M * D) / (d * 2.0)), 2.0);
	double t_2 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((0.5 * t_1) * (h / l)));
	double tmp;
	if (t_2 <= -4e-276) {
		tmp = sqrt((d / h)) * (t_0 * (1.0 - (0.5 * pow(((M * (0.5 * (D / d))) * sqrt((h / l))), 2.0))));
	} else if ((t_2 <= 0.0) || !(t_2 <= 5e+221)) {
		tmp = fabs((d / sqrt((h * l))));
	} else {
		tmp = (t_0 * (1.0 - (0.5 * (h / (l / t_1))))) * (1.0 / sqrt((h / d)));
	}
	return tmp;
}

Fortran

NOTE: D 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) :: tmp
    t_0 = sqrt((d / l))
    t_1 = ((m * d_1) / (d * 2.0d0)) ** 2.0d0
    t_2 = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((0.5d0 * t_1) * (h / l)))
    if (t_2 <= (-4d-276)) then
        tmp = sqrt((d / h)) * (t_0 * (1.0d0 - (0.5d0 * (((m * (0.5d0 * (d_1 / d))) * sqrt((h / l))) ** 2.0d0))))
    else if ((t_2 <= 0.0d0) .or. (.not. (t_2 <= 5d+221))) then
        tmp = abs((d / sqrt((h * l))))
    else
        tmp = (t_0 * (1.0d0 - (0.5d0 * (h / (l / t_1))))) * (1.0d0 / sqrt((h / d)))
    end if
    code = tmp
end function

Java

D = Math.abs(D);
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 t_1 = Math.pow(((M * D) / (d * 2.0)), 2.0);
	double t_2 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((0.5 * t_1) * (h / l)));
	double tmp;
	if (t_2 <= -4e-276) {
		tmp = Math.sqrt((d / h)) * (t_0 * (1.0 - (0.5 * Math.pow(((M * (0.5 * (D / d))) * Math.sqrt((h / l))), 2.0))));
	} else if ((t_2 <= 0.0) || !(t_2 <= 5e+221)) {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	} else {
		tmp = (t_0 * (1.0 - (0.5 * (h / (l / t_1))))) * (1.0 / Math.sqrt((h / d)));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = math.pow(((M * D) / (d * 2.0)), 2.0)
	t_2 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((0.5 * t_1) * (h / l)))
	tmp = 0
	if t_2 <= -4e-276:
		tmp = math.sqrt((d / h)) * (t_0 * (1.0 - (0.5 * math.pow(((M * (0.5 * (D / d))) * math.sqrt((h / l))), 2.0))))
	elif (t_2 <= 0.0) or not (t_2 <= 5e+221):
		tmp = math.fabs((d / math.sqrt((h * l))))
	else:
		tmp = (t_0 * (1.0 - (0.5 * (h / (l / t_1))))) * (1.0 / math.sqrt((h / d)))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0
	t_2 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(0.5 * t_1) * Float64(h / l))))
	tmp = 0.0
	if (t_2 <= -4e-276)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_0 * Float64(1.0 - Float64(0.5 * (Float64(Float64(M * Float64(0.5 * Float64(D / d))) * sqrt(Float64(h / l))) ^ 2.0)))));
	elseif ((t_2 <= 0.0) || !(t_2 <= 5e+221))
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(Float64(t_0 * Float64(1.0 - Float64(0.5 * Float64(h / Float64(l / t_1))))) * Float64(1.0 / sqrt(Float64(h / d))));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = ((M * D) / (d * 2.0)) ^ 2.0;
	t_2 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((0.5 * t_1) * (h / l)));
	tmp = 0.0;
	if (t_2 <= -4e-276)
		tmp = sqrt((d / h)) * (t_0 * (1.0 - (0.5 * (((M * (0.5 * (D / d))) * sqrt((h / l))) ^ 2.0))));
	elseif ((t_2 <= 0.0) || ~((t_2 <= 5e+221)))
		tmp = abs((d / sqrt((h * l))));
	else
		tmp = (t_0 * (1.0 - (0.5 * (h / (l / t_1))))) * (1.0 / sqrt((h / d)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D 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]}, Block[{t$95$1 = N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = 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 * t$95$1), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-276], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(1.0 - N[(0.5 * N[Power[N[(N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$2, 0.0], N[Not[LessEqual[t$95$2, 5e+221]], $MachinePrecision]], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(t$95$0 * N[(1.0 - N[(0.5 * N[(h / N[(l / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $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)))) < -4e-276

   1. Initial program 86.4%

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

   3. Simplified 86.4%

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

      1. add-sqr-sqrt 86.5%

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

      2. pow2 86.5%

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

      3. sqrt-prod 86.4%

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

      4. sqrt-pow1 89.0%

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

      5. metadata-eval 89.0%

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

      6. pow1 89.0%

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

      7. div-inv 89.0%

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

      8. associate-*l* 89.0%

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

      9. metadata-eval 89.0%

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

   5. Applied egg-rr 89.0%

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

   if -4e-276 < (*.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 5.0000000000000002e221 < (*.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 20.9%

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

   2. Taylor expanded in d around inf 29.6%

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

      1. add-sqr-sqrt 28.8%

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

      2. sqrt-unprod 32.8%

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

      3. pow2 32.8%

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

      4. sqrt-div 32.8%

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

      5. metadata-eval 32.8%

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

      6. un-div-inv 32.8%

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

   4. Applied egg-rr 32.8%

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

      1. unpow2 32.8%

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

      2. rem-sqrt-square 55.5%

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

   6. Simplified 55.5%

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

   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)))) < 5.0000000000000002e221

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-udef 99.3%

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

      3. div-inv 99.3%

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

      4. associate-*l* 99.3%

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

      5. metadata-eval 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. expm1-def 99.3%

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

      2. expm1-log1p 99.3%

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

      3. associate-*r/ 99.3%

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

      4. associate-*l/ 99.3%

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

      5. *-commutative 99.3%

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

   7. Simplified 99.3%

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

      1. clear-num 99.3%

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

      2. un-div-inv 99.3%

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

      3. associate-*r* 99.3%

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

      4. metadata-eval 99.3%

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

      5. div-inv 99.3%

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

      6. *-commutative 99.3%

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

      7. frac-times 99.3%

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

   9. Applied egg-rr 99.3%

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

       1. clear-num 99.3%

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

       2. sqrt-div 99.5%

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

       3. metadata-eval 99.5%

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

   11. Applied egg-rr 99.5%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -4 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0 \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+221}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}}}\right)\right) \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \end{array} \]

Alternative 4: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \frac{h}{\frac{\ell}{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}}}\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;h \leq 2.6 \cdot 10^{+189}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot t_0\right) \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: D 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 (/ l (pow (/ (* M D) (* d 2.0)) 2.0)))))))
   (if (<= h -5e-310)
     (*
      (sqrt (/ d h))
      (*
       (- 1.0 (* 0.5 (* h (/ (pow (* M (* 0.5 (/ D d))) 2.0) l))))
       (/ (sqrt (- d)) (sqrt (- l)))))
     (if (<= h 2.6e+189)
       (* (* (/ (sqrt d) (sqrt l)) t_0) (/ 1.0 (sqrt (/ h d))))
       (* (/ (sqrt d) (sqrt h)) (* (sqrt (/ d l)) t_0))))))

C

D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (0.5 * (h / (l / pow(((M * D) / (d * 2.0)), 2.0))));
	double tmp;
	if (h <= -5e-310) {
		tmp = sqrt((d / h)) * ((1.0 - (0.5 * (h * (pow((M * (0.5 * (D / d))), 2.0) / l)))) * (sqrt(-d) / sqrt(-l)));
	} else if (h <= 2.6e+189) {
		tmp = ((sqrt(d) / sqrt(l)) * t_0) * (1.0 / sqrt((h / d)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * t_0);
	}
	return tmp;
}

Fortran

NOTE: D 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 = 1.0d0 - (0.5d0 * (h / (l / (((m * d_1) / (d * 2.0d0)) ** 2.0d0))))
    if (h <= (-5d-310)) then
        tmp = sqrt((d / h)) * ((1.0d0 - (0.5d0 * (h * (((m * (0.5d0 * (d_1 / d))) ** 2.0d0) / l)))) * (sqrt(-d) / sqrt(-l)))
    else if (h <= 2.6d+189) then
        tmp = ((sqrt(d) / sqrt(l)) * t_0) * (1.0d0 / sqrt((h / d)))
    else
        tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * t_0)
    end if
    code = tmp
end function

Java

D = Math.abs(D);
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 / (l / Math.pow(((M * D) / (d * 2.0)), 2.0))));
	double tmp;
	if (h <= -5e-310) {
		tmp = Math.sqrt((d / h)) * ((1.0 - (0.5 * (h * (Math.pow((M * (0.5 * (D / d))), 2.0) / l)))) * (Math.sqrt(-d) / Math.sqrt(-l)));
	} else if (h <= 2.6e+189) {
		tmp = ((Math.sqrt(d) / Math.sqrt(l)) * t_0) * (1.0 / Math.sqrt((h / d)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (Math.sqrt((d / l)) * t_0);
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = 1.0 - (0.5 * (h / (l / math.pow(((M * D) / (d * 2.0)), 2.0))))
	tmp = 0
	if h <= -5e-310:
		tmp = math.sqrt((d / h)) * ((1.0 - (0.5 * (h * (math.pow((M * (0.5 * (D / d))), 2.0) / l)))) * (math.sqrt(-d) / math.sqrt(-l)))
	elif h <= 2.6e+189:
		tmp = ((math.sqrt(d) / math.sqrt(l)) * t_0) * (1.0 / math.sqrt((h / d)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (math.sqrt((d / l)) * t_0)
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(1.0 - Float64(0.5 * Float64(h / Float64(l / (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)))))
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) / l)))) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))));
	elseif (h <= 2.6e+189)
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(l)) * t_0) * Float64(1.0 / sqrt(Float64(h / d))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(Float64(d / l)) * t_0));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 - (0.5 * (h / (l / (((M * D) / (d * 2.0)) ^ 2.0))));
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = sqrt((d / h)) * ((1.0 - (0.5 * (h * (((M * (0.5 * (D / d))) ^ 2.0) / l)))) * (sqrt(-d) / sqrt(-l)));
	elseif (h <= 2.6e+189)
		tmp = ((sqrt(d) / sqrt(l)) * t_0) * (1.0 / sqrt((h / d)));
	else
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D 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[(0.5 * N[(h / N[(l / N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.6e+189], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 62.4%

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

   3. Simplified 63.2%

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

      1. expm1-log1p-u 62.7%

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

      2. expm1-udef 62.7%

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

      3. div-inv 62.7%

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

      4. associate-*l* 62.7%

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

      5. metadata-eval 62.7%

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

   5. Applied egg-rr 62.7%

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

      1. expm1-def 62.7%

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

      2. expm1-log1p 63.2%

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

      3. associate-*r/ 68.1%

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

      4. associate-*l/ 68.8%

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

      5. *-commutative 68.8%

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

   7. Simplified 68.8%

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

      1. frac-2neg 68.8%

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

      2. sqrt-div 77.1%

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

   9. Applied egg-rr 77.1%

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

   if -4.999999999999985e-310 < h < 2.59999999999999981e189

   1. Initial program 67.3%

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

   3. Simplified 66.2%

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

      1. expm1-log1p-u 65.9%

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

      2. expm1-udef 65.9%

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

      3. div-inv 65.9%

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

      4. associate-*l* 65.9%

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

      5. metadata-eval 65.9%

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

   5. Applied egg-rr 65.9%

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

      1. expm1-def 65.9%

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

      2. expm1-log1p 66.2%

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

      3. associate-*r/ 66.5%

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

      4. associate-*l/ 68.6%

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

      5. *-commutative 68.6%

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

   7. Simplified 68.6%

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

      1. clear-num 68.6%

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

      2. un-div-inv 68.6%

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

      3. associate-*r* 68.6%

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

      4. metadata-eval 68.6%

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

      5. div-inv 68.6%

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

      6. *-commutative 68.6%

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

      7. frac-times 69.7%

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

   9. Applied egg-rr 69.7%

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

       1. clear-num 69.6%

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

       2. sqrt-div 70.7%

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

       3. metadata-eval 70.7%

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

   11. Applied egg-rr 70.7%

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

       1. sqrt-div 82.5%

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

       2. div-inv 82.5%

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

   13. Applied egg-rr 83.5%

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

       1. associate-*r/ 82.5%

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

       2. *-rgt-identity 82.5%

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

   15. Simplified 83.5%

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

   if 2.59999999999999981e189 < h

   1. Initial program 38.6%

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

   3. Simplified 38.6%

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

      1. expm1-log1p-u 38.6%

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

      2. expm1-udef 38.6%

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

      3. div-inv 38.6%

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

      4. associate-*l* 38.6%

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

      5. metadata-eval 38.6%

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

   5. Applied egg-rr 38.6%

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

      1. expm1-def 38.6%

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

      2. expm1-log1p 38.6%

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

      3. associate-*r/ 39.1%

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

      4. associate-*l/ 39.1%

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

      5. *-commutative 39.1%

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

   7. Simplified 39.1%

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

      1. clear-num 39.1%

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

      2. un-div-inv 39.1%

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

      3. associate-*r* 39.1%

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

      4. metadata-eval 39.1%

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

      5. div-inv 39.1%

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

      6. *-commutative 39.1%

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

      7. frac-times 39.1%

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

   9. Applied egg-rr 39.1%

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

       1. sqrt-div 80.0%

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

       2. div-inv 80.0%

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

   11. Applied egg-rr 67.8%

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

       1. associate-*r/ 80.0%

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

       2. *-rgt-identity 80.0%

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

   13. Simplified 67.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;h \leq 2.6 \cdot 10^{+189}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}}}\right)\right) \cdot \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}}}\right)\right)\\ \end{array} \]

Alternative 5: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \frac{h}{\frac{\ell}{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}}}\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_1 \cdot \left(\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;h \leq 2.4 \cdot 10^{+189}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot t_0\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: D 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 (/ l (pow (/ (* M D) (* d 2.0)) 2.0))))))
        (t_1 (sqrt (/ d h))))
   (if (<= h -5e-310)
     (*
      t_1
      (*
       (- 1.0 (* 0.5 (* h (/ (pow (* M (* 0.5 (/ D d))) 2.0) l))))
       (/ (sqrt (- d)) (sqrt (- l)))))
     (if (<= h 2.4e+189)
       (* (* (/ (sqrt d) (sqrt l)) t_0) t_1)
       (* (/ (sqrt d) (sqrt h)) (* (sqrt (/ d l)) t_0))))))

C

D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (0.5 * (h / (l / pow(((M * D) / (d * 2.0)), 2.0))));
	double t_1 = sqrt((d / h));
	double tmp;
	if (h <= -5e-310) {
		tmp = t_1 * ((1.0 - (0.5 * (h * (pow((M * (0.5 * (D / d))), 2.0) / l)))) * (sqrt(-d) / sqrt(-l)));
	} else if (h <= 2.4e+189) {
		tmp = ((sqrt(d) / sqrt(l)) * t_0) * t_1;
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * t_0);
	}
	return tmp;
}

Fortran

NOTE: D 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 / (l / (((m * d_1) / (d * 2.0d0)) ** 2.0d0))))
    t_1 = sqrt((d / h))
    if (h <= (-5d-310)) then
        tmp = t_1 * ((1.0d0 - (0.5d0 * (h * (((m * (0.5d0 * (d_1 / d))) ** 2.0d0) / l)))) * (sqrt(-d) / sqrt(-l)))
    else if (h <= 2.4d+189) then
        tmp = ((sqrt(d) / sqrt(l)) * t_0) * t_1
    else
        tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * t_0)
    end if
    code = tmp
end function

Java

D = Math.abs(D);
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 / (l / Math.pow(((M * D) / (d * 2.0)), 2.0))));
	double t_1 = Math.sqrt((d / h));
	double tmp;
	if (h <= -5e-310) {
		tmp = t_1 * ((1.0 - (0.5 * (h * (Math.pow((M * (0.5 * (D / d))), 2.0) / l)))) * (Math.sqrt(-d) / Math.sqrt(-l)));
	} else if (h <= 2.4e+189) {
		tmp = ((Math.sqrt(d) / Math.sqrt(l)) * t_0) * t_1;
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (Math.sqrt((d / l)) * t_0);
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = 1.0 - (0.5 * (h / (l / math.pow(((M * D) / (d * 2.0)), 2.0))))
	t_1 = math.sqrt((d / h))
	tmp = 0
	if h <= -5e-310:
		tmp = t_1 * ((1.0 - (0.5 * (h * (math.pow((M * (0.5 * (D / d))), 2.0) / l)))) * (math.sqrt(-d) / math.sqrt(-l)))
	elif h <= 2.4e+189:
		tmp = ((math.sqrt(d) / math.sqrt(l)) * t_0) * t_1
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (math.sqrt((d / l)) * t_0)
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(1.0 - Float64(0.5 * Float64(h / Float64(l / (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)))))
	t_1 = sqrt(Float64(d / h))
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(t_1 * Float64(Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) / l)))) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))));
	elseif (h <= 2.4e+189)
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(l)) * t_0) * t_1);
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(Float64(d / l)) * t_0));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 - (0.5 * (h / (l / (((M * D) / (d * 2.0)) ^ 2.0))));
	t_1 = sqrt((d / h));
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = t_1 * ((1.0 - (0.5 * (h * (((M * (0.5 * (D / d))) ^ 2.0) / l)))) * (sqrt(-d) / sqrt(-l)));
	elseif (h <= 2.4e+189)
		tmp = ((sqrt(d) / sqrt(l)) * t_0) * t_1;
	else
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D 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[(0.5 * N[(h / N[(l / N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(t$95$1 * N[(N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.4e+189], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 62.4%

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

   3. Simplified 63.2%

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

      1. expm1-log1p-u 62.7%

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

      2. expm1-udef 62.7%

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

      3. div-inv 62.7%

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

      4. associate-*l* 62.7%

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

      5. metadata-eval 62.7%

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

   5. Applied egg-rr 62.7%

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

      1. expm1-def 62.7%

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

      2. expm1-log1p 63.2%

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

      3. associate-*r/ 68.1%

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

      4. associate-*l/ 68.8%

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

      5. *-commutative 68.8%

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

   7. Simplified 68.8%

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

      1. frac-2neg 68.8%

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

      2. sqrt-div 77.1%

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

   9. Applied egg-rr 77.1%

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

   if -4.999999999999985e-310 < h < 2.4000000000000001e189

   1. Initial program 67.3%

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

   3. Simplified 66.2%

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

      1. expm1-log1p-u 65.9%

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

      2. expm1-udef 65.9%

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

      3. div-inv 65.9%

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

      4. associate-*l* 65.9%

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

      5. metadata-eval 65.9%

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

   5. Applied egg-rr 65.9%

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

      1. expm1-def 65.9%

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

      2. expm1-log1p 66.2%

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

      3. associate-*r/ 66.5%

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

      4. associate-*l/ 68.6%

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

      5. *-commutative 68.6%

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

   7. Simplified 68.6%

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

      1. clear-num 68.6%

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

      2. un-div-inv 68.6%

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

      3. associate-*r* 68.6%

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

      4. metadata-eval 68.6%

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

      5. div-inv 68.6%

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

      6. *-commutative 68.6%

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

      7. frac-times 69.7%

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

   9. Applied egg-rr 69.7%

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

       1. sqrt-div 82.5%

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

       2. div-inv 82.5%

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

   11. Applied egg-rr 82.5%

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

       1. associate-*r/ 82.5%

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

       2. *-rgt-identity 82.5%

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

   13. Simplified 82.5%

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

   if 2.4000000000000001e189 < h

   1. Initial program 38.6%

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

   3. Simplified 38.6%

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

      1. expm1-log1p-u 38.6%

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

      2. expm1-udef 38.6%

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

      3. div-inv 38.6%

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

      4. associate-*l* 38.6%

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

      5. metadata-eval 38.6%

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

   5. Applied egg-rr 38.6%

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

      1. expm1-def 38.6%

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

      2. expm1-log1p 38.6%

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

      3. associate-*r/ 39.1%

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

      4. associate-*l/ 39.1%

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

      5. *-commutative 39.1%

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

   7. Simplified 39.1%

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

      1. clear-num 39.1%

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

      2. un-div-inv 39.1%

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

      3. associate-*r* 39.1%

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

      4. metadata-eval 39.1%

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

      5. div-inv 39.1%

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

      6. *-commutative 39.1%

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

      7. frac-times 39.1%

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

   9. Applied egg-rr 39.1%

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

       1. sqrt-div 80.0%

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

       2. div-inv 80.0%

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

   11. Applied egg-rr 67.8%

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

       1. associate-*r/ 80.0%

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

       2. *-rgt-identity 80.0%

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

   13. Simplified 67.9%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;h \leq 2.4 \cdot 10^{+189}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}}}\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}}}\right)\right)\\ \end{array} \]

Alternative 6: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{d}{D} \cdot \left(\frac{2}{M} \cdot \frac{\ell}{\frac{D}{d} \cdot \left(0.5 \cdot M\right)}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: D 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 -1e-310)
     (*
      (/ (sqrt (- d)) (sqrt (- h)))
      (* t_0 (- 1.0 (* 0.5 (* h (/ (pow (* M (* 0.5 (/ D d))) 2.0) l))))))
     (if (<= l 5.4e-191)
       (*
        (sqrt (/ d h))
        (*
         (/ (sqrt d) (sqrt l))
         (-
          1.0
          (*
           0.5
           (/ h (* (/ d D) (* (/ 2.0 M) (/ l (* (/ D d) (* 0.5 M))))))))))
       (*
        (/ (sqrt d) (sqrt h))
        (*
         t_0
         (- 1.0 (* 0.5 (/ h (/ l (pow (/ (* M D) (* d 2.0)) 2.0)))))))))))

C

D = abs(D);
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 <= -1e-310) {
		tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (1.0 - (0.5 * (h * (pow((M * (0.5 * (D / d))), 2.0) / l)))));
	} else if (l <= 5.4e-191) {
		tmp = sqrt((d / h)) * ((sqrt(d) / sqrt(l)) * (1.0 - (0.5 * (h / ((d / D) * ((2.0 / M) * (l / ((D / d) * (0.5 * M)))))))));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0 - (0.5 * (h / (l / pow(((M * D) / (d * 2.0)), 2.0))))));
	}
	return tmp;
}

Fortran

NOTE: D 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 <= (-1d-310)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (1.0d0 - (0.5d0 * (h * (((m * (0.5d0 * (d_1 / d))) ** 2.0d0) / l)))))
    else if (l <= 5.4d-191) then
        tmp = sqrt((d / h)) * ((sqrt(d) / sqrt(l)) * (1.0d0 - (0.5d0 * (h / ((d / d_1) * ((2.0d0 / m) * (l / ((d_1 / d) * (0.5d0 * m)))))))))
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0d0 - (0.5d0 * (h / (l / (((m * d_1) / (d * 2.0d0)) ** 2.0d0))))))
    end if
    code = tmp
end function

Java

D = Math.abs(D);
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 <= -1e-310) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (t_0 * (1.0 - (0.5 * (h * (Math.pow((M * (0.5 * (D / d))), 2.0) / l)))));
	} else if (l <= 5.4e-191) {
		tmp = Math.sqrt((d / h)) * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 - (0.5 * (h / ((d / D) * ((2.0 / M) * (l / ((D / d) * (0.5 * M)))))))));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_0 * (1.0 - (0.5 * (h / (l / Math.pow(((M * D) / (d * 2.0)), 2.0))))));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if l <= -1e-310:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * (t_0 * (1.0 - (0.5 * (h * (math.pow((M * (0.5 * (D / d))), 2.0) / l)))))
	elif l <= 5.4e-191:
		tmp = math.sqrt((d / h)) * ((math.sqrt(d) / math.sqrt(l)) * (1.0 - (0.5 * (h / ((d / D) * ((2.0 / M) * (l / ((D / d) * (0.5 * M)))))))))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_0 * (1.0 - (0.5 * (h / (l / math.pow(((M * D) / (d * 2.0)), 2.0))))))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -1e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(t_0 * Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) / l))))));
	elseif (l <= 5.4e-191)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 - Float64(0.5 * Float64(h / Float64(Float64(d / D) * Float64(Float64(2.0 / M) * Float64(l / Float64(Float64(D / d) * Float64(0.5 * M))))))))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_0 * Float64(1.0 - Float64(0.5 * Float64(h / Float64(l / (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)))))));
	end
	return tmp
end

MATLAB

D = abs(D)
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 <= -1e-310)
		tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (1.0 - (0.5 * (h * (((M * (0.5 * (D / d))) ^ 2.0) / l)))));
	elseif (l <= 5.4e-191)
		tmp = sqrt((d / h)) * ((sqrt(d) / sqrt(l)) * (1.0 - (0.5 * (h / ((d / D) * ((2.0 / M) * (l / ((D / d) * (0.5 * M)))))))));
	else
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0 - (0.5 * (h / (l / (((M * D) / (d * 2.0)) ^ 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D 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, -1e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.4e-191], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h / N[(N[(d / D), $MachinePrecision] * N[(N[(2.0 / M), $MachinePrecision] * N[(l / N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 - N[(0.5 * N[(h / N[(l / N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.999999999999969e-311

   1. Initial program 62.4%

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

   3. Simplified 63.2%

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

      1. expm1-log1p-u 62.7%

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

      2. expm1-udef 62.7%

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

      3. div-inv 62.7%

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

      4. associate-*l* 62.7%

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

      5. metadata-eval 62.7%

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

   5. Applied egg-rr 62.7%

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

      1. expm1-def 62.7%

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

      2. expm1-log1p 63.2%

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

      3. associate-*r/ 68.1%

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

      4. associate-*l/ 68.8%

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

      5. *-commutative 68.8%

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

   7. Simplified 68.8%

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

      1. frac-2neg 68.8%

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

      2. sqrt-div 82.3%

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

   9. Applied egg-rr 82.3%

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

   if -9.999999999999969e-311 < l < 5.39999999999999998e-191

   1. Initial program 62.3%

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

   3. Simplified 62.3%

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

      1. expm1-log1p-u 62.3%

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

      2. expm1-udef 62.3%

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

      3. div-inv 62.3%

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

      4. associate-*l* 62.3%

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

      5. metadata-eval 62.3%

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

   5. Applied egg-rr 62.3%

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

      1. expm1-def 62.3%

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

      2. expm1-log1p 62.3%

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

      3. associate-*r/ 63.1%

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

      4. associate-*l/ 63.1%

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

      5. *-commutative 63.1%

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

   7. Simplified 63.1%

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

      1. clear-num 63.1%

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

      2. un-div-inv 63.1%

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

      3. associate-*r* 63.1%

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

      4. metadata-eval 63.1%

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

      5. div-inv 63.1%

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

      6. *-commutative 63.1%

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

      7. frac-times 63.1%

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

   9. Applied egg-rr 63.1%

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

       1. sqrt-div 83.2%

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

       2. div-inv 83.2%

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

   11. Applied egg-rr 83.2%

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

       1. associate-*r/ 83.2%

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

       2. *-rgt-identity 83.2%

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

   13. Simplified 83.2%

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

       1. *-un-lft-identity 63.1%

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

       2. unpow2 63.1%

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

       3. times-frac 63.5%

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

       4. clear-num 63.5%

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

       5. times-frac 63.4%

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

       6. times-frac 63.4%

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

       7. div-inv 63.4%

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

       8. metadata-eval 63.4%

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

   15. Applied egg-rr 87.6%

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

       1. associate-*l* 87.6%

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

   17. Simplified 87.6%

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

   if 5.39999999999999998e-191 < l

   1. Initial program 61.2%

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

   3. Simplified 60.1%

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

      1. expm1-log1p-u 59.8%

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

      2. expm1-udef 59.8%

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

      3. div-inv 59.8%

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

      4. associate-*l* 59.8%

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

      5. metadata-eval 59.8%

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

   5. Applied egg-rr 59.8%

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

      1. expm1-def 59.8%

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

      2. expm1-log1p 60.1%

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

      3. associate-*r/ 60.3%

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

      4. associate-*l/ 62.4%

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

      5. *-commutative 62.4%

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

   7. Simplified 62.4%

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

      1. clear-num 62.4%

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

      2. un-div-inv 62.4%

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

      3. associate-*r* 62.4%

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

      4. metadata-eval 62.4%

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

      5. div-inv 62.4%

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

      6. *-commutative 62.4%

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

      7. frac-times 63.5%

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

   9. Applied egg-rr 63.5%

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

       1. sqrt-div 89.4%

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

       2. div-inv 89.3%

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

   11. Applied egg-rr 77.6%

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

       1. associate-*r/ 89.4%

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

       2. *-rgt-identity 89.4%

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

   13. Simplified 77.6%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{d}{D} \cdot \left(\frac{2}{M} \cdot \frac{\ell}{\frac{D}{d} \cdot \left(0.5 \cdot M\right)}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}}}\right)\right)\\ \end{array} \]

Alternative 7: 66.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 0.235:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{{\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: D 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 0.235)
   (fabs (/ d (sqrt (* h l))))
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (- 1.0 (* 0.5 (/ h (/ l (pow (* (* M D) (/ 0.5 d)) 2.0)))))))))

C

D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= 0.235) {
		tmp = fabs((d / sqrt((h * l))));
	} else {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * (h / (l / pow(((M * D) * (0.5 / d)), 2.0))))));
	}
	return tmp;
}

Fortran

NOTE: D 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 <= 0.235d0) then
        tmp = abs((d / sqrt((h * l))))
    else
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * (h / (l / (((m * d_1) * (0.5d0 / d)) ** 2.0d0))))))
    end if
    code = tmp
end function

Java

D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= 0.235) {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	} else {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * (h / (l / Math.pow(((M * D) * (0.5 / d)), 2.0))))));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if D <= 0.235:
		tmp = math.fabs((d / math.sqrt((h * l))))
	else:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * (h / (l / math.pow(((M * D) * (0.5 / d)), 2.0))))))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (D <= 0.235)
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(h / Float64(l / (Float64(Float64(M * D) * Float64(0.5 / d)) ^ 2.0)))))));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (D <= 0.235)
		tmp = abs((d / sqrt((h * l))));
	else
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * (h / (l / (((M * D) * (0.5 / d)) ^ 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D 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, 0.235], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h / N[(l / N[Power[N[(N[(M * D), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 0.23499999999999999

   1. Initial program 65.5%

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

   2. Taylor expanded in d around inf 27.6%

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

      1. add-sqr-sqrt 24.6%

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

      2. sqrt-unprod 37.3%

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

      3. pow2 37.3%

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

      4. sqrt-div 37.3%

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

      5. metadata-eval 37.3%

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

      6. un-div-inv 37.3%

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

   4. Applied egg-rr 37.3%

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

      1. unpow2 37.3%

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

      2. rem-sqrt-square 52.2%

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

   6. Simplified 52.2%

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

   if 0.23499999999999999 < D

   1. Initial program 50.8%

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

   3. Simplified 52.4%

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

      1. expm1-log1p-u 51.9%

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

      2. expm1-udef 51.9%

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

      3. div-inv 51.9%

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

      4. associate-*l* 51.9%

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

      5. metadata-eval 51.9%

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

   5. Applied egg-rr 51.9%

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

      1. expm1-def 51.9%

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

      2. expm1-log1p 52.4%

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

      3. associate-*r/ 52.9%

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

      4. associate-*l/ 56.2%

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

      5. *-commutative 56.2%

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

   7. Simplified 56.2%

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

      1. clear-num 56.1%

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

      2. un-div-inv 56.1%

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

      3. associate-*r* 56.1%

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

      4. metadata-eval 56.1%

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

      5. div-inv 56.1%

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

      6. *-commutative 56.1%

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

      7. frac-times 54.5%

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

   9. Applied egg-rr 54.5%

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

       1. clear-num 54.4%

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

       2. associate-/r/ 54.5%

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

       3. *-commutative 54.5%

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

       4. associate-/r* 54.5%

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

       5. metadata-eval 54.5%

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

   11. Applied egg-rr 54.5%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 0.235:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{{\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}}\right)\right)\\ \end{array} \]

Alternative 8: 66.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 0.46:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{\frac{D}{d} \cdot \left(0.5 \cdot M\right)} \cdot \left(\frac{d}{D} \cdot \frac{2}{M}\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: D 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 0.46)
   (fabs (/ d (sqrt (* h l))))
   (*
    (/ 1.0 (sqrt (/ h d)))
    (*
     (sqrt (/ d l))
     (-
      1.0
      (* 0.5 (/ h (* (/ l (* (/ D d) (* 0.5 M))) (* (/ d D) (/ 2.0 M))))))))))

C

D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= 0.46) {
		tmp = fabs((d / sqrt((h * l))));
	} else {
		tmp = (1.0 / sqrt((h / d))) * (sqrt((d / l)) * (1.0 - (0.5 * (h / ((l / ((D / d) * (0.5 * M))) * ((d / D) * (2.0 / M)))))));
	}
	return tmp;
}

Fortran

NOTE: D 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 <= 0.46d0) then
        tmp = abs((d / sqrt((h * l))))
    else
        tmp = (1.0d0 / sqrt((h / d))) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * (h / ((l / ((d_1 / d) * (0.5d0 * m))) * ((d / d_1) * (2.0d0 / m)))))))
    end if
    code = tmp
end function

Java

D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (D <= 0.46) {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	} else {
		tmp = (1.0 / Math.sqrt((h / d))) * (Math.sqrt((d / l)) * (1.0 - (0.5 * (h / ((l / ((D / d) * (0.5 * M))) * ((d / D) * (2.0 / M)))))));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if D <= 0.46:
		tmp = math.fabs((d / math.sqrt((h * l))))
	else:
		tmp = (1.0 / math.sqrt((h / d))) * (math.sqrt((d / l)) * (1.0 - (0.5 * (h / ((l / ((D / d) * (0.5 * M))) * ((d / D) * (2.0 / M)))))))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (D <= 0.46)
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(Float64(1.0 / sqrt(Float64(h / d))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(h / Float64(Float64(l / Float64(Float64(D / d) * Float64(0.5 * M))) * Float64(Float64(d / D) * Float64(2.0 / M))))))));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (D <= 0.46)
		tmp = abs((d / sqrt((h * l))));
	else
		tmp = (1.0 / sqrt((h / d))) * (sqrt((d / l)) * (1.0 - (0.5 * (h / ((l / ((D / d) * (0.5 * M))) * ((d / D) * (2.0 / M)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D 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, 0.46], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h / N[(N[(l / N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(2.0 / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 0.46000000000000002

   1. Initial program 65.5%

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

   2. Taylor expanded in d around inf 27.6%

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

      1. add-sqr-sqrt 24.6%

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

      2. sqrt-unprod 37.3%

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

      3. pow2 37.3%

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

      4. sqrt-div 37.3%

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

      5. metadata-eval 37.3%

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

      6. un-div-inv 37.3%

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

   4. Applied egg-rr 37.3%

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

      1. unpow2 37.3%

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

      2. rem-sqrt-square 52.2%

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

   6. Simplified 52.2%

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

   if 0.46000000000000002 < D

   1. Initial program 50.8%

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

   3. Simplified 52.4%

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

      1. expm1-log1p-u 51.9%

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

      2. expm1-udef 51.9%

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

      3. div-inv 51.9%

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

      4. associate-*l* 51.9%

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

      5. metadata-eval 51.9%

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

   5. Applied egg-rr 51.9%

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

      1. expm1-def 51.9%

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

      2. expm1-log1p 52.4%

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

      3. associate-*r/ 52.9%

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

      4. associate-*l/ 56.2%

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

      5. *-commutative 56.2%

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

   7. Simplified 56.2%

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

      1. clear-num 56.1%

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

      2. un-div-inv 56.1%

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

      3. associate-*r* 56.1%

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

      4. metadata-eval 56.1%

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

      5. div-inv 56.1%

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

      6. *-commutative 56.1%

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

      7. frac-times 54.5%

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

   9. Applied egg-rr 54.5%

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

       1. clear-num 54.4%

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

       2. sqrt-div 55.3%

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

       3. metadata-eval 55.3%

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

   11. Applied egg-rr 55.3%

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

       1. *-un-lft-identity 55.3%

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

       2. unpow2 55.3%

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

       3. times-frac 55.3%

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

       4. clear-num 55.3%

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

       5. times-frac 55.3%

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

       6. times-frac 58.5%

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

       7. div-inv 58.5%

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

       8. metadata-eval 58.5%

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

   13. Applied egg-rr 58.5%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 0.46:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h}{\frac{\ell}{\frac{D}{d} \cdot \left(0.5 \cdot M\right)} \cdot \left(\frac{d}{D} \cdot \frac{2}{M}\right)}\right)\right)\\ \end{array} \]

Alternative 9: 67.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := D \cdot \left(M \cdot \frac{0.5}{d}\right)\\ \mathbf{if}\;D \leq 0.215:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(t_0 \cdot \left(\frac{h}{\ell} \cdot t_0\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: D 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 (* M (/ 0.5 d)))))
   (if (<= D 0.215)
     (fabs (/ d (sqrt (* h l))))
     (*
      (sqrt (/ d h))
      (* (sqrt (/ d l)) (- 1.0 (* 0.5 (* t_0 (* (/ h l) t_0)))))))))

C

D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = D * (M * (0.5 / d));
	double tmp;
	if (D <= 0.215) {
		tmp = fabs((d / sqrt((h * l))));
	} else {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * (t_0 * ((h / l) * t_0)))));
	}
	return tmp;
}

Fortran

NOTE: D 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_1 * (m * (0.5d0 / d))
    if (d_1 <= 0.215d0) then
        tmp = abs((d / sqrt((h * l))))
    else
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * (t_0 * ((h / l) * t_0)))))
    end if
    code = tmp
end function

Java

D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = D * (M * (0.5 / d));
	double tmp;
	if (D <= 0.215) {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	} else {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * (t_0 * ((h / l) * t_0)))));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = D * (M * (0.5 / d))
	tmp = 0
	if D <= 0.215:
		tmp = math.fabs((d / math.sqrt((h * l))))
	else:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * (t_0 * ((h / l) * t_0)))))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(D * Float64(M * Float64(0.5 / d)))
	tmp = 0.0
	if (D <= 0.215)
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(t_0 * Float64(Float64(h / l) * t_0))))));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = D * (M * (0.5 / d));
	tmp = 0.0;
	if (D <= 0.215)
		tmp = abs((d / sqrt((h * l))));
	else
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * (t_0 * ((h / l) * t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D 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[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, 0.215], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 0.214999999999999997

   1. Initial program 65.5%

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

   2. Taylor expanded in d around inf 27.6%

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

      1. add-sqr-sqrt 24.6%

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

      2. sqrt-unprod 37.3%

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

      3. pow2 37.3%

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

      4. sqrt-div 37.3%

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

      5. metadata-eval 37.3%

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

      6. un-div-inv 37.3%

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

   4. Applied egg-rr 37.3%

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

      1. unpow2 37.3%

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

      2. rem-sqrt-square 52.2%

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

   6. Simplified 52.2%

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

   if 0.214999999999999997 < D

   1. Initial program 50.8%

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

   3. Simplified 52.4%

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

      1. expm1-log1p-u 51.9%

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

      2. expm1-udef 51.9%

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

      3. div-inv 51.9%

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

      4. associate-*l* 51.9%

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

      5. metadata-eval 51.9%

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

   5. Applied egg-rr 51.9%

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

      1. expm1-def 51.9%

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

      2. expm1-log1p 52.4%

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

      3. associate-*r/ 52.9%

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

      4. associate-*l/ 56.2%

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

      5. *-commutative 56.2%

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

   7. Simplified 56.2%

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

      1. clear-num 56.1%

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

      2. un-div-inv 56.1%

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

      3. associate-*r* 56.1%

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

      4. metadata-eval 56.1%

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

      5. div-inv 56.1%

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

      6. *-commutative 56.1%

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

      7. frac-times 54.5%

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

   9. Applied egg-rr 54.5%

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

       1. associate-/r/ 50.8%

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

       2. unpow2 50.8%

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

       3. associate-*r* 50.9%

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

       4. div-inv 50.9%

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

       5. associate-*l* 50.9%

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

       6. *-commutative 50.9%

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

       7. associate-/r* 50.9%

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

       8. metadata-eval 50.9%

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

       9. div-inv 50.9%

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

       10. associate-*l* 52.6%

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

       11. *-commutative 52.6%

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

       12. associate-/r* 52.6%

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

       13. metadata-eval 52.6%

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

   11. Applied egg-rr 52.6%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 0.215:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 10: 44.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.9 \cdot 10^{+70}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\frac{1}{{\left(h \cdot \ell\right)}^{3}}\right)}^{0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

NOTE: D 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 1.9e+70)
   (fabs (/ d (sqrt (* h l))))
   (* d (pow (/ 1.0 (pow (* h l) 3.0)) 0.16666666666666666))))

C

D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 1.9e+70) {
		tmp = fabs((d / sqrt((h * l))));
	} else {
		tmp = d * pow((1.0 / pow((h * l), 3.0)), 0.16666666666666666);
	}
	return tmp;
}

Fortran

NOTE: D 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 <= 1.9d+70) then
        tmp = abs((d / sqrt((h * l))))
    else
        tmp = d * ((1.0d0 / ((h * l) ** 3.0d0)) ** 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 1.9e+70) {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	} else {
		tmp = d * Math.pow((1.0 / Math.pow((h * l), 3.0)), 0.16666666666666666);
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if M <= 1.9e+70:
		tmp = math.fabs((d / math.sqrt((h * l))))
	else:
		tmp = d * math.pow((1.0 / math.pow((h * l), 3.0)), 0.16666666666666666)
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= 1.9e+70)
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(d * (Float64(1.0 / (Float64(h * l) ^ 3.0)) ^ 0.16666666666666666));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (M <= 1.9e+70)
		tmp = abs((d / sqrt((h * l))));
	else
		tmp = d * ((1.0 / ((h * l) ^ 3.0)) ^ 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D 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, 1.9e+70], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Power[N[(1.0 / N[Power[N[(h * l), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 0.16666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.8999999999999999e70

   1. Initial program 61.1%

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

   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. add-sqr-sqrt 25.0%

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

      2. sqrt-unprod 37.9%

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

      3. pow2 37.9%

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

      4. sqrt-div 37.9%

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

      5. metadata-eval 37.9%

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

      6. un-div-inv 37.9%

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

   4. Applied egg-rr 37.9%

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

      1. unpow2 37.9%

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

      2. rem-sqrt-square 52.8%

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

   6. Simplified 52.8%

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

   if 1.8999999999999999e70 < M

   1. Initial program 65.8%

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

   2. Taylor expanded in d around inf 16.1%

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

      1. add-cbrt-cube 16.1%

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

      2. pow1/3 16.1%

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

      3. pow3 16.1%

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

      4. associate-/r* 16.1%

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

   4. Applied egg-rr 16.1%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{0.3333333333333333}}} \]
   5. Step-by-step derivation

      1. sqrt-pow1 16.1%

         \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

      2. associate-/r* 16.1%

         \[\leadsto d \cdot {\left({\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      3. cube-div 16.1%

         \[\leadsto d \cdot {\color{blue}{\left(\frac{{1}^{3}}{{\left(h \cdot \ell\right)}^{3}}\right)}}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      4. metadata-eval 16.1%

         \[\leadsto d \cdot {\left(\frac{\color{blue}{1}}{{\left(h \cdot \ell\right)}^{3}}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      5. metadata-eval 16.1%

         \[\leadsto d \cdot {\left(\frac{1}{{\left(h \cdot \ell\right)}^{3}}\right)}^{\color{blue}{0.16666666666666666}} \]

   6. Applied egg-rr 16.1%

      \[\leadsto d \cdot \color{blue}{{\left(\frac{1}{{\left(h \cdot \ell\right)}^{3}}\right)}^{0.16666666666666666}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.9 \cdot 10^{+70}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\frac{1}{{\left(h \cdot \ell\right)}^{3}}\right)}^{0.16666666666666666}\\ \end{array} \]

Alternative 11: 44.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 6.2 \cdot 10^{+26}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{e^{0.5 \cdot \mathsf{log1p}\left(h \cdot \ell + -1\right)}}\\ \end{array} \end{array} \]

FPCore

NOTE: D 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 6.2e+26)
   (fabs (/ d (sqrt (* h l))))
   (/ d (exp (* 0.5 (log1p (+ (* h l) -1.0)))))))

C

D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 6.2e+26) {
		tmp = fabs((d / sqrt((h * l))));
	} else {
		tmp = d / exp((0.5 * log1p(((h * l) + -1.0))));
	}
	return tmp;
}

Java

D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 6.2e+26) {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	} else {
		tmp = d / Math.exp((0.5 * Math.log1p(((h * l) + -1.0))));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if M <= 6.2e+26:
		tmp = math.fabs((d / math.sqrt((h * l))))
	else:
		tmp = d / math.exp((0.5 * math.log1p(((h * l) + -1.0))))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= 6.2e+26)
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(d / exp(Float64(0.5 * log1p(Float64(Float64(h * l) + -1.0)))));
	end
	return tmp
end

Wolfram

NOTE: D 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, 6.2e+26], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d / N[Exp[N[(0.5 * N[Log[1 + N[(N[(h * l), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 6.1999999999999999e26

   1. Initial program 60.9%

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

   2. Taylor expanded in d around inf 28.4%

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

      1. add-sqr-sqrt 25.4%

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

      2. sqrt-unprod 37.0%

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

      3. pow2 37.0%

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

      4. sqrt-div 37.0%

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

      5. metadata-eval 37.0%

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

      6. un-div-inv 37.0%

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

   4. Applied egg-rr 37.0%

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

      1. unpow2 37.0%

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

      2. rem-sqrt-square 52.2%

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

   6. Simplified 52.2%

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

   if 6.1999999999999999e26 < M

   1. Initial program 65.5%

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

   2. Taylor expanded in d around inf 16.7%

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

      1. expm1-log1p-u 12.1%

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

      2. expm1-udef 7.3%

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

      3. sqrt-div 8.2%

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

      4. metadata-eval 8.2%

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

      5. un-div-inv 8.2%

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

   4. Applied egg-rr 8.2%

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

      1. expm1-def 13.1%

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

      2. expm1-log1p 17.7%

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

   6. Simplified 17.7%

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

      1. pow1/2 17.7%

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

      2. pow-to-exp 16.7%

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

   8. Applied egg-rr 16.7%

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

      1. log1p-expm1-u 20.2%

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

      2. expm1-udef 20.2%

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

      3. add-exp-log 20.2%

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

   10. Applied egg-rr 20.2%

       \[\leadsto \frac{d}{e^{\color{blue}{\mathsf{log1p}\left(h \cdot \ell - 1\right)} \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.2 \cdot 10^{+26}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{e^{0.5 \cdot \mathsf{log1p}\left(h \cdot \ell + -1\right)}}\\ \end{array} \]

Alternative 12: 46.6% accurate, 1.6× speedup

Math

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

FPCore

NOTE: D 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 -5e-310)
   (fabs (/ d (sqrt (* h l))))
   (* d (* (pow l -0.5) (pow h -0.5)))))

C

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

Fortran

NOTE: D 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 <= (-5d-310)) then
        tmp = abs((d / sqrt((h * l))))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: D 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, -5e-310], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 62.4%

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

   2. Taylor expanded in d around inf 7.9%

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

      1. add-sqr-sqrt 1.8%

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

      2. sqrt-unprod 35.4%

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

      3. pow2 35.4%

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

      4. sqrt-div 35.4%

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

      5. metadata-eval 35.4%

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

      6. un-div-inv 35.5%

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

   4. Applied egg-rr 35.5%

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

      1. unpow2 35.5%

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

      2. rem-sqrt-square 47.8%

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

   6. Simplified 47.8%

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

   if -4.999999999999985e-310 < h

   1. Initial program 61.4%

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

   2. Taylor expanded in d around inf 46.8%

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

      1. expm1-log1p-u 45.3%

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

      2. expm1-udef 31.6%

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

      3. sqrt-div 32.3%

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

      4. metadata-eval 32.3%

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

      5. un-div-inv 32.3%

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

   4. Applied egg-rr 32.3%

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

      1. expm1-def 46.0%

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

      2. expm1-log1p 47.5%

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

   6. Simplified 47.5%

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

      1. clear-num 47.5%

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

      2. associate-/r/ 47.5%

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

      3. pow1/2 47.5%

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

      4. pow-flip 47.5%

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

      5. metadata-eval 47.5%

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

   8. Applied egg-rr 47.5%

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

      1. *-commutative 47.5%

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

      2. unpow-prod-down 54.9%

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

   10. Applied egg-rr 54.9%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 13: 43.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;M \leq 2.6 \cdot 10^{+70}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: D 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 (* h l))))) (if (<= M 2.6e+70) (fabs t_0) t_0)))

C

D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = d / sqrt((h * l));
	double tmp;
	if (M <= 2.6e+70) {
		tmp = fabs(t_0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: D 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((h * l))
    if (m <= 2.6d+70) then
        tmp = abs(t_0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d / Math.sqrt((h * l));
	double tmp;
	if (M <= 2.6e+70) {
		tmp = Math.abs(t_0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = d / math.sqrt((h * l))
	tmp = 0
	if M <= 2.6e+70:
		tmp = math.fabs(t_0)
	else:
		tmp = t_0
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (M <= 2.6e+70)
		tmp = abs(t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = d / sqrt((h * l));
	tmp = 0.0;
	if (M <= 2.6e+70)
		tmp = abs(t_0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D 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[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 2.6e+70], N[Abs[t$95$0], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if M < 2.6e70

   1. Initial program 61.1%

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

   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. add-sqr-sqrt 25.0%

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

      2. sqrt-unprod 37.9%

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

      3. pow2 37.9%

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

      4. sqrt-div 37.9%

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

      5. metadata-eval 37.9%

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

      6. un-div-inv 37.9%

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

   4. Applied egg-rr 37.9%

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

      1. unpow2 37.9%

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

      2. rem-sqrt-square 52.8%

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

   6. Simplified 52.8%

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

   if 2.6e70 < M

   1. Initial program 65.8%

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

   2. Taylor expanded in d around inf 16.1%

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

      1. expm1-log1p-u 10.8%

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

      2. expm1-udef 6.9%

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

      3. sqrt-div 8.0%

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

      4. metadata-eval 8.0%

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

      5. un-div-inv 8.0%

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

   4. Applied egg-rr 8.0%

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

      1. expm1-def 12.0%

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

      2. expm1-log1p 17.3%

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

   6. Simplified 17.3%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.6 \cdot 10^{+70}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 14: 46.7% accurate, 1.6× speedup

Math

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

FPCore

NOTE: D 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 -5e-310) (fabs (/ d (sqrt (* h l)))) (/ d (* (sqrt h) (sqrt l)))))

C

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

Fortran

NOTE: D 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 <= (-5d-310)) then
        tmp = abs((d / sqrt((h * l))))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: D 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, -5e-310], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 62.4%

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

   2. Taylor expanded in d around inf 7.9%

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

      1. add-sqr-sqrt 1.8%

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

      2. sqrt-unprod 35.4%

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

      3. pow2 35.4%

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

      4. sqrt-div 35.4%

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

      5. metadata-eval 35.4%

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

      6. un-div-inv 35.5%

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

   4. Applied egg-rr 35.5%

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

      1. unpow2 35.5%

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

      2. rem-sqrt-square 47.8%

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

   6. Simplified 47.8%

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

   if -4.999999999999985e-310 < h

   1. Initial program 61.4%

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

   2. Taylor expanded in d around inf 46.8%

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

      1. expm1-log1p-u 45.3%

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

      2. expm1-udef 31.6%

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

      3. sqrt-div 32.3%

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

      4. metadata-eval 32.3%

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

      5. un-div-inv 32.3%

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

   4. Applied egg-rr 32.3%

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

      1. expm1-def 46.0%

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

      2. expm1-log1p 47.5%

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

   6. Simplified 47.5%

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

      1. *-commutative 47.5%

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

      2. sqrt-prod 54.8%

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

   8. Applied egg-rr 54.8%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 15: 27.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot {\left(h \cdot \ell\right)}^{-0.5} \end{array} \]

FPCore

NOTE: D 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 (pow (* h l) -0.5)))

C

D = abs(D);
assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d * pow((h * l), -0.5);
}

Fortran

NOTE: D 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 * ((h * l) ** (-0.5d0))
end function

Java

D = Math.abs(D);
assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d * Math.pow((h * l), -0.5);
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * math.pow((h * l), -0.5)

Julia

D = abs(D)
M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d * (Float64(h * l) ^ -0.5))
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d * ((h * l) ^ -0.5);
end

Wolfram

NOTE: D 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[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in d around inf 25.7%

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

   1. expm1-log1p-u 25.3%

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

   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. inv-pow 17.2%

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

   4. sqrt-pow1 17.5%

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

   5. metadata-eval 17.5%

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

4. Applied egg-rr 17.5%

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

   1. expm1-def 25.6%

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

   2. expm1-log1p 26.0%

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

6. Simplified 26.0%

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

7. Final simplification 26.0%

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

Alternative 16: 27.1% accurate, 3.2× speedup

Math

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

FPCore

NOTE: D 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 (* h l))))

C

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

Fortran

NOTE: D 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((h * l))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: D 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[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in d around inf 25.7%

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

   1. expm1-log1p-u 22.2%

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

   2. expm1-udef 15.9%

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

   3. sqrt-div 16.2%

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

   4. metadata-eval 16.2%

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

   5. un-div-inv 16.2%

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

4. Applied egg-rr 16.2%

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

   1. expm1-def 22.5%

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

   2. expm1-log1p 26.0%

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

6. Simplified 26.0%

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

7. Final simplification 26.0%

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

Reproduce

herbie shell --seed 2023310 
(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)))))