Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.0% → 77.6%

Time: 14.9s

Alternatives: 16

Speedup: 0.5×

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: 66.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 77.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M}{d \cdot 2} \cdot D\_m\\ t_1 := {\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)}\\ \mathbf{if}\;\left(t\_1 \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+245}:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - t\_0 \cdot \left(\frac{h}{\ell} \cdot \left(0.5 \cdot t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(\left(\frac{D\_m}{d} \cdot M\right) \cdot \frac{h}{\ell}\right) \cdot \left(0.5 \cdot D\_m\right)}{4} \cdot \frac{M}{d}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ M (* d 2.0)) D_m)) (t_1 (pow (/ d h) (pow 2.0 -1.0))))
   (if (<=
        (*
         (* t_1 (pow (/ d l) (pow 2.0 -1.0)))
         (-
          1.0
          (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))
        2e+245)
     (* (* t_1 (sqrt (/ d l))) (- 1.0 (* t_0 (* (/ h l) (* 0.5 t_0)))))
     (*
      (- 1.0 (* (/ (* (* (* (/ D_m d) M) (/ h l)) (* 0.5 D_m)) 4.0) (/ M d)))
      (/ (fabs d) (sqrt (* h l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (M / (d * 2.0)) * D_m;
	double t_1 = pow((d / h), pow(2.0, -1.0));
	double tmp;
	if (((t_1 * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= 2e+245) {
		tmp = (t_1 * sqrt((d / l))) * (1.0 - (t_0 * ((h / l) * (0.5 * t_0))));
	} else {
		tmp = (1.0 - ((((((D_m / d) * M) * (h / l)) * (0.5 * D_m)) / 4.0) * (M / d))) * (fabs(d) / sqrt((h * l)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (m / (d * 2.0d0)) * d_m
    t_1 = (d / h) ** (2.0d0 ** (-1.0d0))
    if (((t_1 * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= 2d+245) then
        tmp = (t_1 * sqrt((d / l))) * (1.0d0 - (t_0 * ((h / l) * (0.5d0 * t_0))))
    else
        tmp = (1.0d0 - ((((((d_m / d) * m) * (h / l)) * (0.5d0 * d_m)) / 4.0d0) * (m / d))) * (abs(d) / sqrt((h * l)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (M / (d * 2.0)) * D_m;
	double t_1 = Math.pow((d / h), Math.pow(2.0, -1.0));
	double tmp;
	if (((t_1 * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= 2e+245) {
		tmp = (t_1 * Math.sqrt((d / l))) * (1.0 - (t_0 * ((h / l) * (0.5 * t_0))));
	} else {
		tmp = (1.0 - ((((((D_m / d) * M) * (h / l)) * (0.5 * D_m)) / 4.0) * (M / d))) * (Math.abs(d) / Math.sqrt((h * l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = (M / (d * 2.0)) * D_m
	t_1 = math.pow((d / h), math.pow(2.0, -1.0))
	tmp = 0
	if ((t_1 * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= 2e+245:
		tmp = (t_1 * math.sqrt((d / l))) * (1.0 - (t_0 * ((h / l) * (0.5 * t_0))))
	else:
		tmp = (1.0 - ((((((D_m / d) * M) * (h / l)) * (0.5 * D_m)) / 4.0) * (M / d))) * (math.fabs(d) / math.sqrt((h * l)))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(M / Float64(d * 2.0)) * D_m)
	t_1 = Float64(d / h) ^ (2.0 ^ -1.0)
	tmp = 0.0
	if (Float64(Float64(t_1 * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 2e+245)
		tmp = Float64(Float64(t_1 * sqrt(Float64(d / l))) * Float64(1.0 - Float64(t_0 * Float64(Float64(h / l) * Float64(0.5 * t_0)))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(D_m / d) * M) * Float64(h / l)) * Float64(0.5 * D_m)) / 4.0) * Float64(M / d))) * Float64(abs(d) / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = (M / (d * 2.0)) * D_m;
	t_1 = (d / h) ^ (2.0 ^ -1.0);
	tmp = 0.0;
	if (((t_1 * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= 2e+245)
		tmp = (t_1 * sqrt((d / l))) * (1.0 - (t_0 * ((h / l) * (0.5 * t_0))));
	else
		tmp = (1.0 - ((((((D_m / d) * M) * (h / l)) * (0.5 * D_m)) / 4.0) * (M / d))) * (abs(d) / sqrt((h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+245], N[(N[(t$95$1 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * M), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[(0.5 * D$95$m), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000009e245

   1. Initial program 88.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 86.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 86.5

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 86.5

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

   6. Applied rewrites 86.5%

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

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l/ N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r/ N/A

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

      14. lift-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 N/A

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

   8. Applied rewrites 90.4%

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

   if 2.00000000000000009e245 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 32.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 18.1

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

   4. Applied rewrites 18.1%

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

   6. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 68.0

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

   8. Applied rewrites 68.0%

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

   10. Applied rewrites 70.4%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+245}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{M}{d \cdot 2} \cdot D\right) \cdot \left(\frac{h}{\ell} \cdot \left(0.5 \cdot \left(\frac{M}{d \cdot 2} \cdot D\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{h}{\ell}\right) \cdot \left(0.5 \cdot D\right)}{4} \cdot \frac{M}{d}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ t_2 := \left(\left(-0.125 \cdot \frac{\frac{D\_m \cdot D\_m}{d}}{d}\right) \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+306}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.125 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \frac{\frac{M \cdot M}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+245}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_1 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ (fabs d) (sqrt (* h l))))
        (t_2
         (* (* (* -0.125 (/ (/ (* D_m D_m) d) d)) (* h (/ (* M M) l))) t_1)))
   (if (<= t_0 -1e+306)
     t_2
     (if (<= t_0 0.0)
       (* (fma (* (* -0.125 (* D_m D_m)) (/ (/ (* M M) d) d)) (/ h l) 1.0) t_1)
       (if (<= t_0 2e+245)
         (* (sqrt (/ d l)) (sqrt (/ d h)))
         (if (<= t_0 INFINITY) (* t_1 1.0) t_2))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = fabs(d) / sqrt((h * l));
	double t_2 = ((-0.125 * (((D_m * D_m) / d) / d)) * (h * ((M * M) / l))) * t_1;
	double tmp;
	if (t_0 <= -1e+306) {
		tmp = t_2;
	} else if (t_0 <= 0.0) {
		tmp = fma(((-0.125 * (D_m * D_m)) * (((M * M) / d) / d)), (h / l), 1.0) * t_1;
	} else if (t_0 <= 2e+245) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1 * 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(abs(d) / sqrt(Float64(h * l)))
	t_2 = Float64(Float64(Float64(-0.125 * Float64(Float64(Float64(D_m * D_m) / d) / d)) * Float64(h * Float64(Float64(M * M) / l))) * t_1)
	tmp = 0.0
	if (t_0 <= -1e+306)
		tmp = t_2;
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(Float64(Float64(-0.125 * Float64(D_m * D_m)) * Float64(Float64(Float64(M * M) / d) / d)), Float64(h / l), 1.0) * t_1);
	elseif (t_0 <= 2e+245)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = Float64(t_1 * 1.0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-0.125 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+306], t$95$2, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 2e+245], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$1 * 1.0), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.00000000000000002e306 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 52.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 26.9

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

   4. Applied rewrites 26.9%

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

   6. Applied rewrites 66.3%

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

   7. Taylor expanded in d around 0

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

      1. times-frac N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2}}{{d}^{2}}\right)} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. unpow2 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2}}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. associate-/r* N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \color{blue}{\frac{\frac{{D}^{2}}{d}}{d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \color{blue}{\frac{\frac{{D}^{2}}{d}}{d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\color{blue}{\frac{{D}^{2}}{d}}}{d}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{\color{blue}{D \cdot D}}{d}}{d}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{\color{blue}{D \cdot D}}{d}}{d}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      11. *-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      12. associate-/l* N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \color{blue}{\left(h \cdot \frac{{M}^{2}}{\ell}\right)}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \color{blue}{\left(h \cdot \frac{{M}^{2}}{\ell}\right)}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \left(h \cdot \color{blue}{\frac{{M}^{2}}{\ell}}\right)\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      15. unpow2 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \left(h \cdot \frac{\color{blue}{M \cdot M}}{\ell}\right)\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      16. lower-*.f64 55.6

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

   9. Applied rewrites 55.6%

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

   if -1.00000000000000002e306 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 68.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 37.1

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

   4. Applied rewrites 37.1%

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

   6. Applied rewrites 75.1%

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

   7. Taylor expanded in d around 0

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right)} \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{{M}^{2}}{\color{blue}{d \cdot d}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{\frac{{M}^{2}}{d}}}{d}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{\color{blue}{M \cdot M}}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      12. lower-*.f64 46.8

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

   9. Applied rewrites 46.8%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000009e245

   1. Initial program 99.5%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 34.9

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

   8. Applied rewrites 34.9%

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

   10. Applied rewrites 36.2%

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

   12. Applied rewrites 99.5%

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

   if 2.00000000000000009e245 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 64.6%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 64.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. sqrt-div N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower-fabs.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-*.f64 97.5

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

   7. Applied rewrites 97.5%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{+306}:\\ \;\;\;\;\left(\left(-0.125 \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{M \cdot M}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+245}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.125 \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ t_2 := t\_1 \cdot 1\\ t_3 := \left(\left(-0.125 \cdot \frac{\frac{D\_m \cdot D\_m}{d}}{d}\right) \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_0 \leq -10000000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+245}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ (fabs d) (sqrt (* h l))))
        (t_2 (* t_1 1.0))
        (t_3
         (* (* (* -0.125 (/ (/ (* D_m D_m) d) d)) (* h (/ (* M M) l))) t_1)))
   (if (<= t_0 -10000000000000.0)
     t_3
     (if (<= t_0 0.0)
       t_2
       (if (<= t_0 2e+245)
         (* (sqrt (/ d l)) (sqrt (/ d h)))
         (if (<= t_0 INFINITY) t_2 t_3))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = fabs(d) / sqrt((h * l));
	double t_2 = t_1 * 1.0;
	double t_3 = ((-0.125 * (((D_m * D_m) / d) / d)) * (h * ((M * M) / l))) * t_1;
	double tmp;
	if (t_0 <= -10000000000000.0) {
		tmp = t_3;
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 2e+245) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.abs(d) / Math.sqrt((h * l));
	double t_2 = t_1 * 1.0;
	double t_3 = ((-0.125 * (((D_m * D_m) / d) / d)) * (h * ((M * M) / l))) * t_1;
	double tmp;
	if (t_0 <= -10000000000000.0) {
		tmp = t_3;
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 2e+245) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.fabs(d) / math.sqrt((h * l))
	t_2 = t_1 * 1.0
	t_3 = ((-0.125 * (((D_m * D_m) / d) / d)) * (h * ((M * M) / l))) * t_1
	tmp = 0
	if t_0 <= -10000000000000.0:
		tmp = t_3
	elif t_0 <= 0.0:
		tmp = t_2
	elif t_0 <= 2e+245:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif t_0 <= math.inf:
		tmp = t_2
	else:
		tmp = t_3
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(abs(d) / sqrt(Float64(h * l)))
	t_2 = Float64(t_1 * 1.0)
	t_3 = Float64(Float64(Float64(-0.125 * Float64(Float64(Float64(D_m * D_m) / d) / d)) * Float64(h * Float64(Float64(M * M) / l))) * t_1)
	tmp = 0.0
	if (t_0 <= -10000000000000.0)
		tmp = t_3;
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 2e+245)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = abs(d) / sqrt((h * l));
	t_2 = t_1 * 1.0;
	t_3 = ((-0.125 * (((D_m * D_m) / d) / d)) * (h * ((M * M) / l))) * t_1;
	tmp = 0.0;
	if (t_0 <= -10000000000000.0)
		tmp = t_3;
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 2e+245)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-0.125 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, -10000000000000.0], t$95$3, If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 2e+245], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$2, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1e13 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 56.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 29.8

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

   4. Applied rewrites 29.8%

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

   6. Applied rewrites 67.1%

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

   7. Taylor expanded in d around 0

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

      1. times-frac N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2}}{{d}^{2}}\right)} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. unpow2 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2}}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. associate-/r* N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \color{blue}{\frac{\frac{{D}^{2}}{d}}{d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \color{blue}{\frac{\frac{{D}^{2}}{d}}{d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\color{blue}{\frac{{D}^{2}}{d}}}{d}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{\color{blue}{D \cdot D}}{d}}{d}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{\color{blue}{D \cdot D}}{d}}{d}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      11. *-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      12. associate-/l* N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \color{blue}{\left(h \cdot \frac{{M}^{2}}{\ell}\right)}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \color{blue}{\left(h \cdot \frac{{M}^{2}}{\ell}\right)}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \left(h \cdot \color{blue}{\frac{{M}^{2}}{\ell}}\right)\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      15. unpow2 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \left(h \cdot \frac{\color{blue}{M \cdot M}}{\ell}\right)\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      16. lower-*.f64 52.8

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

   9. Applied rewrites 52.8%

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

   if -1e13 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 2.00000000000000009e245 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 58.7%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 53.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. sqrt-div N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower-fabs.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-*.f64 89.1

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

   7. Applied rewrites 89.1%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000009e245

   1. Initial program 99.5%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 34.9

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

   8. Applied rewrites 34.9%

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

   10. Applied rewrites 36.2%

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

   12. Applied rewrites 99.5%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -10000000000000:\\ \;\;\;\;\left(\left(-0.125 \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+245}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.125 \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+306}:\\ \;\;\;\;\left(\left(-0.125 \cdot \frac{\frac{D\_m \cdot D\_m}{d}}{d}\right) \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.125 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \frac{\frac{M \cdot M}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+257}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\left(\left(-D\_m\right) \cdot M\right) \cdot \left(D\_m \cdot M\right)}{\left(-2 \cdot d\right) \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ (fabs d) (sqrt (* h l)))))
   (if (<= t_0 -1e+306)
     (* (* (* -0.125 (/ (/ (* D_m D_m) d) d)) (* h (/ (* M M) l))) t_1)
     (if (<= t_0 0.0)
       (* (fma (* (* -0.125 (* D_m D_m)) (/ (/ (* M M) d) d)) (/ h l) 1.0) t_1)
       (if (<= t_0 5e+257)
         (* (sqrt (/ d l)) (sqrt (/ d h)))
         (*
          (fma
           (* -0.5 (/ (* (* (- D_m) M) (* D_m M)) (* (* -2.0 d) (* 2.0 d))))
           (/ h l)
           1.0)
          t_1))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = fabs(d) / sqrt((h * l));
	double tmp;
	if (t_0 <= -1e+306) {
		tmp = ((-0.125 * (((D_m * D_m) / d) / d)) * (h * ((M * M) / l))) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = fma(((-0.125 * (D_m * D_m)) * (((M * M) / d) / d)), (h / l), 1.0) * t_1;
	} else if (t_0 <= 5e+257) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = fma((-0.5 * (((-D_m * M) * (D_m * M)) / ((-2.0 * d) * (2.0 * d)))), (h / l), 1.0) * t_1;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(abs(d) / sqrt(Float64(h * l)))
	tmp = 0.0
	if (t_0 <= -1e+306)
		tmp = Float64(Float64(Float64(-0.125 * Float64(Float64(Float64(D_m * D_m) / d) / d)) * Float64(h * Float64(Float64(M * M) / l))) * t_1);
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(Float64(Float64(-0.125 * Float64(D_m * D_m)) * Float64(Float64(Float64(M * M) / d) / d)), Float64(h / l), 1.0) * t_1);
	elseif (t_0 <= 5e+257)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(fma(Float64(-0.5 * Float64(Float64(Float64(Float64(-D_m) * M) * Float64(D_m * M)) / Float64(Float64(-2.0 * d) * Float64(2.0 * d)))), Float64(h / l), 1.0) * t_1);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+306], N[(N[(N[(-0.125 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 5e+257], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5 * N[(N[(N[((-D$95$m) * M), $MachinePrecision] * N[(D$95$m * M), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 * d), $MachinePrecision] * N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.00000000000000002e306

   1. Initial program 83.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 42.7

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

   4. Applied rewrites 42.7%

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

   6. Applied rewrites 82.8%

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

   7. Taylor expanded in d around 0

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

      1. times-frac N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right)} \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2}}{{d}^{2}}\right)} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. unpow2 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2}}{\color{blue}{d \cdot d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. associate-/r* N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \color{blue}{\frac{\frac{{D}^{2}}{d}}{d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \color{blue}{\frac{\frac{{D}^{2}}{d}}{d}}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\color{blue}{\frac{{D}^{2}}{d}}}{d}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{\color{blue}{D \cdot D}}{d}}{d}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{\color{blue}{D \cdot D}}{d}}{d}\right) \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      11. *-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      12. associate-/l* N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \color{blue}{\left(h \cdot \frac{{M}^{2}}{\ell}\right)}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \color{blue}{\left(h \cdot \frac{{M}^{2}}{\ell}\right)}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \left(h \cdot \color{blue}{\frac{{M}^{2}}{\ell}}\right)\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      15. unpow2 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \left(h \cdot \frac{\color{blue}{M \cdot M}}{\ell}\right)\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      16. lower-*.f64 68.4

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

   9. Applied rewrites 68.4%

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

   if -1.00000000000000002e306 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 68.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 37.1

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

   4. Applied rewrites 37.1%

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

   6. Applied rewrites 75.1%

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

   7. Taylor expanded in d around 0

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2}}{{d}^{2}}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right)} \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{{M}^{2}}{{d}^{2}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{{M}^{2}}{\color{blue}{d \cdot d}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{\frac{{M}^{2}}{d}}}{d}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{\color{blue}{M \cdot M}}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      12. lower-*.f64 46.8

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

   9. Applied rewrites 46.8%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000028e257

   1. Initial program 99.5%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 35.8

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

   8. Applied rewrites 35.8%

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

   10. Applied rewrites 37.0%

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

   12. Applied rewrites 99.5%

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

   if 5.00000000000000028e257 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 31.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 18.3

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

   4. Applied rewrites 18.3%

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

   6. Applied rewrites 67.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. unpow2 N/A

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

      9. frac-2neg N/A

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

      10. *-commutative N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-lft-neg-in N/A

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. distribute-lft-neg-in N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   8. Applied rewrites 61.6%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{+306}:\\ \;\;\;\;\left(\left(-0.125 \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{M \cdot M}{d}}{d}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+257}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \frac{\left(\left(-D\right) \cdot M\right) \cdot \left(D \cdot M\right)}{\left(-2 \cdot d\right) \cdot \left(2 \cdot d\right)}, \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-90}:\\ \;\;\;\;\frac{\left(\left(-0.25 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \mathbf{elif}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+245}\right):\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -2e-90)
     (/ (* (* (* -0.25 (* D_m D_m)) (* M M)) (* (/ h l) (sqrt (/ h l)))) h)
     (if (or (<= t_0 0.0) (not (<= t_0 2e+245)))
       (* (/ (fabs d) (sqrt (* h l))) 1.0)
       (* (sqrt (/ d l)) (sqrt (/ d h)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -2e-90) {
		tmp = (((-0.25 * (D_m * D_m)) * (M * M)) * ((h / l) * sqrt((h / l)))) / h;
	} else if ((t_0 <= 0.0) || !(t_0 <= 2e+245)) {
		tmp = (fabs(d) / sqrt((h * l))) * 1.0;
	} else {
		tmp = sqrt((d / l)) * sqrt((d / h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-2d-90)) then
        tmp = ((((-0.25d0) * (d_m * d_m)) * (m * m)) * ((h / l) * sqrt((h / l)))) / h
    else if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 2d+245))) then
        tmp = (abs(d) / sqrt((h * l))) * 1.0d0
    else
        tmp = sqrt((d / l)) * sqrt((d / h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -2e-90) {
		tmp = (((-0.25 * (D_m * D_m)) * (M * M)) * ((h / l) * Math.sqrt((h / l)))) / h;
	} else if ((t_0 <= 0.0) || !(t_0 <= 2e+245)) {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * 1.0;
	} else {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -2e-90:
		tmp = (((-0.25 * (D_m * D_m)) * (M * M)) * ((h / l) * math.sqrt((h / l)))) / h
	elif (t_0 <= 0.0) or not (t_0 <= 2e+245):
		tmp = (math.fabs(d) / math.sqrt((h * l))) * 1.0
	else:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -2e-90)
		tmp = Float64(Float64(Float64(Float64(-0.25 * Float64(D_m * D_m)) * Float64(M * M)) * Float64(Float64(h / l) * sqrt(Float64(h / l)))) / h);
	elseif ((t_0 <= 0.0) || !(t_0 <= 2e+245))
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * 1.0);
	else
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -2e-90)
		tmp = (((-0.25 * (D_m * D_m)) * (M * M)) * ((h / l) * sqrt((h / l)))) / h;
	elseif ((t_0 <= 0.0) || ~((t_0 <= 2e+245)))
		tmp = (abs(d) / sqrt((h * l))) * 1.0;
	else
		tmp = sqrt((d / l)) * sqrt((d / h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-90], N[(N[(N[(N[(-0.25 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+245]], $MachinePrecision]], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.99999999999999999e-90

   1. Initial program 85.7%

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

   4. Applied rewrites 34.8%

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

   5. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   7. Applied rewrites 20.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(0.25 \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot D\right), \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   8. Taylor expanded in h around -inf

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

   10. Applied rewrites 23.1%

       \[\leadsto \frac{\left(\left(-0.25 \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}}{h} \]
   11. Step-by-step derivation

   12. Applied rewrites 24.5%

       \[\leadsto \frac{\left(\left(-0.25 \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   if -1.99999999999999999e-90 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 2.00000000000000009e245 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 30.9%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 34.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. sqrt-div N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower-fabs.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-*.f64 60.9

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

   7. Applied rewrites 60.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000009e245

   1. Initial program 99.5%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 34.9

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

   8. Applied rewrites 34.9%

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

   10. Applied rewrites 36.2%

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

   12. Applied rewrites 99.5%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-90}:\\ \;\;\;\;\frac{\left(\left(-0.25 \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0 \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+245}\right):\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-90}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+245}\right):\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -2e-90)
     (/ (* (- d) (sqrt (/ h l))) h)
     (if (or (<= t_0 0.0) (not (<= t_0 2e+245)))
       (* (/ (fabs d) (sqrt (* h l))) 1.0)
       (* (sqrt (/ d l)) (sqrt (/ d h)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -2e-90) {
		tmp = (-d * sqrt((h / l))) / h;
	} else if ((t_0 <= 0.0) || !(t_0 <= 2e+245)) {
		tmp = (fabs(d) / sqrt((h * l))) * 1.0;
	} else {
		tmp = sqrt((d / l)) * sqrt((d / h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-2d-90)) then
        tmp = (-d * sqrt((h / l))) / h
    else if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 2d+245))) then
        tmp = (abs(d) / sqrt((h * l))) * 1.0d0
    else
        tmp = sqrt((d / l)) * sqrt((d / h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -2e-90) {
		tmp = (-d * Math.sqrt((h / l))) / h;
	} else if ((t_0 <= 0.0) || !(t_0 <= 2e+245)) {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * 1.0;
	} else {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -2e-90:
		tmp = (-d * math.sqrt((h / l))) / h
	elif (t_0 <= 0.0) or not (t_0 <= 2e+245):
		tmp = (math.fabs(d) / math.sqrt((h * l))) * 1.0
	else:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -2e-90)
		tmp = Float64(Float64(Float64(-d) * sqrt(Float64(h / l))) / h);
	elseif ((t_0 <= 0.0) || !(t_0 <= 2e+245))
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * 1.0);
	else
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -2e-90)
		tmp = (-d * sqrt((h / l))) / h;
	elseif ((t_0 <= 0.0) || ~((t_0 <= 2e+245)))
		tmp = (abs(d) / sqrt((h * l))) * 1.0;
	else
		tmp = sqrt((d / l)) * sqrt((d / h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-90], N[(N[((-d) * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+245]], $MachinePrecision]], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.99999999999999999e-90

   1. Initial program 85.7%

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

   4. Applied rewrites 34.8%

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

   5. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   7. Applied rewrites 20.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(0.25 \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot D\right), \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   8. Taylor expanded in l around -inf

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

   10. Applied rewrites 18.8%

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

   if -1.99999999999999999e-90 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 2.00000000000000009e245 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 30.9%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 34.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. sqrt-div N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower-fabs.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-*.f64 60.9

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

   7. Applied rewrites 60.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000009e245

   1. Initial program 99.5%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 34.9

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

   8. Applied rewrites 34.9%

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

   10. Applied rewrites 36.2%

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

   12. Applied rewrites 99.5%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-90}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0 \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+245}\right):\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m \cdot M}{-2 \cdot d}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(t\_0 \cdot t\_0\right), \frac{h}{\ell}, 1\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+245}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(\left(\frac{D\_m}{d} \cdot M\right) \cdot \frac{h}{\ell}\right) \cdot \left(0.5 \cdot D\_m\right)}{4} \cdot \frac{M}{d}\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (/ (* D_m M) (* -2.0 d)))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (/ (fabs d) (sqrt (* h l)))))
   (if (<= t_1 0.0)
     (* (fma (* -0.5 (* t_0 t_0)) (/ h l) 1.0) t_2)
     (if (<= t_1 2e+245)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (*
        (- 1.0 (* (/ (* (* (* (/ D_m d) M) (/ h l)) (* 0.5 D_m)) 4.0) (/ M d)))
        t_2)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (D_m * M) / (-2.0 * d);
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = fabs(d) / sqrt((h * l));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = fma((-0.5 * (t_0 * t_0)), (h / l), 1.0) * t_2;
	} else if (t_1 <= 2e+245) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = (1.0 - ((((((D_m / d) * M) * (h / l)) * (0.5 * D_m)) / 4.0) * (M / d))) * t_2;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(D_m * M) / Float64(-2.0 * d))
	t_1 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(abs(d) / sqrt(Float64(h * l)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(fma(Float64(-0.5 * Float64(t_0 * t_0)), Float64(h / l), 1.0) * t_2);
	elseif (t_1 <= 2e+245)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(D_m / d) * M) * Float64(h / l)) * Float64(0.5 * D_m)) / 4.0) * Float64(M / d))) * t_2);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m * M), $MachinePrecision] / N[(-2.0 * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(N[(-0.5 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 2e+245], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * M), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[(0.5 * D$95$m), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 79.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 41.3

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

   4. Applied rewrites 41.3%

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

   6. Applied rewrites 80.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. unpow2 N/A

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

      9. frac-2neg N/A

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

      10. frac-2neg N/A

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

      11. distribute-frac-neg2 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. sqr-neg N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 82.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000009e245

   1. Initial program 99.5%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 34.9

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

   8. Applied rewrites 34.9%

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

   10. Applied rewrites 36.2%

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

   12. Applied rewrites 99.5%

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

   if 2.00000000000000009e245 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 32.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 18.1

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

   4. Applied rewrites 18.1%

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

   6. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 68.0

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

   8. Applied rewrites 68.0%

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

   10. Applied rewrites 70.4%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+245}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{h}{\ell}\right) \cdot \left(0.5 \cdot D\right)}{4} \cdot \frac{M}{d}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{M}{d \cdot 2}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+245}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(\left(\left(t\_1 \cdot D\_m\right) \cdot t\_1\right) \cdot D\_m\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ M (* d 2.0))))
   (if (or (<= t_0 0.0) (not (<= t_0 2e+245)))
     (*
      (fma (* -0.5 (* (* (* t_1 D_m) t_1) D_m)) (/ h l) 1.0)
      (/ (fabs d) (sqrt (* h l))))
     (* (sqrt (/ d l)) (sqrt (/ d h))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = M / (d * 2.0);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+245)) {
		tmp = fma((-0.5 * (((t_1 * D_m) * t_1) * D_m)), (h / l), 1.0) * (fabs(d) / sqrt((h * l)));
	} else {
		tmp = sqrt((d / l)) * sqrt((d / h));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(M / Float64(d * 2.0))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 2e+245))
		tmp = Float64(fma(Float64(-0.5 * Float64(Float64(Float64(t_1 * D_m) * t_1) * D_m)), Float64(h / l), 1.0) * Float64(abs(d) / sqrt(Float64(h * l))));
	else
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+245]], $MachinePrecision]], N[(N[(N[(-0.5 * N[(N[(N[(t$95$1 * D$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 2.00000000000000009e245 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 57.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 30.4

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

   4. Applied rewrites 30.4%

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

   6. Applied rewrites 74.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 75.4

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

   8. Applied rewrites 75.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 73.8

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 73.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 73.8

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 73.8

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

   10. Applied rewrites 73.8%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000009e245

   1. Initial program 99.5%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 34.9

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

   8. Applied rewrites 34.9%

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

   10. Applied rewrites 36.2%

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

   12. Applied rewrites 99.5%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0 \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+245}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(\left(\left(\frac{M}{d \cdot 2} \cdot D\right) \cdot \frac{M}{d \cdot 2}\right) \cdot D\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m \cdot M}{-2 \cdot d}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ t_3 := \frac{M}{d \cdot 2}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(t\_0 \cdot t\_0\right), \frac{h}{\ell}, 1\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+245}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(\left(\left(t\_3 \cdot D\_m\right) \cdot t\_3\right) \cdot D\_m\right), \frac{h}{\ell}, 1\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (/ (* D_m M) (* -2.0 d)))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (/ (fabs d) (sqrt (* h l))))
        (t_3 (/ M (* d 2.0))))
   (if (<= t_1 0.0)
     (* (fma (* -0.5 (* t_0 t_0)) (/ h l) 1.0) t_2)
     (if (<= t_1 2e+245)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (* (fma (* -0.5 (* (* (* t_3 D_m) t_3) D_m)) (/ h l) 1.0) t_2)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (D_m * M) / (-2.0 * d);
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = fabs(d) / sqrt((h * l));
	double t_3 = M / (d * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = fma((-0.5 * (t_0 * t_0)), (h / l), 1.0) * t_2;
	} else if (t_1 <= 2e+245) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = fma((-0.5 * (((t_3 * D_m) * t_3) * D_m)), (h / l), 1.0) * t_2;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(D_m * M) / Float64(-2.0 * d))
	t_1 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(abs(d) / sqrt(Float64(h * l)))
	t_3 = Float64(M / Float64(d * 2.0))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(fma(Float64(-0.5 * Float64(t_0 * t_0)), Float64(h / l), 1.0) * t_2);
	elseif (t_1 <= 2e+245)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(fma(Float64(-0.5 * Float64(Float64(Float64(t_3 * D_m) * t_3) * D_m)), Float64(h / l), 1.0) * t_2);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m * M), $MachinePrecision] / N[(-2.0 * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(N[(-0.5 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 2e+245], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5 * N[(N[(N[(t$95$3 * D$95$m), $MachinePrecision] * t$95$3), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 79.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 41.3

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

   4. Applied rewrites 41.3%

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

   6. Applied rewrites 80.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. unpow2 N/A

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

      9. frac-2neg N/A

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

      10. frac-2neg N/A

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

      11. distribute-frac-neg2 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. sqr-neg N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 82.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000009e245

   1. Initial program 99.5%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 34.9

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

   8. Applied rewrites 34.9%

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

   10. Applied rewrites 36.2%

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

   12. Applied rewrites 99.5%

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

   if 2.00000000000000009e245 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 32.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 18.1

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

   4. Applied rewrites 18.1%

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

   6. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 68.0

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

   8. Applied rewrites 68.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 68.0

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 68.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 68.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 68.0

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

   10. Applied rewrites 68.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(\frac{D \cdot M}{-2 \cdot d} \cdot \frac{D \cdot M}{-2 \cdot d}\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+245}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(\left(\left(\frac{M}{d \cdot 2} \cdot D\right) \cdot \frac{M}{d \cdot 2}\right) \cdot D\right), \frac{h}{\ell}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-90}:\\ \;\;\;\;\frac{\left(-d\right) \cdot t\_0}{h}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 -2e-90)
     (/ (* (- d) t_0) h)
     (if (<= t_1 5e+295)
       (/ (* t_0 d) h)
       (* (/ (fabs d) (sqrt (* h l))) 1.0)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((h / l));
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -2e-90) {
		tmp = (-d * t_0) / h;
	} else if (t_1 <= 5e+295) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (fabs(d) / sqrt((h * l))) * 1.0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((h / l))
    t_1 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_1 <= (-2d-90)) then
        tmp = (-d * t_0) / h
    else if (t_1 <= 5d+295) then
        tmp = (t_0 * d) / h
    else
        tmp = (abs(d) / sqrt((h * l))) * 1.0d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -2e-90) {
		tmp = (-d * t_0) / h;
	} else if (t_1 <= 5e+295) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * 1.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((h / l))
	t_1 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= -2e-90:
		tmp = (-d * t_0) / h
	elif t_1 <= 5e+295:
		tmp = (t_0 * d) / h
	else:
		tmp = (math.fabs(d) / math.sqrt((h * l))) * 1.0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(h / l))
	t_1 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= -2e-90)
		tmp = Float64(Float64(Float64(-d) * t_0) / h);
	elseif (t_1 <= 5e+295)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * 1.0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((h / l));
	t_1 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= -2e-90)
		tmp = (-d * t_0) / h;
	elseif (t_1 <= 5e+295)
		tmp = (t_0 * d) / h;
	else
		tmp = (abs(d) / sqrt((h * l))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-90], N[(N[((-d) * t$95$0), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$1, 5e+295], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.99999999999999999e-90

   1. Initial program 85.7%

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

   4. Applied rewrites 34.8%

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

   5. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   7. Applied rewrites 20.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(0.25 \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot D\right), \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   8. Taylor expanded in l around -inf

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

   10. Applied rewrites 18.8%

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

   if -1.99999999999999999e-90 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999991e295

   1. Initial program 91.3%

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

   4. Applied rewrites 86.6%

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

   5. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   7. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(0.25 \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot D\right), \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   8. Taylor expanded in d around inf

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

   10. Applied rewrites 87.6%

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

   if 4.99999999999999991e295 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 29.0%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 32.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. sqrt-div N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower-fabs.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-*.f64 57.0

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

   7. Applied rewrites 57.0%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-90}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 0.0)
     (* (sqrt (pow (* l h) -1.0)) d)
     (if (<= t_0 5e+295)
       (/ (* (sqrt (/ h l)) d) h)
       (* (/ (fabs d) (sqrt (* h l))) 1.0)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt(pow((l * h), -1.0)) * d;
	} else if (t_0 <= 5e+295) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = (fabs(d) / sqrt((h * l))) * 1.0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= 0.0d0) then
        tmp = sqrt(((l * h) ** (-1.0d0))) * d
    else if (t_0 <= 5d+295) then
        tmp = (sqrt((h / l)) * d) / h
    else
        tmp = (abs(d) / sqrt((h * l))) * 1.0d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt(Math.pow((l * h), -1.0)) * d;
	} else if (t_0 <= 5e+295) {
		tmp = (Math.sqrt((h / l)) * d) / h;
	} else {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * 1.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt(math.pow((l * h), -1.0)) * d
	elif t_0 <= 5e+295:
		tmp = (math.sqrt((h / l)) * d) / h
	else:
		tmp = (math.fabs(d) / math.sqrt((h * l))) * 1.0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt((Float64(l * h) ^ -1.0)) * d);
	elseif (t_0 <= 5e+295)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * 1.0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt(((l * h) ^ -1.0)) * d;
	elseif (t_0 <= 5e+295)
		tmp = (sqrt((h / l)) * d) / h;
	else
		tmp = (abs(d) / sqrt((h * l))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], If[LessEqual[t$95$0, 5e+295], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 79.3%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 13.9

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

   8. Applied rewrites 13.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999991e295

   1. Initial program 99.5%

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   7. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(0.25 \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot D\right), \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   8. Taylor expanded in d around inf

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

   10. Applied rewrites 93.3%

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

   if 4.99999999999999991e295 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 29.0%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 32.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. sqrt-div N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower-fabs.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-*.f64 57.0

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

   7. Applied rewrites 57.0%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+245}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{\frac{D\_m}{d}}{2} \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(0.5 \cdot \frac{D\_m}{2}\right) \cdot \frac{M}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(\left(\frac{D\_m}{d} \cdot M\right) \cdot \frac{h}{\ell}\right) \cdot \left(0.5 \cdot D\_m\right)}{4} \cdot \frac{M}{d}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
       (-
        1.0
        (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))
      2e+245)
   (*
    (* (sqrt (/ d h)) (sqrt (/ d l)))
    (-
     1.0
     (* (* (/ (/ D_m d) 2.0) (* M (/ h l))) (* (* 0.5 (/ D_m 2.0)) (/ M d)))))
   (*
    (- 1.0 (* (/ (* (* (* (/ D_m d) M) (/ h l)) (* 0.5 D_m)) 4.0) (/ M d)))
    (/ (fabs d) (sqrt (* h l))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= 2e+245) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - ((((D_m / d) / 2.0) * (M * (h / l))) * ((0.5 * (D_m / 2.0)) * (M / d))));
	} else {
		tmp = (1.0 - ((((((D_m / d) * M) * (h / l)) * (0.5 * D_m)) / 4.0) * (M / d))) * (fabs(d) / sqrt((h * l)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (((((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= 2d+245) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - ((((d_m / d) / 2.0d0) * (m * (h / l))) * ((0.5d0 * (d_m / 2.0d0)) * (m / d))))
    else
        tmp = (1.0d0 - ((((((d_m / d) * m) * (h / l)) * (0.5d0 * d_m)) / 4.0d0) * (m / d))) * (abs(d) / sqrt((h * l)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= 2e+245) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - ((((D_m / d) / 2.0) * (M * (h / l))) * ((0.5 * (D_m / 2.0)) * (M / d))));
	} else {
		tmp = (1.0 - ((((((D_m / d) * M) * (h / l)) * (0.5 * D_m)) / 4.0) * (M / d))) * (Math.abs(d) / Math.sqrt((h * l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if ((math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= 2e+245:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - ((((D_m / d) / 2.0) * (M * (h / l))) * ((0.5 * (D_m / 2.0)) * (M / d))))
	else:
		tmp = (1.0 - ((((((D_m / d) * M) * (h / l)) * (0.5 * D_m)) / 4.0) * (M / d))) * (math.fabs(d) / math.sqrt((h * l)))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 2e+245)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(Float64(Float64(Float64(D_m / d) / 2.0) * Float64(M * Float64(h / l))) * Float64(Float64(0.5 * Float64(D_m / 2.0)) * Float64(M / d)))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(D_m / d) * M) * Float64(h / l)) * Float64(0.5 * D_m)) / 4.0) * Float64(M / d))) * Float64(abs(d) / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (((((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= 2e+245)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - ((((D_m / d) / 2.0) * (M * (h / l))) * ((0.5 * (D_m / 2.0)) * (M / d))));
	else
		tmp = (1.0 - ((((((D_m / d) * M) * (h / l)) * (0.5 * D_m)) / 4.0) * (M / d))) * (abs(d) / sqrt((h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+245], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * N[(M * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * M), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[(0.5 * D$95$m), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000009e245

   1. Initial program 88.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 86.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 86.5

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 86.5

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

   6. Applied rewrites 86.5%

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

   7. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 86.5

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

   9. Applied rewrites 86.5%

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

   if 2.00000000000000009e245 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 32.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 18.1

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

   4. Applied rewrites 18.1%

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

   6. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 68.0

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

   8. Applied rewrites 68.0%

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

   10. Applied rewrites 70.4%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+245}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\frac{\frac{D}{d}}{2} \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(0.5 \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{h}{\ell}\right) \cdot \left(0.5 \cdot D\right)}{4} \cdot \frac{M}{d}\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-53}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
       (-
        1.0
        (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))
      -1e-53)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (* (/ (fabs d) (sqrt (* h l))) 1.0)))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-53) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = (fabs(d) / sqrt((h * l))) * 1.0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (((((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-1d-53)) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = (abs(d) / sqrt((h * l))) * 1.0d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-53) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * 1.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if ((math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-53:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = (math.fabs(d) / math.sqrt((h * l))) * 1.0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -1e-53)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * 1.0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (((((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -1e-53)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = (abs(d) / sqrt((h * l))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-53], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.00000000000000003e-53

   1. Initial program 85.6%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 0.8%

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

   6. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 11.3

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

   8. Applied rewrites 11.3%

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

   if -1.00000000000000003e-53 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 62.2%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 63.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1/2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-times N/A

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

      13. sqrt-div N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower-fabs.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-*.f64 66.7

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

   7. Applied rewrites 66.7%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-53}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.55 \cdot 10^{-236}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d 1.55e-236)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (/ d (sqrt (* l h)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= 1.55e-236) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (d <= 1.55d-236) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= 1.55e-236) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if d <= 1.55e-236:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= 1.55e-236)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (d <= 1.55e-236)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, 1.55e-236], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.5499999999999999e-236

   1. Initial program 71.6%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 40.9%

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

   6. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 40.1

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

   8. Applied rewrites 40.1%

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

   if 1.5499999999999999e-236 < d

   1. Initial program 68.1%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 44.5%

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

   6. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 49.4

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

   8. Applied rewrites 49.4%

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

   10. Applied rewrites 50.2%

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

   12. Applied rewrites 50.2%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.55 \cdot 10^{-236}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 26.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m) :precision binary64 (* (sqrt (pow (* l h) -1.0)) d))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	return sqrt(pow((l * h), -1.0)) * d;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    code = sqrt(((l * h) ** (-1.0d0))) * d
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	return Math.sqrt(Math.pow((l * h), -1.0)) * d;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	return math.sqrt(math.pow((l * h), -1.0)) * d

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	return Float64(sqrt((Float64(l * h) ^ -1.0)) * d)
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp = code(d, h, l, M, D_m)
	tmp = sqrt(((l * h) ^ -1.0)) * d;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := N[(N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 69.9%

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

3. Taylor expanded in d around inf

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

5. Applied rewrites 42.6%

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

6. Taylor expanded in d around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 26.9

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

8. Applied rewrites 26.9%

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

9. Final simplification 26.9%

   \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]
10. Add Preprocessing

Alternative 16: 26.2% accurate, 15.3× speedup

Math

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

FPCore

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

C

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

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    code = d / sqrt((l * h))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

3. Taylor expanded in d around inf

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

5. Applied rewrites 42.6%

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

6. Taylor expanded in d around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 26.9

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

8. Applied rewrites 26.9%

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

10. Applied rewrites 26.6%

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

12. Applied rewrites 26.6%

    \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024329 
(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)))))