Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.7% → 80.3%

Time: 13.7s

Alternatives: 21

Speedup: 3.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\\ t_1 := \frac{\frac{D\_m}{d}}{2} \cdot M\_m\\ t_2 := 1 - \frac{t\_1 \cdot \left(t\_1 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\\ t_3 := {\left(\frac{h}{\ell}\right)}^{0.75}\\ \mathbf{if}\;d \leq -7.6 \cdot 10^{-106}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\right) \cdot t\_2\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-181}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot D\_m, D\_m \cdot \left(t\_3 \cdot \left(t\_3 \cdot \left(\frac{M\_m}{d} \cdot M\_m\right)\right)\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t\_0\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (/ d l) (/ 1.0 2.0)))
        (t_1 (* (/ (/ D_m d) 2.0) M_m))
        (t_2 (- 1.0 (/ (* t_1 (* t_1 (* 0.5 h))) l)))
        (t_3 (pow (/ h l) 0.75)))
   (if (<= d -7.6e-106)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) t_0) t_2)
     (if (<= d 2.1e-181)
       (/
        (fma
         (* -0.125 D_m)
         (* D_m (* t_3 (* t_3 (* (/ M_m d) M_m))))
         (* (sqrt (/ h l)) d))
        h)
       (* (* (/ (sqrt d) (sqrt h)) t_0) t_2)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((d / l), (1.0 / 2.0));
	double t_1 = ((D_m / d) / 2.0) * M_m;
	double t_2 = 1.0 - ((t_1 * (t_1 * (0.5 * h))) / l);
	double t_3 = pow((h / l), 0.75);
	double tmp;
	if (d <= -7.6e-106) {
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * t_2;
	} else if (d <= 2.1e-181) {
		tmp = fma((-0.125 * D_m), (D_m * (t_3 * (t_3 * ((M_m / d) * M_m)))), (sqrt((h / l)) * d)) / h;
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * t_0) * t_2;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d / l) ^ Float64(1.0 / 2.0)
	t_1 = Float64(Float64(Float64(D_m / d) / 2.0) * M_m)
	t_2 = Float64(1.0 - Float64(Float64(t_1 * Float64(t_1 * Float64(0.5 * h))) / l))
	t_3 = Float64(h / l) ^ 0.75
	tmp = 0.0
	if (d <= -7.6e-106)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0) * t_2);
	elseif (d <= 2.1e-181)
		tmp = Float64(fma(Float64(-0.125 * D_m), Float64(D_m * Float64(t_3 * Float64(t_3 * Float64(Float64(M_m / d) * M_m)))), Float64(sqrt(Float64(h / l)) * d)) / h);
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_0) * t_2);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(N[(t$95$1 * N[(t$95$1 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(h / l), $MachinePrecision], 0.75], $MachinePrecision]}, If[LessEqual[d, -7.6e-106], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[d, 2.1e-181], N[(N[(N[(-0.125 * D$95$m), $MachinePrecision] * N[(D$95$m * N[(t$95$3 * N[(t$95$3 * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.5999999999999999e-106

   1. Initial program 78.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 79.4

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 79.4

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

   4. Applied rewrites 79.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 83.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 83.5

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 91.4

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

   8. Applied rewrites 91.4%

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

   if -7.5999999999999999e-106 < d < 2.10000000000000003e-181

   1. Initial program 42.4%

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

      1. lift-*.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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 40.9

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 40.9

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

   4. Applied rewrites 40.9%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   7. Applied rewrites 42.3%

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

   9. Applied rewrites 57.6%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot D, D \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\frac{M}{d} \cdot M\right)\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]
   10. Step-by-step derivation

   11. Applied rewrites 62.3%

       \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot D, D \cdot \left({\left(\frac{h}{\ell}\right)}^{0.75} \cdot \left({\left(\frac{h}{\ell}\right)}^{0.75} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

   if 2.10000000000000003e-181 < d

   1. Initial program 78.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 80.1

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 80.1

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

   4. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 82.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 82.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 91.8

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

   8. Applied rewrites 91.8%

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

Alternative 2: 47.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-115}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -5e-115)
     (/ (* (- d) (sqrt (/ h l))) h)
     (if (<= t_0 0.0)
       (* (- (sqrt (/ 1.0 (* l h)))) d)
       (* (sqrt (/ d h)) (sqrt (/ d l)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -5e-115) {
		tmp = (-d * sqrt((h / l))) / h;
	} else if (t_0 <= 0.0) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = sqrt((d / h)) * sqrt((d / l));
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-5d-115)) then
        tmp = (-d * sqrt((h / l))) / h
    else if (t_0 <= 0.0d0) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else
        tmp = sqrt((d / h)) * sqrt((d / l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -5e-115) {
		tmp = (-d * Math.sqrt((h / l))) / h;
	} else if (t_0 <= 0.0) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -5e-115:
		tmp = (-d * math.sqrt((h / l))) / h
	elif t_0 <= 0.0:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	else:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(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_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -5e-115)
		tmp = Float64(Float64(Float64(-d) * sqrt(Float64(h / l))) / h);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	else
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -5e-115)
		tmp = (-d * sqrt((h / l))) / h;
	elseif (t_0 <= 0.0)
		tmp = -sqrt((1.0 / (l * h))) * d;
	else
		tmp = sqrt((d / h)) * sqrt((d / l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(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$95$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, -5e-115], N[(N[((-d) * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $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)))) < -5.0000000000000003e-115

   1. Initial program 89.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 86.5

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 86.5

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

   4. Applied rewrites 86.5%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   7. Applied rewrites 45.7%

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

   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 27.3%

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

   if -5.0000000000000003e-115 < (*.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 37.0%

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

   3. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in h around -inf

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

   8. Applied rewrites 67.0%

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

   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))))

   1. Initial program 61.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 60.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 60.3

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

   6. Applied rewrites 60.3%

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

   7. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. associate-*l* N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. *-lft-identity N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 63.7

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

   9. Applied rewrites 63.7%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-115}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\\ t_1 := \frac{\frac{D\_m}{d}}{2} \cdot M\_m\\ t_2 := 1 - \frac{t\_1 \cdot \left(t\_1 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\\ \mathbf{if}\;d \leq -7.6 \cdot 10^{-106}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\right) \cdot t\_2\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-185}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot D\_m, \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \frac{M\_m}{d}\right) \cdot \left(M\_m \cdot D\_m\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t\_0\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (/ d l) (/ 1.0 2.0)))
        (t_1 (* (/ (/ D_m d) 2.0) M_m))
        (t_2 (- 1.0 (/ (* t_1 (* t_1 (* 0.5 h))) l))))
   (if (<= d -7.6e-106)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) t_0) t_2)
     (if (<= d 1.6e-185)
       (/
        (fma
         (* -0.125 D_m)
         (* (* (pow (/ h l) 1.5) (/ M_m d)) (* M_m D_m))
         (* (sqrt (/ h l)) d))
        h)
       (* (* (/ (sqrt d) (sqrt h)) t_0) t_2)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((d / l), (1.0 / 2.0));
	double t_1 = ((D_m / d) / 2.0) * M_m;
	double t_2 = 1.0 - ((t_1 * (t_1 * (0.5 * h))) / l);
	double tmp;
	if (d <= -7.6e-106) {
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * t_2;
	} else if (d <= 1.6e-185) {
		tmp = fma((-0.125 * D_m), ((pow((h / l), 1.5) * (M_m / d)) * (M_m * D_m)), (sqrt((h / l)) * d)) / h;
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * t_0) * t_2;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d / l) ^ Float64(1.0 / 2.0)
	t_1 = Float64(Float64(Float64(D_m / d) / 2.0) * M_m)
	t_2 = Float64(1.0 - Float64(Float64(t_1 * Float64(t_1 * Float64(0.5 * h))) / l))
	tmp = 0.0
	if (d <= -7.6e-106)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0) * t_2);
	elseif (d <= 1.6e-185)
		tmp = Float64(fma(Float64(-0.125 * D_m), Float64(Float64((Float64(h / l) ^ 1.5) * Float64(M_m / d)) * Float64(M_m * D_m)), Float64(sqrt(Float64(h / l)) * d)) / h);
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_0) * t_2);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(N[(t$95$1 * N[(t$95$1 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -7.6e-106], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[d, 1.6e-185], N[(N[(N[(-0.125 * D$95$m), $MachinePrecision] * N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.5999999999999999e-106

   1. Initial program 78.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 79.4

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 79.4

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

   4. Applied rewrites 79.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 83.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 83.5

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 91.4

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

   8. Applied rewrites 91.4%

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

   if -7.5999999999999999e-106 < d < 1.5999999999999999e-185

   1. Initial program 42.4%

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

      1. lift-*.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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 40.9

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 40.9

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

   4. Applied rewrites 40.9%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   7. Applied rewrites 42.3%

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

   9. Applied rewrites 57.6%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot D, D \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\frac{M}{d} \cdot M\right)\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]
   10. Step-by-step derivation

   11. Applied rewrites 59.3%

       \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot D, \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

   if 1.5999999999999999e-185 < d

   1. Initial program 78.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 80.1

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 80.1

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

   4. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 82.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 82.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 91.8

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

   8. Applied rewrites 91.8%

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

Alternative 4: 78.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\_m\\ t_1 := 1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\\ \mathbf{if}\;d \leq -2.05 \cdot 10^{-197}:\\ \;\;\;\;\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot t\_1\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-185}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot D\_m, \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \frac{M\_m}{d}\right) \cdot \left(M\_m \cdot D\_m\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ (/ D_m d) 2.0) M_m))
        (t_1 (- 1.0 (/ (* t_0 (* t_0 (* 0.5 h))) l))))
   (if (<= d -2.05e-197)
     (* (* (- d) (sqrt (/ 1.0 (* l h)))) t_1)
     (if (<= d 1.6e-185)
       (/
        (fma
         (* -0.125 D_m)
         (* (* (pow (/ h l) 1.5) (/ M_m d)) (* M_m D_m))
         (* (sqrt (/ h l)) d))
        h)
       (* (* (/ (sqrt d) (sqrt h)) (pow (/ d l) (/ 1.0 2.0))) t_1)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M_m;
	double t_1 = 1.0 - ((t_0 * (t_0 * (0.5 * h))) / l);
	double tmp;
	if (d <= -2.05e-197) {
		tmp = (-d * sqrt((1.0 / (l * h)))) * t_1;
	} else if (d <= 1.6e-185) {
		tmp = fma((-0.125 * D_m), ((pow((h / l), 1.5) * (M_m / d)) * (M_m * D_m)), (sqrt((h / l)) * d)) / h;
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), (1.0 / 2.0))) * t_1;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(Float64(D_m / d) / 2.0) * M_m)
	t_1 = Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(0.5 * h))) / l))
	tmp = 0.0
	if (d <= -2.05e-197)
		tmp = Float64(Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h)))) * t_1);
	elseif (d <= 1.6e-185)
		tmp = Float64(fma(Float64(-0.125 * D_m), Float64(Float64((Float64(h / l) ^ 1.5) * Float64(M_m / d)) * Float64(M_m * D_m)), Float64(sqrt(Float64(h / l)) * d)) / h);
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * t_1);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.05e-197], N[(N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 1.6e-185], N[(N[(N[(-0.125 * D$95$m), $MachinePrecision] * N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.05e-197

   1. Initial program 75.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 75.7

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 75.7

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

   4. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 79.0%

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

   7. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 84.6

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

   9. Applied rewrites 84.6%

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

   if -2.05e-197 < d < 1.5999999999999999e-185

   1. Initial program 32.4%

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

      1. lift-*.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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 30.0

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 30.0

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

   4. Applied rewrites 30.0%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   7. Applied rewrites 36.8%

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

   9. Applied rewrites 51.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot D, D \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\frac{M}{d} \cdot M\right)\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]
   10. Step-by-step derivation

   11. Applied rewrites 53.6%

       \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot D, \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

   if 1.5999999999999999e-185 < d

   1. Initial program 78.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 80.1

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 80.1

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

   4. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 82.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 82.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 91.8

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

   8. Applied rewrites 91.8%

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

Alternative 5: 76.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\_m\\ t_1 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq -2.05 \cdot 10^{-197}:\\ \;\;\;\;\left(\left(-d\right) \cdot t\_1\right) \cdot \left(1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-178}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot D\_m, \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \frac{M\_m}{d}\right) \cdot \left(M\_m \cdot D\_m\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{+38}:\\ \;\;\;\;\left(t\_1 \cdot d\right) \cdot \left(1 - \frac{{\left(\frac{D\_m}{2} \cdot \frac{M\_m}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {t\_0}^{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right)}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ (/ D_m d) 2.0) M_m)) (t_1 (sqrt (/ 1.0 (* l h)))))
   (if (<= d -2.05e-197)
     (* (* (- d) t_1) (- 1.0 (/ (* t_0 (* t_0 (* 0.5 h))) l)))
     (if (<= d 2.05e-178)
       (/
        (fma
         (* -0.125 D_m)
         (* (* (pow (/ h l) 1.5) (/ M_m d)) (* M_m D_m))
         (* (sqrt (/ h l)) d))
        h)
       (if (<= d 1.55e+38)
         (*
          (* t_1 d)
          (- 1.0 (/ (* (pow (* (/ D_m 2.0) (/ M_m d)) 2.0) (* 0.5 h)) l)))
         (/
          (*
           (sqrt d)
           (* (fma (* (/ h l) -0.5) (pow t_0 2.0) 1.0) (sqrt (/ d l))))
          (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M_m;
	double t_1 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -2.05e-197) {
		tmp = (-d * t_1) * (1.0 - ((t_0 * (t_0 * (0.5 * h))) / l));
	} else if (d <= 2.05e-178) {
		tmp = fma((-0.125 * D_m), ((pow((h / l), 1.5) * (M_m / d)) * (M_m * D_m)), (sqrt((h / l)) * d)) / h;
	} else if (d <= 1.55e+38) {
		tmp = (t_1 * d) * (1.0 - ((pow(((D_m / 2.0) * (M_m / d)), 2.0) * (0.5 * h)) / l));
	} else {
		tmp = (sqrt(d) * (fma(((h / l) * -0.5), pow(t_0, 2.0), 1.0) * sqrt((d / l)))) / sqrt(h);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(Float64(D_m / d) / 2.0) * M_m)
	t_1 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= -2.05e-197)
		tmp = Float64(Float64(Float64(-d) * t_1) * Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(0.5 * h))) / l)));
	elseif (d <= 2.05e-178)
		tmp = Float64(fma(Float64(-0.125 * D_m), Float64(Float64((Float64(h / l) ^ 1.5) * Float64(M_m / d)) * Float64(M_m * D_m)), Float64(sqrt(Float64(h / l)) * d)) / h);
	elseif (d <= 1.55e+38)
		tmp = Float64(Float64(t_1 * d) * Float64(1.0 - Float64(Float64((Float64(Float64(D_m / 2.0) * Float64(M_m / d)) ^ 2.0) * Float64(0.5 * h)) / l)));
	else
		tmp = Float64(Float64(sqrt(d) * Float64(fma(Float64(Float64(h / l) * -0.5), (t_0 ^ 2.0), 1.0) * sqrt(Float64(d / l)))) / sqrt(h));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2.05e-197], N[(N[((-d) * t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.05e-178], N[(N[(N[(-0.125 * D$95$m), $MachinePrecision] * N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 1.55e+38], N[(N[(t$95$1 * d), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] * N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.05e-197

   1. Initial program 75.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 75.7

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 75.7

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

   4. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 79.0%

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

   7. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 84.6

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

   9. Applied rewrites 84.6%

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

   if -2.05e-197 < d < 2.05e-178

   1. Initial program 34.0%

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

      1. lift-*.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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 31.7

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 31.7

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

   4. Applied rewrites 31.7%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   7. Applied rewrites 38.2%

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

   9. Applied rewrites 52.3%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot D, D \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\frac{M}{d} \cdot M\right)\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]
   10. Step-by-step derivation

   11. Applied rewrites 54.6%

       \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot D, \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

   if 2.05e-178 < d < 1.55000000000000009e38

   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. 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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 68.6

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 68.6

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

   4. Applied rewrites 68.6%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 78.9

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

   7. Applied rewrites 78.9%

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

   if 1.55000000000000009e38 < d

   1. Initial program 83.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 86.9

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 86.9

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

   4. Applied rewrites 86.9%

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

   6. Applied rewrites 96.3%

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

Alternative 6: 77.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\_m\\ \mathbf{if}\;d \leq -2.05 \cdot 10^{-197}:\\ \;\;\;\;\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-185}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot D\_m, \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \frac{M\_m}{d}\right) \cdot \left(M\_m \cdot D\_m\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{d} \cdot \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{h}}\right) \cdot \left(1 - \frac{{\left(\frac{D\_m}{2} \cdot \frac{M\_m}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ (/ D_m d) 2.0) M_m)))
   (if (<= d -2.05e-197)
     (*
      (* (- d) (sqrt (/ 1.0 (* l h))))
      (- 1.0 (/ (* t_0 (* t_0 (* 0.5 h))) l)))
     (if (<= d 1.6e-185)
       (/
        (fma
         (* -0.125 D_m)
         (* (* (pow (/ h l) 1.5) (/ M_m d)) (* M_m D_m))
         (* (sqrt (/ h l)) d))
        h)
       (*
        (* (sqrt d) (/ (sqrt (/ d l)) (sqrt h)))
        (- 1.0 (/ (* (pow (* (/ D_m 2.0) (/ M_m d)) 2.0) (* 0.5 h)) l)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M_m;
	double tmp;
	if (d <= -2.05e-197) {
		tmp = (-d * sqrt((1.0 / (l * h)))) * (1.0 - ((t_0 * (t_0 * (0.5 * h))) / l));
	} else if (d <= 1.6e-185) {
		tmp = fma((-0.125 * D_m), ((pow((h / l), 1.5) * (M_m / d)) * (M_m * D_m)), (sqrt((h / l)) * d)) / h;
	} else {
		tmp = (sqrt(d) * (sqrt((d / l)) / sqrt(h))) * (1.0 - ((pow(((D_m / 2.0) * (M_m / d)), 2.0) * (0.5 * h)) / l));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(Float64(D_m / d) / 2.0) * M_m)
	tmp = 0.0
	if (d <= -2.05e-197)
		tmp = Float64(Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(0.5 * h))) / l)));
	elseif (d <= 1.6e-185)
		tmp = Float64(fma(Float64(-0.125 * D_m), Float64(Float64((Float64(h / l) ^ 1.5) * Float64(M_m / d)) * Float64(M_m * D_m)), Float64(sqrt(Float64(h / l)) * d)) / h);
	else
		tmp = Float64(Float64(sqrt(d) * Float64(sqrt(Float64(d / l)) / sqrt(h))) * Float64(1.0 - Float64(Float64((Float64(Float64(D_m / 2.0) * Float64(M_m / d)) ^ 2.0) * Float64(0.5 * h)) / l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[d, -2.05e-197], N[(N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.6e-185], N[(N[(N[(-0.125 * D$95$m), $MachinePrecision] * N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.05e-197

   1. Initial program 75.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 75.7

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 75.7

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

   4. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 79.0%

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

   7. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 84.6

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

   9. Applied rewrites 84.6%

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

   if -2.05e-197 < d < 1.5999999999999999e-185

   1. Initial program 32.4%

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

      1. lift-*.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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 30.0

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 30.0

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

   4. Applied rewrites 30.0%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   7. Applied rewrites 36.8%

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

   9. Applied rewrites 51.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot D, D \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\frac{M}{d} \cdot M\right)\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]
   10. Step-by-step derivation

   11. Applied rewrites 53.6%

       \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot D, \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

   if 1.5999999999999999e-185 < d

   1. Initial program 78.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 80.1

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 80.1

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

   4. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-div N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 89.7

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

      14. lift-/.f64 N/A

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

      15. metadata-eval 89.7

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

      16. lift-pow.f64 N/A

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

      17. pow1/2 N/A

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

      18. lift-sqrt.f64 89.7

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

   6. Applied rewrites 89.7%

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

Alternative 7: 74.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\_m\\ t_1 := 1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\\ t_2 := \frac{M\_m}{d} \cdot D\_m\\ t_3 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq -2.05 \cdot 10^{-197}:\\ \;\;\;\;\left(\left(-d\right) \cdot t\_3\right) \cdot t\_1\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-178}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot D\_m, \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \frac{M\_m}{d}\right) \cdot \left(M\_m \cdot D\_m\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+172}:\\ \;\;\;\;\left(t\_3 \cdot d\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot \frac{t\_2 \cdot t\_2}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ (/ D_m d) 2.0) M_m))
        (t_1 (- 1.0 (/ (* t_0 (* t_0 (* 0.5 h))) l)))
        (t_2 (* (/ M_m d) D_m))
        (t_3 (sqrt (/ 1.0 (* l h)))))
   (if (<= d -2.05e-197)
     (* (* (- d) t_3) t_1)
     (if (<= d 2e-178)
       (/
        (fma
         (* -0.125 D_m)
         (* (* (pow (/ h l) 1.5) (/ M_m d)) (* M_m D_m))
         (* (sqrt (/ h l)) d))
        h)
       (if (<= d 7e+172)
         (* (* t_3 d) t_1)
         (*
          (* (fma (* -0.5 (/ (* t_2 t_2) 4.0)) (/ h l) 1.0) (sqrt (/ d h)))
          (/ (sqrt d) (sqrt l))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M_m;
	double t_1 = 1.0 - ((t_0 * (t_0 * (0.5 * h))) / l);
	double t_2 = (M_m / d) * D_m;
	double t_3 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -2.05e-197) {
		tmp = (-d * t_3) * t_1;
	} else if (d <= 2e-178) {
		tmp = fma((-0.125 * D_m), ((pow((h / l), 1.5) * (M_m / d)) * (M_m * D_m)), (sqrt((h / l)) * d)) / h;
	} else if (d <= 7e+172) {
		tmp = (t_3 * d) * t_1;
	} else {
		tmp = (fma((-0.5 * ((t_2 * t_2) / 4.0)), (h / l), 1.0) * sqrt((d / h))) * (sqrt(d) / sqrt(l));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(Float64(D_m / d) / 2.0) * M_m)
	t_1 = Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(0.5 * h))) / l))
	t_2 = Float64(Float64(M_m / d) * D_m)
	t_3 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= -2.05e-197)
		tmp = Float64(Float64(Float64(-d) * t_3) * t_1);
	elseif (d <= 2e-178)
		tmp = Float64(fma(Float64(-0.125 * D_m), Float64(Float64((Float64(h / l) ^ 1.5) * Float64(M_m / d)) * Float64(M_m * D_m)), Float64(sqrt(Float64(h / l)) * d)) / h);
	elseif (d <= 7e+172)
		tmp = Float64(Float64(t_3 * d) * t_1);
	else
		tmp = Float64(Float64(fma(Float64(-0.5 * Float64(Float64(t_2 * t_2) / 4.0)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * Float64(sqrt(d) / sqrt(l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2.05e-197], N[(N[((-d) * t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 2e-178], N[(N[(N[(-0.125 * D$95$m), $MachinePrecision] * N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 7e+172], N[(N[(t$95$3 * d), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[(-0.5 * N[(N[(t$95$2 * t$95$2), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.05e-197

   1. Initial program 75.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 75.7

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 75.7

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

   4. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 79.0%

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

   7. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 84.6

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

   9. Applied rewrites 84.6%

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

   if -2.05e-197 < d < 1.9999999999999999e-178

   1. Initial program 34.0%

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

      1. lift-*.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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 31.7

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 31.7

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

   4. Applied rewrites 31.7%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   7. Applied rewrites 38.2%

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

   9. Applied rewrites 52.3%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot D, D \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\frac{M}{d} \cdot M\right)\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]
   10. Step-by-step derivation

   11. Applied rewrites 54.6%

       \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot D, \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \frac{M}{d}\right) \cdot \left(M \cdot D\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

   if 1.9999999999999999e-178 < d < 6.99999999999999955e172

   1. Initial program 72.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 73.0

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 73.0

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

   4. Applied rewrites 73.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 76.0%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 84.5

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

   9. Applied rewrites 84.5%

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

   if 6.99999999999999955e172 < d

   1. Initial program 89.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 92.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      18. metadata-eval 92.7

          \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 92.7%

      \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

      3. sqrt-div N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}} \]

      6. lower-sqrt.f64 92.9

         \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

   8. Applied rewrites 92.9%

      \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 74.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\_m\\ t_1 := 1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\\ t_2 := \frac{M\_m}{d} \cdot D\_m\\ t_3 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{-197}:\\ \;\;\;\;\left(\left(-d\right) \cdot t\_3\right) \cdot t\_1\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-179}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot D\_m, D\_m \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\frac{M\_m}{d} \cdot M\_m\right)\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+172}:\\ \;\;\;\;\left(t\_3 \cdot d\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot \frac{t\_2 \cdot t\_2}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ (/ D_m d) 2.0) M_m))
        (t_1 (- 1.0 (/ (* t_0 (* t_0 (* 0.5 h))) l)))
        (t_2 (* (/ M_m d) D_m))
        (t_3 (sqrt (/ 1.0 (* l h)))))
   (if (<= d -1.3e-197)
     (* (* (- d) t_3) t_1)
     (if (<= d 1.35e-179)
       (/
        (fma
         (* -0.125 D_m)
         (* D_m (* (pow (/ h l) 1.5) (* (/ M_m d) M_m)))
         (* (sqrt (/ h l)) d))
        h)
       (if (<= d 7e+172)
         (* (* t_3 d) t_1)
         (*
          (* (fma (* -0.5 (/ (* t_2 t_2) 4.0)) (/ h l) 1.0) (sqrt (/ d h)))
          (/ (sqrt d) (sqrt l))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M_m;
	double t_1 = 1.0 - ((t_0 * (t_0 * (0.5 * h))) / l);
	double t_2 = (M_m / d) * D_m;
	double t_3 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -1.3e-197) {
		tmp = (-d * t_3) * t_1;
	} else if (d <= 1.35e-179) {
		tmp = fma((-0.125 * D_m), (D_m * (pow((h / l), 1.5) * ((M_m / d) * M_m))), (sqrt((h / l)) * d)) / h;
	} else if (d <= 7e+172) {
		tmp = (t_3 * d) * t_1;
	} else {
		tmp = (fma((-0.5 * ((t_2 * t_2) / 4.0)), (h / l), 1.0) * sqrt((d / h))) * (sqrt(d) / sqrt(l));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(Float64(D_m / d) / 2.0) * M_m)
	t_1 = Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(0.5 * h))) / l))
	t_2 = Float64(Float64(M_m / d) * D_m)
	t_3 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= -1.3e-197)
		tmp = Float64(Float64(Float64(-d) * t_3) * t_1);
	elseif (d <= 1.35e-179)
		tmp = Float64(fma(Float64(-0.125 * D_m), Float64(D_m * Float64((Float64(h / l) ^ 1.5) * Float64(Float64(M_m / d) * M_m))), Float64(sqrt(Float64(h / l)) * d)) / h);
	elseif (d <= 7e+172)
		tmp = Float64(Float64(t_3 * d) * t_1);
	else
		tmp = Float64(Float64(fma(Float64(-0.5 * Float64(Float64(t_2 * t_2) / 4.0)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * Float64(sqrt(d) / sqrt(l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.3e-197], N[(N[((-d) * t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 1.35e-179], N[(N[(N[(-0.125 * D$95$m), $MachinePrecision] * N[(D$95$m * N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 7e+172], N[(N[(t$95$3 * d), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[(-0.5 * N[(N[(t$95$2 * t$95$2), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.3000000000000001e-197

   1. Initial program 75.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 75.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 75.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   4. Applied rewrites 75.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   6. Applied rewrites 79.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]

   7. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right)\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot \color{blue}{-1}\right)\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \cdot -1\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot -1\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right)\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(d \cdot \color{blue}{\left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. mul-1-neg N/A

         \[\leadsto \left(d \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      9. distribute-rgt-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      16. lower-*.f64 84.6

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   9. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   if -1.3000000000000001e-197 < d < 1.34999999999999994e-179

   1. Initial program 34.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.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. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 31.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 31.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   4. Applied rewrites 31.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]

   7. Applied rewrites 38.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot \frac{M}{d}\right), \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]
   8. Step-by-step derivation

   9. Applied rewrites 52.3%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot D, D \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\frac{M}{d} \cdot M\right)\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h} \]

   if 1.34999999999999994e-179 < d < 6.99999999999999955e172

   1. Initial program 72.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 73.0

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 73.0

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   4. Applied rewrites 73.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   6. Applied rewrites 76.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]

   7. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. lower-*.f64 84.5

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   9. Applied rewrites 84.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   if 6.99999999999999955e172 < d

   1. Initial program 89.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 92.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. associate-*l/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{D \cdot \frac{M}{d}}{2}} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. associate-*l/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\frac{D \cdot \frac{M}{d}}{2}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. frac-times N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{2 \cdot 2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{4}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      18. metadata-eval 92.7

          \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 92.7%

      \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

      3. sqrt-div N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}} \]

      6. lower-sqrt.f64 92.9

         \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

   8. Applied rewrites 92.9%

      \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 77.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\_m\\ t_1 := 1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\\ t_2 := \frac{M\_m}{d} \cdot D\_m\\ t_3 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-d\right) \cdot t\_3\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+36}:\\ \;\;\;\;\left(t\_3 \cdot d\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5 \cdot \frac{t\_2 \cdot t\_2}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ (/ D_m d) 2.0) M_m))
        (t_1 (- 1.0 (/ (* t_0 (* t_0 (* 0.5 h))) l)))
        (t_2 (* (/ M_m d) D_m))
        (t_3 (sqrt (/ 1.0 (* l h)))))
   (if (<= l -5e-310)
     (* (* (- d) t_3) t_1)
     (if (<= l 6.2e+36)
       (* (* t_3 d) t_1)
       (*
        (* (fma (* -0.5 (/ (* t_2 t_2) 4.0)) (/ h l) 1.0) (sqrt (/ d h)))
        (/ (sqrt d) (sqrt l)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M_m;
	double t_1 = 1.0 - ((t_0 * (t_0 * (0.5 * h))) / l);
	double t_2 = (M_m / d) * D_m;
	double t_3 = sqrt((1.0 / (l * h)));
	double tmp;
	if (l <= -5e-310) {
		tmp = (-d * t_3) * t_1;
	} else if (l <= 6.2e+36) {
		tmp = (t_3 * d) * t_1;
	} else {
		tmp = (fma((-0.5 * ((t_2 * t_2) / 4.0)), (h / l), 1.0) * sqrt((d / h))) * (sqrt(d) / sqrt(l));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(Float64(D_m / d) / 2.0) * M_m)
	t_1 = Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(0.5 * h))) / l))
	t_2 = Float64(Float64(M_m / d) * D_m)
	t_3 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(Float64(-d) * t_3) * t_1);
	elseif (l <= 6.2e+36)
		tmp = Float64(Float64(t_3 * d) * t_1);
	else
		tmp = Float64(Float64(fma(Float64(-0.5 * Float64(Float64(t_2 * t_2) / 4.0)), Float64(h / l), 1.0) * sqrt(Float64(d / h))) * Float64(sqrt(d) / sqrt(l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(N[((-d) * t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 6.2e+36], N[(N[(t$95$3 * d), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[(-0.5 * N[(N[(t$95$2 * t$95$2), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 68.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 68.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   4. Applied rewrites 68.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   6. Applied rewrites 71.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]

   7. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right)\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot \color{blue}{-1}\right)\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \cdot -1\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot -1\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right)\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(d \cdot \color{blue}{\left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. mul-1-neg N/A

         \[\leadsto \left(d \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      9. distribute-rgt-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      16. lower-*.f64 80.7

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   9. Applied rewrites 80.7%

      \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   if -4.999999999999985e-310 < l < 6.1999999999999999e36

   1. Initial program 75.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.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. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 80.0

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 80.0

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   4. Applied rewrites 80.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   6. Applied rewrites 80.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]

   7. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. lower-*.f64 85.0

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   9. Applied rewrites 85.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   if 6.1999999999999999e36 < l

   1. Initial program 63.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 61.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. associate-*l/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{D \cdot \frac{M}{d}}{2}} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. associate-*l/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\frac{D \cdot \frac{M}{d}}{2}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. frac-times N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{2 \cdot 2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{4}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      18. metadata-eval 61.7

          \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 61.7%

      \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

      3. sqrt-div N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}} \]

      6. lower-sqrt.f64 70.8

         \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \]

   8. Applied rewrites 70.8%

      \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 77.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\_m\\ t_1 := 1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\\ t_2 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-d\right) \cdot t\_2\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+138}:\\ \;\;\;\;\left(t\_2 \cdot d\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ (/ D_m d) 2.0) M_m))
        (t_1 (- 1.0 (/ (* t_0 (* t_0 (* 0.5 h))) l)))
        (t_2 (sqrt (/ 1.0 (* l h)))))
   (if (<= l -5e-310)
     (* (* (- d) t_2) t_1)
     (if (<= l 1.25e+138) (* (* t_2 d) t_1) (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M_m;
	double t_1 = 1.0 - ((t_0 * (t_0 * (0.5 * h))) / l);
	double t_2 = sqrt((1.0 / (l * h)));
	double tmp;
	if (l <= -5e-310) {
		tmp = (-d * t_2) * t_1;
	} else if (l <= 1.25e+138) {
		tmp = (t_2 * d) * t_1;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((d_m / d) / 2.0d0) * m_m
    t_1 = 1.0d0 - ((t_0 * (t_0 * (0.5d0 * h))) / l)
    t_2 = sqrt((1.0d0 / (l * h)))
    if (l <= (-5d-310)) then
        tmp = (-d * t_2) * t_1
    else if (l <= 1.25d+138) then
        tmp = (t_2 * d) * t_1
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M_m;
	double t_1 = 1.0 - ((t_0 * (t_0 * (0.5 * h))) / l);
	double t_2 = Math.sqrt((1.0 / (l * h)));
	double tmp;
	if (l <= -5e-310) {
		tmp = (-d * t_2) * t_1;
	} else if (l <= 1.25e+138) {
		tmp = (t_2 * d) * t_1;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = ((D_m / d) / 2.0) * M_m
	t_1 = 1.0 - ((t_0 * (t_0 * (0.5 * h))) / l)
	t_2 = math.sqrt((1.0 / (l * h)))
	tmp = 0
	if l <= -5e-310:
		tmp = (-d * t_2) * t_1
	elif l <= 1.25e+138:
		tmp = (t_2 * d) * t_1
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(Float64(D_m / d) / 2.0) * M_m)
	t_1 = Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(0.5 * h))) / l))
	t_2 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(Float64(-d) * t_2) * t_1);
	elseif (l <= 1.25e+138)
		tmp = Float64(Float64(t_2 * d) * t_1);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = ((D_m / d) / 2.0) * M_m;
	t_1 = 1.0 - ((t_0 * (t_0 * (0.5 * h))) / l);
	t_2 = sqrt((1.0 / (l * h)));
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = (-d * t_2) * t_1;
	elseif (l <= 1.25e+138)
		tmp = (t_2 * d) * t_1;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(N[((-d) * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 1.25e+138], N[(N[(t$95$2 * d), $MachinePrecision] * t$95$1), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 68.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 68.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   4. Applied rewrites 68.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   6. Applied rewrites 71.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]

   7. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right)\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot \color{blue}{-1}\right)\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \cdot -1\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot -1\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right)\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(d \cdot \color{blue}{\left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      8. mul-1-neg N/A

         \[\leadsto \left(d \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      9. distribute-rgt-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      16. lower-*.f64 80.7

          \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   9. Applied rewrites 80.7%

      \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   if -4.999999999999985e-310 < l < 1.25000000000000004e138

   1. Initial program 77.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.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. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 80.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 80.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   4. Applied rewrites 80.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   6. Applied rewrites 82.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]

   7. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. lower-*.f64 84.6

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   9. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   if 1.25000000000000004e138 < l

   1. Initial program 53.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      15. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      17. lower-*.f64 45.8

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.8%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 64.3%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+138}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\_m\\ t_1 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-t\_1\right) \cdot d\right) \cdot \left(1 - \left(t\_0 \cdot \left(0.25 \cdot \left(\frac{M\_m}{d} \cdot D\_m\right)\right)\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+138}:\\ \;\;\;\;\left(t\_1 \cdot d\right) \cdot \left(1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ (/ D_m d) 2.0) M_m)) (t_1 (sqrt (/ 1.0 (* l h)))))
   (if (<= l -5e-310)
     (* (* (- t_1) d) (- 1.0 (* (* t_0 (* 0.25 (* (/ M_m d) D_m))) (/ h l))))
     (if (<= l 1.25e+138)
       (* (* t_1 d) (- 1.0 (/ (* t_0 (* t_0 (* 0.5 h))) l)))
       (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M_m;
	double t_1 = sqrt((1.0 / (l * h)));
	double tmp;
	if (l <= -5e-310) {
		tmp = (-t_1 * d) * (1.0 - ((t_0 * (0.25 * ((M_m / d) * D_m))) * (h / l)));
	} else if (l <= 1.25e+138) {
		tmp = (t_1 * d) * (1.0 - ((t_0 * (t_0 * (0.5 * h))) / l));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((d_m / d) / 2.0d0) * m_m
    t_1 = sqrt((1.0d0 / (l * h)))
    if (l <= (-5d-310)) then
        tmp = (-t_1 * d) * (1.0d0 - ((t_0 * (0.25d0 * ((m_m / d) * d_m))) * (h / l)))
    else if (l <= 1.25d+138) then
        tmp = (t_1 * d) * (1.0d0 - ((t_0 * (t_0 * (0.5d0 * h))) / l))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M_m;
	double t_1 = Math.sqrt((1.0 / (l * h)));
	double tmp;
	if (l <= -5e-310) {
		tmp = (-t_1 * d) * (1.0 - ((t_0 * (0.25 * ((M_m / d) * D_m))) * (h / l)));
	} else if (l <= 1.25e+138) {
		tmp = (t_1 * d) * (1.0 - ((t_0 * (t_0 * (0.5 * h))) / l));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = ((D_m / d) / 2.0) * M_m
	t_1 = math.sqrt((1.0 / (l * h)))
	tmp = 0
	if l <= -5e-310:
		tmp = (-t_1 * d) * (1.0 - ((t_0 * (0.25 * ((M_m / d) * D_m))) * (h / l)))
	elif l <= 1.25e+138:
		tmp = (t_1 * d) * (1.0 - ((t_0 * (t_0 * (0.5 * h))) / l))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(Float64(D_m / d) / 2.0) * M_m)
	t_1 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(Float64(-t_1) * d) * Float64(1.0 - Float64(Float64(t_0 * Float64(0.25 * Float64(Float64(M_m / d) * D_m))) * Float64(h / l))));
	elseif (l <= 1.25e+138)
		tmp = Float64(Float64(t_1 * d) * Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(0.5 * h))) / l)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = ((D_m / d) / 2.0) * M_m;
	t_1 = sqrt((1.0 / (l * h)));
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = (-t_1 * d) * (1.0 - ((t_0 * (0.25 * ((M_m / d) * D_m))) * (h / l)));
	elseif (l <= 1.25e+138)
		tmp = (t_1 * d) * (1.0 - ((t_0 * (t_0 * (0.5 * h))) / l));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(N[((-t$95$1) * d), $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 * N[(0.25 * N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.25e+138], N[(N[(t$95$1 * d), $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 2.9

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 2.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. frac-times N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 2.9%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot 0.5\right)\right)} \cdot \frac{h}{\ell}\right) \]

   8. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot M}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot M}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-/l* N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. lower-/.f64 2.9

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.25 \cdot \left(\color{blue}{\frac{M}{d}} \cdot D\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

   10. Applied rewrites 2.9%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(0.25 \cdot \left(\frac{M}{d} \cdot D\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

   11. Taylor expanded in h around -inf

       \[\leadsto \left(\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot \frac{h}{\ell}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 70.7%

       \[\leadsto \left(\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.25 \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

   if -4.999999999999985e-310 < l < 1.25000000000000004e138

   1. Initial program 77.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.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. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot h}{\ell}\right) \]

      7. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      13. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      17. lower-*.f64 80.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      18. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{\ell}\right) \]

      19. metadata-eval 80.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot h\right)}{\ell}\right) \]

   4. Applied rewrites 80.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      6. frac-times N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{\ell}\right) \]

      12. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{\ell}\right) \]

   6. Applied rewrites 82.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}}{\ell}\right) \]

   7. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      6. lower-*.f64 84.6

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   9. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right) \]

   if 1.25000000000000004e138 < l

   1. Initial program 53.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      15. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      17. lower-*.f64 45.8

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.8%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 64.3%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.25 \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+138}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.5 \cdot h\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2}\\ t_1 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-t\_1\right) \cdot d\right) \cdot \left(1 - \left(\left(t\_0 \cdot M\_m\right) \cdot \left(0.25 \cdot \left(\frac{M\_m}{d} \cdot D\_m\right)\right)\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+138}:\\ \;\;\;\;\left(t\_1 \cdot d\right) \cdot \left(1 - \frac{\left(\left(0.25 \cdot D\_m\right) \cdot \frac{M\_m}{d}\right) \cdot \left(\left(M\_m \cdot t\_0\right) \cdot h\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (/ D_m d) 2.0)) (t_1 (sqrt (/ 1.0 (* l h)))))
   (if (<= l -5e-310)
     (*
      (* (- t_1) d)
      (- 1.0 (* (* (* t_0 M_m) (* 0.25 (* (/ M_m d) D_m))) (/ h l))))
     (if (<= l 1.25e+138)
       (*
        (* t_1 d)
        (- 1.0 (/ (* (* (* 0.25 D_m) (/ M_m d)) (* (* M_m t_0) h)) l)))
       (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m / d) / 2.0;
	double t_1 = sqrt((1.0 / (l * h)));
	double tmp;
	if (l <= -5e-310) {
		tmp = (-t_1 * d) * (1.0 - (((t_0 * M_m) * (0.25 * ((M_m / d) * D_m))) * (h / l)));
	} else if (l <= 1.25e+138) {
		tmp = (t_1 * d) * (1.0 - ((((0.25 * D_m) * (M_m / d)) * ((M_m * t_0) * h)) / l));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (d_m / d) / 2.0d0
    t_1 = sqrt((1.0d0 / (l * h)))
    if (l <= (-5d-310)) then
        tmp = (-t_1 * d) * (1.0d0 - (((t_0 * m_m) * (0.25d0 * ((m_m / d) * d_m))) * (h / l)))
    else if (l <= 1.25d+138) then
        tmp = (t_1 * d) * (1.0d0 - ((((0.25d0 * d_m) * (m_m / d)) * ((m_m * t_0) * h)) / l))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m / d) / 2.0;
	double t_1 = Math.sqrt((1.0 / (l * h)));
	double tmp;
	if (l <= -5e-310) {
		tmp = (-t_1 * d) * (1.0 - (((t_0 * M_m) * (0.25 * ((M_m / d) * D_m))) * (h / l)));
	} else if (l <= 1.25e+138) {
		tmp = (t_1 * d) * (1.0 - ((((0.25 * D_m) * (M_m / d)) * ((M_m * t_0) * h)) / l));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (D_m / d) / 2.0
	t_1 = math.sqrt((1.0 / (l * h)))
	tmp = 0
	if l <= -5e-310:
		tmp = (-t_1 * d) * (1.0 - (((t_0 * M_m) * (0.25 * ((M_m / d) * D_m))) * (h / l)))
	elif l <= 1.25e+138:
		tmp = (t_1 * d) * (1.0 - ((((0.25 * D_m) * (M_m / d)) * ((M_m * t_0) * h)) / l))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(D_m / d) / 2.0)
	t_1 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(Float64(-t_1) * d) * Float64(1.0 - Float64(Float64(Float64(t_0 * M_m) * Float64(0.25 * Float64(Float64(M_m / d) * D_m))) * Float64(h / l))));
	elseif (l <= 1.25e+138)
		tmp = Float64(Float64(t_1 * d) * Float64(1.0 - Float64(Float64(Float64(Float64(0.25 * D_m) * Float64(M_m / d)) * Float64(Float64(M_m * t_0) * h)) / l)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (D_m / d) / 2.0;
	t_1 = sqrt((1.0 / (l * h)));
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = (-t_1 * d) * (1.0 - (((t_0 * M_m) * (0.25 * ((M_m / d) * D_m))) * (h / l)));
	elseif (l <= 1.25e+138)
		tmp = (t_1 * d) * (1.0 - ((((0.25 * D_m) * (M_m / d)) * ((M_m * t_0) * h)) / l));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(N[((-t$95$1) * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(t$95$0 * M$95$m), $MachinePrecision] * N[(0.25 * N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.25e+138], N[(N[(t$95$1 * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * t$95$0), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 2.9

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 2.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. frac-times N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 2.9%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot 0.5\right)\right)} \cdot \frac{h}{\ell}\right) \]

   8. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot M}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot M}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-/l* N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. lower-/.f64 2.9

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.25 \cdot \left(\color{blue}{\frac{M}{d}} \cdot D\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

   10. Applied rewrites 2.9%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(0.25 \cdot \left(\frac{M}{d} \cdot D\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

   11. Taylor expanded in h around -inf

       \[\leadsto \left(\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot \frac{h}{\ell}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 70.7%

       \[\leadsto \left(\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.25 \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

   if -4.999999999999985e-310 < l < 1.25000000000000004e138

   1. Initial program 77.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 74.7

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. frac-times N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot 0.5\right)\right)} \cdot \frac{h}{\ell}\right) \]

   8. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot M}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot M}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-/l* N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. lower-/.f64 75.8

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.25 \cdot \left(\color{blue}{\frac{M}{d}} \cdot D\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

   10. Applied rewrites 75.8%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(0.25 \cdot \left(\frac{M}{d} \cdot D\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot \frac{h}{\ell}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

       3. associate-*r/ N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot h}{\ell}}\right) \]

       4. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot h}{\ell}}\right) \]

   12. Applied rewrites 84.4%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right) \cdot \left(\left(M \cdot \frac{\frac{D}{d}}{2}\right) \cdot h\right)}{\ell}}\right) \]

   if 1.25000000000000004e138 < l

   1. Initial program 53.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      15. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      17. lower-*.f64 45.8

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.8%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 64.3%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.25 \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+138}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right) \cdot \left(\left(M \cdot \frac{\frac{D}{d}}{2}\right) \cdot h\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq -3.65 \cdot 10^{+127}:\\ \;\;\;\;\left(-t\_0\right) \cdot d\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-200}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(M\_m \cdot M\_m\right) \cdot -0.125, \frac{h}{d \cdot d} \cdot \frac{D\_m \cdot D\_m}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot d\right) \cdot \left(1 - \frac{\left(\left(0.25 \cdot D\_m\right) \cdot \frac{M\_m}{d}\right) \cdot \left(\left(M\_m \cdot \frac{\frac{D\_m}{d}}{2}\right) \cdot h\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= d -3.65e+127)
     (* (- t_0) d)
     (if (<= d -2.3e-200)
       (*
        (*
         (fma (* (* M_m M_m) -0.125) (* (/ h (* d d)) (/ (* D_m D_m) l)) 1.0)
         (sqrt (/ d h)))
        (sqrt (/ d l)))
       (*
        (* t_0 d)
        (-
         1.0
         (/
          (* (* (* 0.25 D_m) (/ M_m d)) (* (* M_m (/ (/ D_m d) 2.0)) h))
          l)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -3.65e+127) {
		tmp = -t_0 * d;
	} else if (d <= -2.3e-200) {
		tmp = (fma(((M_m * M_m) * -0.125), ((h / (d * d)) * ((D_m * D_m) / l)), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = (t_0 * d) * (1.0 - ((((0.25 * D_m) * (M_m / d)) * ((M_m * ((D_m / d) / 2.0)) * h)) / l));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= -3.65e+127)
		tmp = Float64(Float64(-t_0) * d);
	elseif (d <= -2.3e-200)
		tmp = Float64(Float64(fma(Float64(Float64(M_m * M_m) * -0.125), Float64(Float64(h / Float64(d * d)) * Float64(Float64(D_m * D_m) / l)), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(t_0 * d) * Float64(1.0 - Float64(Float64(Float64(Float64(0.25 * D_m) * Float64(M_m / d)) * Float64(Float64(M_m * Float64(Float64(D_m / d) / 2.0)) * h)) / l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -3.65e+127], N[((-t$95$0) * d), $MachinePrecision], If[LessEqual[d, -2.3e-200], N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.6499999999999998e127

   1. Initial program 80.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      15. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      17. lower-*.f64 8.7

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 8.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   6. Taylor expanded in h around -inf

      \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d \]
   7. Step-by-step derivation

   8. Applied rewrites 70.8%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if -3.6499999999999998e127 < d < -2.30000000000000007e-200

   1. Initial program 72.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 71.5%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. associate-*l/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{D \cdot \frac{M}{d}}{2}} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. associate-*l/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\frac{D \cdot \frac{M}{d}}{2}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. frac-times N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{2 \cdot 2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{4}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      18. metadata-eval 71.5

          \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 71.5%

      \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   7. Taylor expanded in M around inf

      \[\leadsto \left(\color{blue}{\left({M}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell} + \frac{1}{{M}^{2}}\right)\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   8. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \left(\color{blue}{\left({M}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) + {M}^{2} \cdot \frac{1}{{M}^{2}}\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left({M}^{2} \cdot \frac{-1}{8}\right) \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}} + {M}^{2} \cdot \frac{1}{{M}^{2}}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. rgt-mult-inverse N/A

         \[\leadsto \left(\left(\left({M}^{2} \cdot \frac{-1}{8}\right) \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell} + \color{blue}{1}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({M}^{2} \cdot \frac{-1}{8}, \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{{M}^{2} \cdot \frac{-1}{8}}, \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{-1}{8}, \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{-1}{8}, \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot \frac{-1}{8}, \frac{\color{blue}{h \cdot {D}^{2}}}{{d}^{2} \cdot \ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. times-frac N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot \frac{-1}{8}, \color{blue}{\frac{h}{{d}^{2}} \cdot \frac{{D}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot \frac{-1}{8}, \color{blue}{\frac{h}{{d}^{2}} \cdot \frac{{D}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot \frac{-1}{8}, \color{blue}{\frac{h}{{d}^{2}}} \cdot \frac{{D}^{2}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot \frac{-1}{8}, \frac{h}{\color{blue}{d \cdot d}} \cdot \frac{{D}^{2}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot \frac{-1}{8}, \frac{h}{\color{blue}{d \cdot d}} \cdot \frac{{D}^{2}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot \frac{-1}{8}, \frac{h}{d \cdot d} \cdot \color{blue}{\frac{{D}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot \frac{-1}{8}, \frac{h}{d \cdot d} \cdot \frac{\color{blue}{D \cdot D}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. lower-*.f64 54.3

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot -0.125, \frac{h}{d \cdot d} \cdot \frac{\color{blue}{D \cdot D}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 54.3%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(M \cdot M\right) \cdot -0.125, \frac{h}{d \cdot d} \cdot \frac{D \cdot D}{\ell}, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -2.30000000000000007e-200 < d

   1. Initial program 64.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 58.5

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. frac-times N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 59.2%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot 0.5\right)\right)} \cdot \frac{h}{\ell}\right) \]

   8. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot M}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot M}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-/l* N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. lower-/.f64 58.3

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.25 \cdot \left(\color{blue}{\frac{M}{d}} \cdot D\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

   10. Applied rewrites 58.3%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(0.25 \cdot \left(\frac{M}{d} \cdot D\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot \frac{h}{\ell}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

       3. associate-*r/ N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot h}{\ell}}\right) \]

       4. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot h}{\ell}}\right) \]

   12. Applied rewrites 63.4%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right) \cdot \left(\left(M \cdot \frac{\frac{D}{d}}{2}\right) \cdot h\right)}{\ell}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.65 \cdot 10^{+127}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-200}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot -0.125, \frac{h}{d \cdot d} \cdot \frac{D \cdot D}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right) \cdot \left(\left(M \cdot \frac{\frac{D}{d}}{2}\right) \cdot h\right)}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq -3.65 \cdot 10^{+127}:\\ \;\;\;\;\left(-t\_0\right) \cdot d\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-200}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(M\_m \cdot M\_m\right) \cdot -0.125, \frac{h}{d \cdot d} \cdot \frac{D\_m \cdot D\_m}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot d\right) \cdot \left(1 - \frac{\left(M\_m \cdot D\_m\right) \cdot \left(\frac{h}{\ell} \cdot \left(\left(0.25 \cdot D\_m\right) \cdot \frac{M\_m}{d}\right)\right)}{2 \cdot d}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= d -3.65e+127)
     (* (- t_0) d)
     (if (<= d -2.3e-200)
       (*
        (*
         (fma (* (* M_m M_m) -0.125) (* (/ h (* d d)) (/ (* D_m D_m) l)) 1.0)
         (sqrt (/ d h)))
        (sqrt (/ d l)))
       (*
        (* t_0 d)
        (-
         1.0
         (/
          (* (* M_m D_m) (* (/ h l) (* (* 0.25 D_m) (/ M_m d))))
          (* 2.0 d))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -3.65e+127) {
		tmp = -t_0 * d;
	} else if (d <= -2.3e-200) {
		tmp = (fma(((M_m * M_m) * -0.125), ((h / (d * d)) * ((D_m * D_m) / l)), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = (t_0 * d) * (1.0 - (((M_m * D_m) * ((h / l) * ((0.25 * D_m) * (M_m / d)))) / (2.0 * d)));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= -3.65e+127)
		tmp = Float64(Float64(-t_0) * d);
	elseif (d <= -2.3e-200)
		tmp = Float64(Float64(fma(Float64(Float64(M_m * M_m) * -0.125), Float64(Float64(h / Float64(d * d)) * Float64(Float64(D_m * D_m) / l)), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(t_0 * d) * Float64(1.0 - Float64(Float64(Float64(M_m * D_m) * Float64(Float64(h / l) * Float64(Float64(0.25 * D_m) * Float64(M_m / d)))) / Float64(2.0 * d))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -3.65e+127], N[((-t$95$0) * d), $MachinePrecision], If[LessEqual[d, -2.3e-200], N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.6499999999999998e127

   1. Initial program 80.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      15. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      17. lower-*.f64 8.7

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 8.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   6. Taylor expanded in h around -inf

      \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d \]
   7. Step-by-step derivation

   8. Applied rewrites 70.8%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if -3.6499999999999998e127 < d < -2.30000000000000007e-200

   1. Initial program 72.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   4. Applied rewrites 71.5%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5 \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\left(\left(\frac{D}{2} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right)}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. associate-*l/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\color{blue}{\frac{D \cdot \frac{M}{d}}{2}} \cdot \left(\frac{D}{2} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. associate-*l/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \left(\frac{D \cdot \frac{M}{d}}{2} \cdot \color{blue}{\frac{D \cdot \frac{M}{d}}{2}}\right), \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. frac-times N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{2 \cdot 2}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{4}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      18. metadata-eval 71.5

          \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\color{blue}{4}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 71.5%

      \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{4}}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   7. Taylor expanded in M around inf

      \[\leadsto \left(\color{blue}{\left({M}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell} + \frac{1}{{M}^{2}}\right)\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   8. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \left(\color{blue}{\left({M}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) + {M}^{2} \cdot \frac{1}{{M}^{2}}\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\color{blue}{\left({M}^{2} \cdot \frac{-1}{8}\right) \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}} + {M}^{2} \cdot \frac{1}{{M}^{2}}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      3. rgt-mult-inverse N/A

         \[\leadsto \left(\left(\left({M}^{2} \cdot \frac{-1}{8}\right) \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell} + \color{blue}{1}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({M}^{2} \cdot \frac{-1}{8}, \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{{M}^{2} \cdot \frac{-1}{8}}, \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{-1}{8}, \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{-1}{8}, \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot \frac{-1}{8}, \frac{\color{blue}{h \cdot {D}^{2}}}{{d}^{2} \cdot \ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      9. times-frac N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot \frac{-1}{8}, \color{blue}{\frac{h}{{d}^{2}} \cdot \frac{{D}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot \frac{-1}{8}, \color{blue}{\frac{h}{{d}^{2}} \cdot \frac{{D}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot \frac{-1}{8}, \color{blue}{\frac{h}{{d}^{2}}} \cdot \frac{{D}^{2}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      12. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot \frac{-1}{8}, \frac{h}{\color{blue}{d \cdot d}} \cdot \frac{{D}^{2}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot \frac{-1}{8}, \frac{h}{\color{blue}{d \cdot d}} \cdot \frac{{D}^{2}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      14. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot \frac{-1}{8}, \frac{h}{d \cdot d} \cdot \color{blue}{\frac{{D}^{2}}{\ell}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      15. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot \frac{-1}{8}, \frac{h}{d \cdot d} \cdot \frac{\color{blue}{D \cdot D}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

      16. lower-*.f64 54.3

          \[\leadsto \left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot -0.125, \frac{h}{d \cdot d} \cdot \frac{\color{blue}{D \cdot D}}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 54.3%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(M \cdot M\right) \cdot -0.125, \frac{h}{d \cdot d} \cdot \frac{D \cdot D}{\ell}, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]

   if -2.30000000000000007e-200 < d

   1. Initial program 64.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 58.5

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. frac-times N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 59.2%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot 0.5\right)\right)} \cdot \frac{h}{\ell}\right) \]

   8. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot M}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot M}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-/l* N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. lower-/.f64 58.3

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.25 \cdot \left(\color{blue}{\frac{M}{d}} \cdot D\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

   10. Applied rewrites 58.3%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(0.25 \cdot \left(\frac{M}{d} \cdot D\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot \frac{h}{\ell}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right)\right)} \cdot \frac{h}{\ell}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right)} \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)} \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

       6. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(M \cdot \color{blue}{\frac{\frac{D}{d}}{2}}\right) \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

       7. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(M \cdot \frac{\color{blue}{\frac{D}{d}}}{2}\right) \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

       8. associate-/l/ N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(M \cdot \color{blue}{\frac{D}{d \cdot 2}}\right) \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right) \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

       10. associate-/l* N/A

           \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

       11. associate-*l/ N/A

           \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)}{2 \cdot d}}\right) \]

       12. lower-/.f64 N/A

           \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)}{2 \cdot d}}\right) \]

   12. Applied rewrites 59.5%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{h}{\ell} \cdot \left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right)\right)}{2 \cdot d}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.65 \cdot 10^{+127}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-200}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(M \cdot M\right) \cdot -0.125, \frac{h}{d \cdot d} \cdot \frac{D \cdot D}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\frac{h}{\ell} \cdot \left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right)\right)}{2 \cdot d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;h \leq 2.9 \cdot 10^{-303}:\\ \;\;\;\;\left(-t\_0\right) \cdot d\\ \mathbf{elif}\;h \leq 1.45 \cdot 10^{+221}:\\ \;\;\;\;\left(t\_0 \cdot d\right) \cdot \left(1 - \frac{\left(M\_m \cdot D\_m\right) \cdot \left(\frac{h}{\ell} \cdot \left(\left(0.25 \cdot D\_m\right) \cdot \frac{M\_m}{d}\right)\right)}{2 \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= h 2.9e-303)
     (* (- t_0) d)
     (if (<= h 1.45e+221)
       (*
        (* t_0 d)
        (-
         1.0
         (/ (* (* M_m D_m) (* (/ h l) (* (* 0.25 D_m) (/ M_m d)))) (* 2.0 d))))
       (* (/ (/ 1.0 (sqrt l)) (sqrt h)) d)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (h <= 2.9e-303) {
		tmp = -t_0 * d;
	} else if (h <= 1.45e+221) {
		tmp = (t_0 * d) * (1.0 - (((M_m * D_m) * ((h / l) * ((0.25 * D_m) * (M_m / d)))) / (2.0 * d)));
	} else {
		tmp = ((1.0 / sqrt(l)) / sqrt(h)) * d;
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (l * h)))
    if (h <= 2.9d-303) then
        tmp = -t_0 * d
    else if (h <= 1.45d+221) then
        tmp = (t_0 * d) * (1.0d0 - (((m_m * d_m) * ((h / l) * ((0.25d0 * d_m) * (m_m / d)))) / (2.0d0 * d)))
    else
        tmp = ((1.0d0 / sqrt(l)) / sqrt(h)) * d
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((1.0 / (l * h)));
	double tmp;
	if (h <= 2.9e-303) {
		tmp = -t_0 * d;
	} else if (h <= 1.45e+221) {
		tmp = (t_0 * d) * (1.0 - (((M_m * D_m) * ((h / l) * ((0.25 * D_m) * (M_m / d)))) / (2.0 * d)));
	} else {
		tmp = ((1.0 / Math.sqrt(l)) / Math.sqrt(h)) * d;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((1.0 / (l * h)))
	tmp = 0
	if h <= 2.9e-303:
		tmp = -t_0 * d
	elif h <= 1.45e+221:
		tmp = (t_0 * d) * (1.0 - (((M_m * D_m) * ((h / l) * ((0.25 * D_m) * (M_m / d)))) / (2.0 * d)))
	else:
		tmp = ((1.0 / math.sqrt(l)) / math.sqrt(h)) * d
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (h <= 2.9e-303)
		tmp = Float64(Float64(-t_0) * d);
	elseif (h <= 1.45e+221)
		tmp = Float64(Float64(t_0 * d) * Float64(1.0 - Float64(Float64(Float64(M_m * D_m) * Float64(Float64(h / l) * Float64(Float64(0.25 * D_m) * Float64(M_m / d)))) / Float64(2.0 * d))));
	else
		tmp = Float64(Float64(Float64(1.0 / sqrt(l)) / sqrt(h)) * d);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((1.0 / (l * h)));
	tmp = 0.0;
	if (h <= 2.9e-303)
		tmp = -t_0 * d;
	elseif (h <= 1.45e+221)
		tmp = (t_0 * d) * (1.0 - (((M_m * D_m) * ((h / l) * ((0.25 * D_m) * (M_m / d)))) / (2.0 * d)));
	else
		tmp = ((1.0 / sqrt(l)) / sqrt(h)) * d;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, 2.9e-303], N[((-t$95$0) * d), $MachinePrecision], If[LessEqual[h, 1.45e+221], N[(N[(t$95$0 * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < 2.90000000000000014e-303

   1. Initial program 68.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      15. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      17. lower-*.f64 10.2

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 10.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   6. Taylor expanded in h around -inf

      \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d \]
   7. Step-by-step derivation

   8. Applied rewrites 44.2%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if 2.90000000000000014e-303 < h < 1.4499999999999999e221

   1. Initial program 71.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 75.1

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. frac-times N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\color{blue}{\frac{D}{2}} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\left(\frac{D}{2} \cdot \color{blue}{\frac{M}{d}}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left({\color{blue}{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 76.1%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot 0.5\right)\right)} \cdot \frac{h}{\ell}\right) \]

   8. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot M}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot M}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-/l* N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      5. lower-/.f64 75.0

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(0.25 \cdot \left(\color{blue}{\frac{M}{d}} \cdot D\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

   10. Applied rewrites 75.0%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \color{blue}{\left(0.25 \cdot \left(\frac{M}{d} \cdot D\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \cdot \frac{h}{\ell}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right)\right)} \cdot \frac{h}{\ell}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right) \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{\frac{D}{d}}{2} \cdot M\right)} \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(M \cdot \frac{\frac{D}{d}}{2}\right)} \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

       6. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(M \cdot \color{blue}{\frac{\frac{D}{d}}{2}}\right) \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

       7. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(M \cdot \frac{\color{blue}{\frac{D}{d}}}{2}\right) \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

       8. associate-/l/ N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(M \cdot \color{blue}{\frac{D}{d \cdot 2}}\right) \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right) \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

       10. associate-/l* N/A

           \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)\right) \]

       11. associate-*l/ N/A

           \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)}{2 \cdot d}}\right) \]

       12. lower-/.f64 N/A

           \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\left(\frac{1}{4} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \frac{h}{\ell}\right)}{2 \cdot d}}\right) \]

   12. Applied rewrites 76.6%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\frac{h}{\ell} \cdot \left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right)\right)}{2 \cdot d}}\right) \]

   if 1.4499999999999999e221 < h

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      15. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      17. lower-*.f64 18.8

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 18.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.1%

      \[\leadsto \frac{\sqrt{{\ell}^{-1}}}{\sqrt{h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 56.2%

      \[\leadsto \frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 2.9 \cdot 10^{-303}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;h \leq 1.45 \cdot 10^{+221}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot \left(\frac{h}{\ell} \cdot \left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right)\right)}{2 \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 53.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq 2.25 \cdot 10^{-278}:\\ \;\;\;\;\left(-t\_0\right) \cdot d\\ \mathbf{elif}\;d \leq 60000000000000:\\ \;\;\;\;\left(t\_0 \cdot d\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \frac{M\_m \cdot \frac{M\_m}{d}}{d}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= d 2.25e-278)
     (* (- t_0) d)
     (if (<= d 60000000000000.0)
       (*
        (* t_0 d)
        (- 1.0 (* (* (* 0.125 (* D_m D_m)) (/ (* M_m (/ M_m d)) d)) (/ h l))))
       (* (/ (/ 1.0 (sqrt l)) (sqrt h)) d)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= 2.25e-278) {
		tmp = -t_0 * d;
	} else if (d <= 60000000000000.0) {
		tmp = (t_0 * d) * (1.0 - (((0.125 * (D_m * D_m)) * ((M_m * (M_m / d)) / d)) * (h / l)));
	} else {
		tmp = ((1.0 / sqrt(l)) / sqrt(h)) * d;
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (l * h)))
    if (d <= 2.25d-278) then
        tmp = -t_0 * d
    else if (d <= 60000000000000.0d0) then
        tmp = (t_0 * d) * (1.0d0 - (((0.125d0 * (d_m * d_m)) * ((m_m * (m_m / d)) / d)) * (h / l)))
    else
        tmp = ((1.0d0 / sqrt(l)) / sqrt(h)) * d
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= 2.25e-278) {
		tmp = -t_0 * d;
	} else if (d <= 60000000000000.0) {
		tmp = (t_0 * d) * (1.0 - (((0.125 * (D_m * D_m)) * ((M_m * (M_m / d)) / d)) * (h / l)));
	} else {
		tmp = ((1.0 / Math.sqrt(l)) / Math.sqrt(h)) * d;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((1.0 / (l * h)))
	tmp = 0
	if d <= 2.25e-278:
		tmp = -t_0 * d
	elif d <= 60000000000000.0:
		tmp = (t_0 * d) * (1.0 - (((0.125 * (D_m * D_m)) * ((M_m * (M_m / d)) / d)) * (h / l)))
	else:
		tmp = ((1.0 / math.sqrt(l)) / math.sqrt(h)) * d
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= 2.25e-278)
		tmp = Float64(Float64(-t_0) * d);
	elseif (d <= 60000000000000.0)
		tmp = Float64(Float64(t_0 * d) * Float64(1.0 - Float64(Float64(Float64(0.125 * Float64(D_m * D_m)) * Float64(Float64(M_m * Float64(M_m / d)) / d)) * Float64(h / l))));
	else
		tmp = Float64(Float64(Float64(1.0 / sqrt(l)) / sqrt(h)) * d);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((1.0 / (l * h)));
	tmp = 0.0;
	if (d <= 2.25e-278)
		tmp = -t_0 * d;
	elseif (d <= 60000000000000.0)
		tmp = (t_0 * d) * (1.0 - (((0.125 * (D_m * D_m)) * ((M_m * (M_m / d)) / d)) * (h / l)));
	else
		tmp = ((1.0 / sqrt(l)) / sqrt(h)) * d;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, 2.25e-278], N[((-t$95$0) * d), $MachinePrecision], If[LessEqual[d, 60000000000000.0], N[(N[(t$95$0 * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < 2.2499999999999999e-278

   1. Initial program 66.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      15. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      17. lower-*.f64 10.0

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 10.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   6. Taylor expanded in h around -inf

      \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d \]
   7. Step-by-step derivation

   8. Applied rewrites 42.8%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if 2.2499999999999999e-278 < d < 6e13

   1. Initial program 61.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 65.5

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{8} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{1}{8} \cdot {D}^{2}\right)} \cdot \frac{{M}^{2}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{1}{8} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{{M}^{2}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{1}{8} \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{{M}^{2}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{{M}^{2}}{\color{blue}{d \cdot d}}\right) \cdot \frac{h}{\ell}\right) \]

      8. associate-/r* N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{{M}^{2}}{d}}{d}}\right) \cdot \frac{h}{\ell}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{\color{blue}{M \cdot M}}{d}}{d}\right) \cdot \frac{h}{\ell}\right) \]

      11. associate-/l* N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{M \cdot \frac{M}{d}}}{d}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(\frac{1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{\color{blue}{M \cdot \frac{M}{d}}}{d}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-/.f64 52.7

          \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \color{blue}{\frac{M}{d}}}{d}\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 52.7%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right)} \cdot \frac{h}{\ell}\right) \]

   if 6e13 < d

   1. Initial program 82.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      15. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      17. lower-*.f64 56.1

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.5%

      \[\leadsto \frac{\sqrt{{\ell}^{-1}}}{\sqrt{h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 72.6%

      \[\leadsto \frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.25 \cdot 10^{-278}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;d \leq 60000000000000:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\left(0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot \frac{M}{d}}{d}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 47.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 9.5 \cdot 10^{-250}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d 9.5e-250)
   (* (- (sqrt (/ 1.0 (* l h)))) d)
   (* (/ (/ 1.0 (sqrt l)) (sqrt h)) d)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 9.5e-250) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = ((1.0 / sqrt(l)) / sqrt(h)) * d;
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 9.5d-250) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else
        tmp = ((1.0d0 / sqrt(l)) / sqrt(h)) * d
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 9.5e-250) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = ((1.0 / Math.sqrt(l)) / Math.sqrt(h)) * d;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= 9.5e-250:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	else:
		tmp = ((1.0 / math.sqrt(l)) / math.sqrt(h)) * d
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= 9.5e-250)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	else
		tmp = Float64(Float64(Float64(1.0 / sqrt(l)) / sqrt(h)) * d);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= 9.5e-250)
		tmp = -sqrt((1.0 / (l * h))) * d;
	else
		tmp = ((1.0 / sqrt(l)) / sqrt(h)) * d;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, 9.5e-250], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], N[(N[(N[(1.0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 9.5000000000000002e-250

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      15. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      17. lower-*.f64 9.7

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 9.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   6. Taylor expanded in h around -inf

      \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d \]
   7. Step-by-step derivation

   8. Applied rewrites 42.1%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if 9.5000000000000002e-250 < d

   1. Initial program 75.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      15. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      17. lower-*.f64 46.9

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 46.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.4%

      \[\leadsto \frac{\sqrt{{\ell}^{-1}}}{\sqrt{h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 58.5%

      \[\leadsto \frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}} \cdot d \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 9.5 \cdot 10^{-250}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{h}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 47.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 9.5 \cdot 10^{-250}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d 9.5e-250)
   (* (- (sqrt (/ 1.0 (* l h)))) d)
   (/ d (* (sqrt l) (sqrt h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 9.5e-250) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 9.5d-250) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 9.5e-250) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= 9.5e-250:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= 9.5e-250)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= 9.5e-250)
		tmp = -sqrt((1.0 / (l * h))) * d;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, 9.5e-250], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 9.5000000000000002e-250

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      15. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      17. lower-*.f64 9.7

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 9.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   6. Taylor expanded in h around -inf

      \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d \]
   7. Step-by-step derivation

   8. Applied rewrites 42.1%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if 9.5000000000000002e-250 < d

   1. Initial program 75.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      15. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      17. lower-*.f64 46.9

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 46.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 46.9%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 46.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 58.4%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 9.5 \cdot 10^{-250}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 43.3% accurate, 10.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{-266}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -4.2e-266) (* (- (sqrt (/ 1.0 (* l h)))) d) (/ d (sqrt (* l h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -4.2e-266) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-4.2d-266)) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -4.2e-266) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -4.2e-266:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -4.2e-266)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -4.2e-266)
		tmp = -sqrt((1.0 / (l * h))) * d;
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -4.2e-266], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.19999999999999994e-266

   1. Initial program 67.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      15. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      17. lower-*.f64 7.3

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 7.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   6. Taylor expanded in h around -inf

      \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d \]
   7. Step-by-step derivation

   8. Applied rewrites 46.5%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if -4.19999999999999994e-266 < l

   1. Initial program 70.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      15. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      17. lower-*.f64 41.9

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 41.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.9%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 41.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{-266}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 27.0% accurate, 12.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (* (sqrt (/ 1.0 (* l h))) d))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return sqrt((1.0 / (l * h))) * d;
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = sqrt((1.0d0 / (l * h))) * d
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return Math.sqrt((1.0 / (l * h))) * d;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return math.sqrt((1.0 / (l * h))) * d

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(sqrt(Float64(1.0 / Float64(l * h))) * d)
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = sqrt((1.0 / (l * h))) * d;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 69.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

   7. rem-square-sqrt N/A

      \[\leadsto \mathsf{neg}\left(\left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \]

   11. remove-double-neg N/A

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   12. *-commutative N/A

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   13. lower-*.f64 N/A

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   14. lower-sqrt.f64 N/A

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   15. lower-/.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   16. *-commutative N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   17. lower-*.f64 25.1

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

5. Applied rewrites 25.1%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

6. Final simplification 25.1%

   \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
7. Add Preprocessing

Alternative 21: 27.0% accurate, 15.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (/ d (sqrt (* l h))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d / sqrt((l * h));
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d / sqrt((l * h))
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d / Math.sqrt((l * h));
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d / math.sqrt((l * h))

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d / sqrt((l * h));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

   7. rem-square-sqrt N/A

      \[\leadsto \mathsf{neg}\left(\left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}\right) \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}\right) \]

   11. remove-double-neg N/A

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   12. *-commutative N/A

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   13. lower-*.f64 N/A

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   14. lower-sqrt.f64 N/A

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   15. lower-/.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   16. *-commutative N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   17. lower-*.f64 25.1

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

5. Applied rewrites 25.1%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
6. Step-by-step derivation

7. Applied rewrites 24.8%

   \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
8. Step-by-step derivation

9. Applied rewrites 24.8%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

10. Final simplification 24.8%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025017 
(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)))))