Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.8% → 76.1%

Time: 9.6s

Alternatives: 19

Speedup: 1.7×

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: 66.8% 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: 76.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D}{d + d}\\ \mathbf{if}\;d \leq 2.45 \cdot 10^{-306}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right) \cdot \left(1 - \frac{\left(\left(t\_0 \cdot t\_0\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* M_m (/ D (+ d d)))))
   (if (<= d 2.45e-306)
     (*
      (* d (- (sqrt (/ 1.0 (* l h)))))
      (- 1.0 (/ (* (* (* t_0 t_0) 0.5) h) l)))
     (*
      (* (/ (sqrt d) (sqrt h)) (pow (/ d l) (/ 1.0 2.0)))
      (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = M_m * (D / (d + d));
	double tmp;
	if (d <= 2.45e-306) {
		tmp = (d * -sqrt((1.0 / (l * h)))) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: d, h, l, M_m, and D 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_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_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m_m * (d_1 / (d + d))
    if (d <= 2.45d-306) then
        tmp = (d * -sqrt((1.0d0 / (l * h)))) * (1.0d0 - ((((t_0 * t_0) * 0.5d0) * h) / l))
    else
        tmp = ((sqrt(d) / sqrt(h)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = M_m * (D / (d + d));
	double tmp;
	if (d <= 2.45e-306) {
		tmp = (d * -Math.sqrt((1.0 / (l * h)))) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = M_m * (D / (d + d))
	tmp = 0
	if d <= 2.45e-306:
		tmp = (d * -math.sqrt((1.0 / (l * h)))) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l))
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(M_m * Float64(D / Float64(d + d)))
	tmp = 0.0
	if (d <= 2.45e-306)
		tmp = Float64(Float64(d * Float64(-sqrt(Float64(1.0 / Float64(l * h))))) * Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) * 0.5) * h) / l)));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = M_m * (D / (d + d));
	tmp = 0.0;
	if (d <= 2.45e-306)
		tmp = (d * -sqrt((1.0 / (l * h)))) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	else
		tmp = ((sqrt(d) / sqrt(h)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(M$95$m * N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 2.45e-306], N[(N[(d * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $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] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 2.45000000000000012e-306

   1. Initial program 66.2%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 67.7%

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

   4. Taylor expanded in h around -inf

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

      1. metadata-eval N/A

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

      2. pow1/2 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. sqrt-pow2 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 74.0%

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

   if 2.45000000000000012e-306 < d

   1. Initial program 67.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 76.8

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

   3. Applied rewrites 76.8%

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

Alternative 2: 75.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D}{d + d}\\ t_1 := 1 - \frac{\left(\left(t\_0 \cdot t\_0\right) \cdot 0.5\right) \cdot h}{\ell}\\ \mathbf{if}\;d \leq 2.45 \cdot 10^{-306}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{h}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* M_m (/ D (+ d d))))
        (t_1 (- 1.0 (/ (* (* (* t_0 t_0) 0.5) h) l))))
   (if (<= d 2.45e-306)
     (* (* d (- (sqrt (/ 1.0 (* l h))))) t_1)
     (* (/ (* (sqrt d) (sqrt (/ d l))) (sqrt h)) t_1))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = M_m * (D / (d + d));
	double t_1 = 1.0 - ((((t_0 * t_0) * 0.5) * h) / l);
	double tmp;
	if (d <= 2.45e-306) {
		tmp = (d * -sqrt((1.0 / (l * h)))) * t_1;
	} else {
		tmp = ((sqrt(d) * sqrt((d / l))) / sqrt(h)) * t_1;
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: d, h, l, M_m, and D 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_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_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = m_m * (d_1 / (d + d))
    t_1 = 1.0d0 - ((((t_0 * t_0) * 0.5d0) * h) / l)
    if (d <= 2.45d-306) then
        tmp = (d * -sqrt((1.0d0 / (l * h)))) * t_1
    else
        tmp = ((sqrt(d) * sqrt((d / l))) / sqrt(h)) * t_1
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = M_m * (D / (d + d));
	double t_1 = 1.0 - ((((t_0 * t_0) * 0.5) * h) / l);
	double tmp;
	if (d <= 2.45e-306) {
		tmp = (d * -Math.sqrt((1.0 / (l * h)))) * t_1;
	} else {
		tmp = ((Math.sqrt(d) * Math.sqrt((d / l))) / Math.sqrt(h)) * t_1;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = M_m * (D / (d + d))
	t_1 = 1.0 - ((((t_0 * t_0) * 0.5) * h) / l)
	tmp = 0
	if d <= 2.45e-306:
		tmp = (d * -math.sqrt((1.0 / (l * h)))) * t_1
	else:
		tmp = ((math.sqrt(d) * math.sqrt((d / l))) / math.sqrt(h)) * t_1
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(M_m * Float64(D / Float64(d + d)))
	t_1 = Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) * 0.5) * h) / l))
	tmp = 0.0
	if (d <= 2.45e-306)
		tmp = Float64(Float64(d * Float64(-sqrt(Float64(1.0 / Float64(l * h))))) * t_1);
	else
		tmp = Float64(Float64(Float64(sqrt(d) * sqrt(Float64(d / l))) / sqrt(h)) * t_1);
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = M_m * (D / (d + d));
	t_1 = 1.0 - ((((t_0 * t_0) * 0.5) * h) / l);
	tmp = 0.0;
	if (d <= 2.45e-306)
		tmp = (d * -sqrt((1.0 / (l * h)))) * t_1;
	else
		tmp = ((sqrt(d) * sqrt((d / l))) / sqrt(h)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(M$95$m * N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 2.45e-306], N[(N[(d * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if d < 2.45000000000000012e-306

   1. Initial program 66.2%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 67.7%

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

   4. Taylor expanded in h around -inf

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

      1. metadata-eval N/A

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

      2. pow1/2 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. sqrt-pow2 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 74.0%

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

   if 2.45000000000000012e-306 < d

   1. Initial program 67.4%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 68.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. pow1/2 N/A

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

      7. sqrt-div N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift-pow.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. pow1/2 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-sqrt.f64 78.2

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

   5. Applied rewrites 78.2%

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

Alternative 3: 75.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D}{d + d}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot 0.5\\ t_2 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq 1.2 \cdot 10^{-282}:\\ \;\;\;\;\left(d \cdot \left(-t\_2\right)\right) \cdot \left(1 - \frac{t\_1 \cdot h}{\ell}\right)\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-155}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(D \cdot \left(D \cdot \left(\frac{M\_m}{d} \cdot M\_m\right)\right)\right), -0.125, t\_2 \cdot d\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - t\_1 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* M_m (/ D (+ d d))))
        (t_1 (* (* t_0 t_0) 0.5))
        (t_2 (sqrt (/ 1.0 (* l h)))))
   (if (<= d 1.2e-282)
     (* (* d (- t_2)) (- 1.0 (/ (* t_1 h) l)))
     (if (<= d 1.85e-155)
       (fma
        (* (sqrt (/ h (* (* l l) l))) (* D (* D (* (/ M_m d) M_m))))
        -0.125
        (* t_2 d))
       (* (sqrt (/ d h)) (* (/ (sqrt d) (sqrt l)) (- 1.0 (* t_1 (/ h l)))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = M_m * (D / (d + d));
	double t_1 = (t_0 * t_0) * 0.5;
	double t_2 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= 1.2e-282) {
		tmp = (d * -t_2) * (1.0 - ((t_1 * h) / l));
	} else if (d <= 1.85e-155) {
		tmp = fma((sqrt((h / ((l * l) * l))) * (D * (D * ((M_m / d) * M_m)))), -0.125, (t_2 * d));
	} else {
		tmp = sqrt((d / h)) * ((sqrt(d) / sqrt(l)) * (1.0 - (t_1 * (h / l))));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(M_m * Float64(D / Float64(d + d)))
	t_1 = Float64(Float64(t_0 * t_0) * 0.5)
	t_2 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= 1.2e-282)
		tmp = Float64(Float64(d * Float64(-t_2)) * Float64(1.0 - Float64(Float64(t_1 * h) / l)));
	elseif (d <= 1.85e-155)
		tmp = fma(Float64(sqrt(Float64(h / Float64(Float64(l * l) * l))) * Float64(D * Float64(D * Float64(Float64(M_m / d) * M_m)))), -0.125, Float64(t_2 * d));
	else
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 - Float64(t_1 * Float64(h / l)))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(M$95$m * N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, 1.2e-282], N[(N[(d * (-t$95$2)), $MachinePrecision] * N[(1.0 - N[(N[(t$95$1 * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.85e-155], N[(N[(N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * N[(D * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(t$95$2 * d), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$1 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < 1.19999999999999998e-282

   1. Initial program 64.9%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 66.4%

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

   4. Taylor expanded in h around -inf

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

      1. metadata-eval N/A

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

      2. pow1/2 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. sqrt-pow2 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 71.7%

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

   if 1.19999999999999998e-282 < d < 1.85e-155

   1. Initial program 47.8%

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

   2. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 46.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), -0.125, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 51.3

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

   6. Applied rewrites 51.3%

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right), -0.125, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 53.4

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

   8. Applied rewrites 53.4%

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

   if 1.85e-155 < d

   1. Initial program 74.5%

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

   3. Applied rewrites 74.2%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 77.9

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

   5. Applied rewrites 77.9%

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

Alternative 4: 72.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D}{d + d}\\ t_1 := 1 - \frac{\left(\left(t\_0 \cdot t\_0\right) \cdot 0.5\right) \cdot h}{\ell}\\ t_2 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(d \cdot \left(-t\_2\right)\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-86}:\\ \;\;\;\;\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(D \cdot \left(D \cdot \left(\frac{M\_m}{d} \cdot M\_m\right)\right)\right), -0.125, t\_2 \cdot d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* M_m (/ D (+ d d))))
        (t_1 (- 1.0 (/ (* (* (* t_0 t_0) 0.5) h) l)))
        (t_2 (sqrt (/ 1.0 (* l h)))))
   (if (<= l -2e-311)
     (* (* d (- t_2)) t_1)
     (if (<= l 1.1e-86)
       (* (* (sqrt (/ 1.0 (* h l))) d) t_1)
       (if (<= l 2.15e+102)
         (fma
          (* (sqrt (/ h (* (* l l) l))) (* D (* D (* (/ M_m d) M_m))))
          -0.125
          (* t_2 d))
         (/ (* 1.0 d) (* (sqrt h) (sqrt l))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = M_m * (D / (d + d));
	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 <= -2e-311) {
		tmp = (d * -t_2) * t_1;
	} else if (l <= 1.1e-86) {
		tmp = (sqrt((1.0 / (h * l))) * d) * t_1;
	} else if (l <= 2.15e+102) {
		tmp = fma((sqrt((h / ((l * l) * l))) * (D * (D * ((M_m / d) * M_m)))), -0.125, (t_2 * d));
	} else {
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(M_m * Float64(D / Float64(d + d)))
	t_1 = Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) * 0.5) * h) / l))
	t_2 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (l <= -2e-311)
		tmp = Float64(Float64(d * Float64(-t_2)) * t_1);
	elseif (l <= 1.1e-86)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(h * l))) * d) * t_1);
	elseif (l <= 2.15e+102)
		tmp = fma(Float64(sqrt(Float64(h / Float64(Float64(l * l) * l))) * Float64(D * Float64(D * Float64(Float64(M_m / d) * M_m)))), -0.125, Float64(t_2 * d));
	else
		tmp = Float64(Float64(1.0 * d) / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(M$95$m * N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2e-311], N[(N[(d * (-t$95$2)), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 1.1e-86], N[(N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 2.15e+102], N[(N[(N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * N[(D * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(t$95$2 * d), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.9999999999999e-311

   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. 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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 67.9%

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

   4. Taylor expanded in h around -inf

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

      1. metadata-eval N/A

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

      2. pow1/2 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. sqrt-pow2 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 74.2%

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

   if -1.9999999999999e-311 < l < 1.1000000000000001e-86

   1. Initial program 73.0%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. pow1/2 N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sqrt.f64 85.7

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 85.7

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

   8. Applied rewrites 85.7%

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

   if 1.1000000000000001e-86 < l < 2.15e102

   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. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 66.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), -0.125, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.0

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

   6. Applied rewrites 69.0%

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right), -0.125, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 77.8

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

   8. Applied rewrites 77.8%

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

   if 2.15e102 < l

   1. Initial program 54.5%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.3

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

   4. Applied rewrites 51.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 51.3

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

   6. Applied rewrites 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-sqrt.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 51.3

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

   8. Applied rewrites 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 64.7

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

   10. Applied rewrites 64.7%

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

Alternative 5: 71.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D}{d + d}\\ t_1 := 1 - \frac{\left(\left(t\_0 \cdot t\_0\right) \cdot 0.5\right) \cdot h}{\ell}\\ t_2 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq 1.2 \cdot 10^{-282}:\\ \;\;\;\;\left(d \cdot \left(-t\_2\right)\right) \cdot t\_1\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-155}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(D \cdot \left(D \cdot \left(\frac{M\_m}{d} \cdot M\_m\right)\right)\right), -0.125, t\_2 \cdot d\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* M_m (/ D (+ d d))))
        (t_1 (- 1.0 (/ (* (* (* t_0 t_0) 0.5) h) l)))
        (t_2 (sqrt (/ 1.0 (* l h)))))
   (if (<= d 1.2e-282)
     (* (* d (- t_2)) t_1)
     (if (<= d 2.2e-155)
       (fma
        (* (sqrt (/ h (* (* l l) l))) (* D (* D (* (/ M_m d) M_m))))
        -0.125
        (* t_2 d))
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) t_1)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = M_m * (D / (d + d));
	double t_1 = 1.0 - ((((t_0 * t_0) * 0.5) * h) / l);
	double t_2 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= 1.2e-282) {
		tmp = (d * -t_2) * t_1;
	} else if (d <= 2.2e-155) {
		tmp = fma((sqrt((h / ((l * l) * l))) * (D * (D * ((M_m / d) * M_m)))), -0.125, (t_2 * d));
	} else {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * t_1;
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(M_m * Float64(D / Float64(d + d)))
	t_1 = Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) * 0.5) * h) / l))
	t_2 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= 1.2e-282)
		tmp = Float64(Float64(d * Float64(-t_2)) * t_1);
	elseif (d <= 2.2e-155)
		tmp = fma(Float64(sqrt(Float64(h / Float64(Float64(l * l) * l))) * Float64(D * Float64(D * Float64(Float64(M_m / d) * M_m)))), -0.125, Float64(t_2 * d));
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * t_1);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(M$95$m * N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, 1.2e-282], N[(N[(d * (-t$95$2)), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 2.2e-155], N[(N[(N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * N[(D * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(t$95$2 * d), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < 1.19999999999999998e-282

   1. Initial program 64.9%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 66.4%

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

   4. Taylor expanded in h around -inf

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

      1. metadata-eval N/A

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

      2. pow1/2 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. sqrt-pow2 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 71.7%

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

   if 1.19999999999999998e-282 < d < 2.1999999999999999e-155

   1. Initial program 47.8%

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

   2. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 46.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), -0.125, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 51.3

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

   6. Applied rewrites 51.3%

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right), -0.125, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 53.4

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

   8. Applied rewrites 53.4%

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

   if 2.1999999999999999e-155 < d

   1. Initial program 74.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 76.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 76.2

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

   5. Applied rewrites 76.2%

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

Alternative 6: 70.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D}{d + d}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot 0.5\\ t_2 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq 1.2 \cdot 10^{-282}:\\ \;\;\;\;\left(d \cdot \left(-t\_2\right)\right) \cdot \left(1 - \frac{t\_1 \cdot h}{\ell}\right)\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-155}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(D \cdot \left(D \cdot \left(\frac{M\_m}{d} \cdot M\_m\right)\right)\right), -0.125, t\_2 \cdot d\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - t\_1 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* M_m (/ D (+ d d))))
        (t_1 (* (* t_0 t_0) 0.5))
        (t_2 (sqrt (/ 1.0 (* l h)))))
   (if (<= d 1.2e-282)
     (* (* d (- t_2)) (- 1.0 (/ (* t_1 h) l)))
     (if (<= d 2.2e-155)
       (fma
        (* (sqrt (/ h (* (* l l) l))) (* D (* D (* (/ M_m d) M_m))))
        -0.125
        (* t_2 d))
       (* (sqrt (/ d h)) (* (sqrt (/ d l)) (- 1.0 (* t_1 (/ h l)))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = M_m * (D / (d + d));
	double t_1 = (t_0 * t_0) * 0.5;
	double t_2 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= 1.2e-282) {
		tmp = (d * -t_2) * (1.0 - ((t_1 * h) / l));
	} else if (d <= 2.2e-155) {
		tmp = fma((sqrt((h / ((l * l) * l))) * (D * (D * ((M_m / d) * M_m)))), -0.125, (t_2 * d));
	} else {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (t_1 * (h / l))));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(M_m * Float64(D / Float64(d + d)))
	t_1 = Float64(Float64(t_0 * t_0) * 0.5)
	t_2 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= 1.2e-282)
		tmp = Float64(Float64(d * Float64(-t_2)) * Float64(1.0 - Float64(Float64(t_1 * h) / l)));
	elseif (d <= 2.2e-155)
		tmp = fma(Float64(sqrt(Float64(h / Float64(Float64(l * l) * l))) * Float64(D * Float64(D * Float64(Float64(M_m / d) * M_m)))), -0.125, Float64(t_2 * d));
	else
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(t_1 * Float64(h / l)))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(M$95$m * N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, 1.2e-282], N[(N[(d * (-t$95$2)), $MachinePrecision] * N[(1.0 - N[(N[(t$95$1 * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.2e-155], N[(N[(N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * N[(D * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(t$95$2 * d), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(t$95$1 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < 1.19999999999999998e-282

   1. Initial program 64.9%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 66.4%

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

   4. Taylor expanded in h around -inf

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

      1. metadata-eval N/A

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

      2. pow1/2 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. sqrt-pow2 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 71.7%

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

   if 1.19999999999999998e-282 < d < 2.1999999999999999e-155

   1. Initial program 47.8%

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

   2. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 46.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), -0.125, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 51.3

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

   6. Applied rewrites 51.3%

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right), -0.125, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 53.4

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

   8. Applied rewrites 53.4%

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

   if 2.1999999999999999e-155 < d

   1. Initial program 74.5%

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

   3. Applied rewrites 74.2%

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

Alternative 7: 68.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ t_1 := M\_m \cdot \frac{D}{d + d}\\ t_2 := \left(M\_m \cdot \left(D \cdot M\_m\right)\right) \cdot D\\ t_3 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{t\_2}{d \cdot d} \cdot 0.125\right) \cdot h}{\ell}\right)\\ \mathbf{if}\;d \leq -9.8 \cdot 10^{+91}:\\ \;\;\;\;\left(-t\_0\right) \cdot d\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-144}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t\_2 \cdot -1}{d} \cdot \sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}}\right) \cdot -0.125\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-155}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(D \cdot \left(D \cdot \left(\frac{M\_m}{d} \cdot M\_m\right)\right)\right), -0.125, t\_0 \cdot d\right)\\ \mathbf{elif}\;d \leq 10^{+136}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(t\_1 \cdot t\_1\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h))))
        (t_1 (* M_m (/ D (+ d d))))
        (t_2 (* (* M_m (* D M_m)) D))
        (t_3
         (*
          (* (sqrt (/ d l)) (sqrt (/ d h)))
          (- 1.0 (/ (* (* (/ t_2 (* d d)) 0.125) h) l)))))
   (if (<= d -9.8e+91)
     (* (- t_0) d)
     (if (<= d -1.9e-144)
       t_3
       (if (<= d -5e-310)
         (* (* (/ (* t_2 -1.0) d) (sqrt (/ (/ h (* l l)) l))) -0.125)
         (if (<= d 2.5e-155)
           (fma
            (* (sqrt (/ h (* (* l l) l))) (* D (* D (* (/ M_m d) M_m))))
            -0.125
            (* t_0 d))
           (if (<= d 1e+136)
             t_3
             (*
              (* (sqrt (/ 1.0 (* h l))) d)
              (- 1.0 (/ (* (* (* t_1 t_1) 0.5) h) l))))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt((1.0 / (l * h)));
	double t_1 = M_m * (D / (d + d));
	double t_2 = (M_m * (D * M_m)) * D;
	double t_3 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((t_2 / (d * d)) * 0.125) * h) / l));
	double tmp;
	if (d <= -9.8e+91) {
		tmp = -t_0 * d;
	} else if (d <= -1.9e-144) {
		tmp = t_3;
	} else if (d <= -5e-310) {
		tmp = (((t_2 * -1.0) / d) * sqrt(((h / (l * l)) / l))) * -0.125;
	} else if (d <= 2.5e-155) {
		tmp = fma((sqrt((h / ((l * l) * l))) * (D * (D * ((M_m / d) * M_m)))), -0.125, (t_0 * d));
	} else if (d <= 1e+136) {
		tmp = t_3;
	} else {
		tmp = (sqrt((1.0 / (h * l))) * d) * (1.0 - ((((t_1 * t_1) * 0.5) * h) / l));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	t_1 = Float64(M_m * Float64(D / Float64(d + d)))
	t_2 = Float64(Float64(M_m * Float64(D * M_m)) * D)
	t_3 = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(t_2 / Float64(d * d)) * 0.125) * h) / l)))
	tmp = 0.0
	if (d <= -9.8e+91)
		tmp = Float64(Float64(-t_0) * d);
	elseif (d <= -1.9e-144)
		tmp = t_3;
	elseif (d <= -5e-310)
		tmp = Float64(Float64(Float64(Float64(t_2 * -1.0) / d) * sqrt(Float64(Float64(h / Float64(l * l)) / l))) * -0.125);
	elseif (d <= 2.5e-155)
		tmp = fma(Float64(sqrt(Float64(h / Float64(Float64(l * l) * l))) * Float64(D * Float64(D * Float64(Float64(M_m / d) * M_m)))), -0.125, Float64(t_0 * d));
	elseif (d <= 1e+136)
		tmp = t_3;
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(h * l))) * d) * Float64(1.0 - Float64(Float64(Float64(Float64(t_1 * t_1) * 0.5) * h) / l)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(M$95$m * N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(M$95$m * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$2 / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -9.8e+91], N[((-t$95$0) * d), $MachinePrecision], If[LessEqual[d, -1.9e-144], t$95$3, If[LessEqual[d, -5e-310], N[(N[(N[(N[(t$95$2 * -1.0), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(N[(h / N[(l * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], If[LessEqual[d, 2.5e-155], N[(N[(N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * N[(D * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(t$95$0 * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1e+136], t$95$3, N[(N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -9.8000000000000006e91

   1. Initial program 74.7%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 0.4%

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

   5. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 66.9

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

   7. Applied rewrites 66.9%

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

   if -9.8000000000000006e91 < d < -1.89999999999999996e-144 or 2.4999999999999999e-155 < d < 1.00000000000000006e136

   1. Initial program 74.7%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 76.5

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*l* N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 74.6

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

   8. Applied rewrites 74.6%

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

   if -1.89999999999999996e-144 < d < -4.999999999999985e-310

   1. Initial program 44.1%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 42.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 42.9

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in h around -inf

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

   8. Applied rewrites 47.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 49.7

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

   10. Applied rewrites 49.7%

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

   if -4.999999999999985e-310 < d < 2.4999999999999999e-155

   1. Initial program 45.1%

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

   2. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 46.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), -0.125, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 50.9

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

   6. Applied rewrites 50.9%

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right), -0.125, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 53.3

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

   8. Applied rewrites 53.3%

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

   if 1.00000000000000006e136 < d

   1. Initial program 73.1%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 75.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 75.5

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. pow1/2 N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sqrt.f64 85.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 85.4

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

   8. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 8: 65.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := -\left(\left(0.125 \cdot \frac{\left(M\_m \cdot \left(D \cdot M\_m\right)\right) \cdot D}{d}\right) \cdot \sqrt{\frac{1}{\left(\ell \cdot \ell\right) \cdot \left(h \cdot \ell\right)}}\right) \cdot h\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1
         (-
          (*
           (*
            (* 0.125 (/ (* (* M_m (* D M_m)) D) d))
            (sqrt (/ 1.0 (* (* l l) (* h l)))))
           h))))
   (if (<= t_0 -4e-31)
     t_1
     (if (<= t_0 INFINITY) (* (sqrt (/ d h)) (sqrt (/ d l))) t_1))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = -(((0.125 * (((M_m * (D * M_m)) * D) / d)) * sqrt((1.0 / ((l * l) * (h * l))))) * h);
	double tmp;
	if (t_0 <= -4e-31) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (Math.pow((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) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = -(((0.125 * (((M_m * (D * M_m)) * D) / d)) * Math.sqrt((1.0 / ((l * l) * (h * l))))) * h);
	double tmp;
	if (t_0 <= -4e-31) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = (math.pow((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) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = -(((0.125 * (((M_m * (D * M_m)) * D) / d)) * math.sqrt((1.0 / ((l * l) * (h * l))))) * h)
	tmp = 0
	if t_0 <= -4e-31:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = t_1
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64((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) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(-Float64(Float64(Float64(0.125 * Float64(Float64(Float64(M_m * Float64(D * M_m)) * D) / d)) * sqrt(Float64(1.0 / Float64(Float64(l * l) * Float64(h * l))))) * h))
	tmp = 0.0
	if (t_0 <= -4e-31)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = -(((0.125 * (((M_m * (D * M_m)) * D) / d)) * sqrt((1.0 / ((l * l) * (h * l))))) * h);
	tmp = 0.0;
	if (t_0 <= -4e-31)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(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), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(N[(N[(0.125 * N[(N[(N[(M$95$m * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(l * l), $MachinePrecision] * N[(h * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision])}, If[LessEqual[t$95$0, -4e-31], t$95$1, If[LessEqual[t$95$0, Infinity], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4e-31 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 55.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 31.5

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

   3. Applied rewrites 31.5%

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

   4. Taylor expanded in h around -inf

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

   6. Applied rewrites 31.1%

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

   7. Taylor expanded in d around 0

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

   9. Applied rewrites 56.2%

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

   if -4e-31 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 78.7%

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

   3. Applied rewrites 78.0%

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

   4. Taylor expanded in d around inf

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 75.7

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

   6. Applied rewrites 75.7%

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

Alternative 9: 51.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left(M\_m \cdot \left(D \cdot M\_m\right)\right) \cdot D\\ \mathbf{if}\;\ell \leq -1.18 \cdot 10^{-17}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;\ell \leq -4.8 \cdot 10^{-301}:\\ \;\;\;\;\left(\frac{t\_0 \cdot -1}{d} \cdot \sqrt{\frac{\frac{h}{\ell \cdot \ell}}{\ell}}\right) \cdot -0.125\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+48}:\\ \;\;\;\;\left(-0.125 \cdot \frac{t\_0}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (* M_m (* D M_m)) D)))
   (if (<= l -1.18e-17)
     (* (- (sqrt (/ 1.0 (* l h)))) d)
     (if (<= l -4.8e-301)
       (* (* (/ (* t_0 -1.0) d) (sqrt (/ (/ h (* l l)) l))) -0.125)
       (if (<= l 7e+48)
         (* (* -0.125 (/ t_0 d)) (sqrt (/ h (* (* l l) l))))
         (/ (* 1.0 d) (* (sqrt h) (sqrt l))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (M_m * (D * M_m)) * D;
	double tmp;
	if (l <= -1.18e-17) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else if (l <= -4.8e-301) {
		tmp = (((t_0 * -1.0) / d) * sqrt(((h / (l * l)) / l))) * -0.125;
	} else if (l <= 7e+48) {
		tmp = (-0.125 * (t_0 / d)) * sqrt((h / ((l * l) * l)));
	} else {
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: d, h, l, M_m, and D 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_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_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m_m * (d_1 * m_m)) * d_1
    if (l <= (-1.18d-17)) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else if (l <= (-4.8d-301)) then
        tmp = (((t_0 * (-1.0d0)) / d) * sqrt(((h / (l * l)) / l))) * (-0.125d0)
    else if (l <= 7d+48) then
        tmp = ((-0.125d0) * (t_0 / d)) * sqrt((h / ((l * l) * l)))
    else
        tmp = (1.0d0 * d) / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (M_m * (D * M_m)) * D;
	double tmp;
	if (l <= -1.18e-17) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else if (l <= -4.8e-301) {
		tmp = (((t_0 * -1.0) / d) * Math.sqrt(((h / (l * l)) / l))) * -0.125;
	} else if (l <= 7e+48) {
		tmp = (-0.125 * (t_0 / d)) * Math.sqrt((h / ((l * l) * l)));
	} else {
		tmp = (1.0 * d) / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = (M_m * (D * M_m)) * D
	tmp = 0
	if l <= -1.18e-17:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	elif l <= -4.8e-301:
		tmp = (((t_0 * -1.0) / d) * math.sqrt(((h / (l * l)) / l))) * -0.125
	elif l <= 7e+48:
		tmp = (-0.125 * (t_0 / d)) * math.sqrt((h / ((l * l) * l)))
	else:
		tmp = (1.0 * d) / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64(M_m * Float64(D * M_m)) * D)
	tmp = 0.0
	if (l <= -1.18e-17)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	elseif (l <= -4.8e-301)
		tmp = Float64(Float64(Float64(Float64(t_0 * -1.0) / d) * sqrt(Float64(Float64(h / Float64(l * l)) / l))) * -0.125);
	elseif (l <= 7e+48)
		tmp = Float64(Float64(-0.125 * Float64(t_0 / d)) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	else
		tmp = Float64(Float64(1.0 * d) / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (M_m * (D * M_m)) * D;
	tmp = 0.0;
	if (l <= -1.18e-17)
		tmp = -sqrt((1.0 / (l * h))) * d;
	elseif (l <= -4.8e-301)
		tmp = (((t_0 * -1.0) / d) * sqrt(((h / (l * l)) / l))) * -0.125;
	elseif (l <= 7e+48)
		tmp = (-0.125 * (t_0 / d)) * sqrt((h / ((l * l) * l)));
	else
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(M$95$m * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]}, If[LessEqual[l, -1.18e-17], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], If[LessEqual[l, -4.8e-301], N[(N[(N[(N[(t$95$0 * -1.0), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(N[(h / N[(l * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], If[LessEqual[l, 7e+48], N[(N[(-0.125 * N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.18000000000000004e-17

   1. Initial program 60.8%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 49.6

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

   7. Applied rewrites 49.6%

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

   if -1.18000000000000004e-17 < l < -4.79999999999999982e-301

   1. Initial program 72.1%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 76.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in h around -inf

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

   8. Applied rewrites 46.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 46.8

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

   10. Applied rewrites 46.8%

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

   if -4.79999999999999982e-301 < l < 6.9999999999999995e48

   1. Initial program 74.3%

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

   2. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 43.7%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 46.6%

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

   if 6.9999999999999995e48 < l

   1. Initial program 58.0%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 52.3

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

   4. Applied rewrites 52.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 52.2

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

   6. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-sqrt.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 52.3

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

   8. Applied rewrites 52.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 63.9

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

   10. Applied rewrites 63.9%

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

Alternative 10: 51.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left(M\_m \cdot \left(D \cdot M\_m\right)\right) \cdot D\\ t_1 := \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{if}\;\ell \leq -1.18 \cdot 10^{-17}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;\ell \leq -4.8 \cdot 10^{-301}:\\ \;\;\;\;\left(\frac{t\_0 \cdot -1}{d} \cdot t\_1\right) \cdot -0.125\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+48}:\\ \;\;\;\;\left(-0.125 \cdot \frac{t\_0}{d}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (* M_m (* D M_m)) D)) (t_1 (sqrt (/ h (* (* l l) l)))))
   (if (<= l -1.18e-17)
     (* (- (sqrt (/ 1.0 (* l h)))) d)
     (if (<= l -4.8e-301)
       (* (* (/ (* t_0 -1.0) d) t_1) -0.125)
       (if (<= l 7e+48)
         (* (* -0.125 (/ t_0 d)) t_1)
         (/ (* 1.0 d) (* (sqrt h) (sqrt l))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (M_m * (D * M_m)) * D;
	double t_1 = sqrt((h / ((l * l) * l)));
	double tmp;
	if (l <= -1.18e-17) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else if (l <= -4.8e-301) {
		tmp = (((t_0 * -1.0) / d) * t_1) * -0.125;
	} else if (l <= 7e+48) {
		tmp = (-0.125 * (t_0 / d)) * t_1;
	} else {
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: d, h, l, M_m, and D 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_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_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (m_m * (d_1 * m_m)) * d_1
    t_1 = sqrt((h / ((l * l) * l)))
    if (l <= (-1.18d-17)) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else if (l <= (-4.8d-301)) then
        tmp = (((t_0 * (-1.0d0)) / d) * t_1) * (-0.125d0)
    else if (l <= 7d+48) then
        tmp = ((-0.125d0) * (t_0 / d)) * t_1
    else
        tmp = (1.0d0 * d) / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (M_m * (D * M_m)) * D;
	double t_1 = Math.sqrt((h / ((l * l) * l)));
	double tmp;
	if (l <= -1.18e-17) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else if (l <= -4.8e-301) {
		tmp = (((t_0 * -1.0) / d) * t_1) * -0.125;
	} else if (l <= 7e+48) {
		tmp = (-0.125 * (t_0 / d)) * t_1;
	} else {
		tmp = (1.0 * d) / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = (M_m * (D * M_m)) * D
	t_1 = math.sqrt((h / ((l * l) * l)))
	tmp = 0
	if l <= -1.18e-17:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	elif l <= -4.8e-301:
		tmp = (((t_0 * -1.0) / d) * t_1) * -0.125
	elif l <= 7e+48:
		tmp = (-0.125 * (t_0 / d)) * t_1
	else:
		tmp = (1.0 * d) / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64(M_m * Float64(D * M_m)) * D)
	t_1 = sqrt(Float64(h / Float64(Float64(l * l) * l)))
	tmp = 0.0
	if (l <= -1.18e-17)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	elseif (l <= -4.8e-301)
		tmp = Float64(Float64(Float64(Float64(t_0 * -1.0) / d) * t_1) * -0.125);
	elseif (l <= 7e+48)
		tmp = Float64(Float64(-0.125 * Float64(t_0 / d)) * t_1);
	else
		tmp = Float64(Float64(1.0 * d) / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (M_m * (D * M_m)) * D;
	t_1 = sqrt((h / ((l * l) * l)));
	tmp = 0.0;
	if (l <= -1.18e-17)
		tmp = -sqrt((1.0 / (l * h))) * d;
	elseif (l <= -4.8e-301)
		tmp = (((t_0 * -1.0) / d) * t_1) * -0.125;
	elseif (l <= 7e+48)
		tmp = (-0.125 * (t_0 / d)) * t_1;
	else
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(M$95$m * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.18e-17], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], If[LessEqual[l, -4.8e-301], N[(N[(N[(N[(t$95$0 * -1.0), $MachinePrecision] / d), $MachinePrecision] * t$95$1), $MachinePrecision] * -0.125), $MachinePrecision], If[LessEqual[l, 7e+48], N[(N[(-0.125 * N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(1.0 * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.18000000000000004e-17

   1. Initial program 60.8%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 49.6

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

   7. Applied rewrites 49.6%

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

   if -1.18000000000000004e-17 < l < -4.79999999999999982e-301

   1. Initial program 72.1%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 76.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in h around -inf

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

   8. Applied rewrites 46.2%

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

   if -4.79999999999999982e-301 < l < 6.9999999999999995e48

   1. Initial program 74.3%

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

   2. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 43.7%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 46.6%

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

   if 6.9999999999999995e48 < l

   1. Initial program 58.0%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 52.3

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

   4. Applied rewrites 52.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 52.2

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

   6. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-sqrt.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 52.3

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

   8. Applied rewrites 52.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 63.9

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

   10. Applied rewrites 63.9%

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

Alternative 11: 50.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{if}\;\ell \leq -1.18 \cdot 10^{-17}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;\ell \leq -4.8 \cdot 10^{-301}:\\ \;\;\;\;\left(0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\_m\right) \cdot \frac{M\_m}{d}\right)\right) \cdot t\_0\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+48}:\\ \;\;\;\;\left(-0.125 \cdot \frac{\left(M\_m \cdot \left(D \cdot M\_m\right)\right) \cdot D}{d}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h (* (* l l) l)))))
   (if (<= l -1.18e-17)
     (* (- (sqrt (/ 1.0 (* l h)))) d)
     (if (<= l -4.8e-301)
       (* (* 0.125 (* (* (* D D) M_m) (/ M_m d))) t_0)
       (if (<= l 7e+48)
         (* (* -0.125 (/ (* (* M_m (* D M_m)) D) d)) t_0)
         (/ (* 1.0 d) (* (sqrt h) (sqrt l))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt((h / ((l * l) * l)));
	double tmp;
	if (l <= -1.18e-17) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else if (l <= -4.8e-301) {
		tmp = (0.125 * (((D * D) * M_m) * (M_m / d))) * t_0;
	} else if (l <= 7e+48) {
		tmp = (-0.125 * (((M_m * (D * M_m)) * D) / d)) * t_0;
	} else {
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: d, h, l, M_m, and D 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_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_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((h / ((l * l) * l)))
    if (l <= (-1.18d-17)) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else if (l <= (-4.8d-301)) then
        tmp = (0.125d0 * (((d_1 * d_1) * m_m) * (m_m / d))) * t_0
    else if (l <= 7d+48) then
        tmp = ((-0.125d0) * (((m_m * (d_1 * m_m)) * d_1) / d)) * t_0
    else
        tmp = (1.0d0 * d) / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.sqrt((h / ((l * l) * l)));
	double tmp;
	if (l <= -1.18e-17) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else if (l <= -4.8e-301) {
		tmp = (0.125 * (((D * D) * M_m) * (M_m / d))) * t_0;
	} else if (l <= 7e+48) {
		tmp = (-0.125 * (((M_m * (D * M_m)) * D) / d)) * t_0;
	} else {
		tmp = (1.0 * d) / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.sqrt((h / ((l * l) * l)))
	tmp = 0
	if l <= -1.18e-17:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	elif l <= -4.8e-301:
		tmp = (0.125 * (((D * D) * M_m) * (M_m / d))) * t_0
	elif l <= 7e+48:
		tmp = (-0.125 * (((M_m * (D * M_m)) * D) / d)) * t_0
	else:
		tmp = (1.0 * d) / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(h / Float64(Float64(l * l) * l)))
	tmp = 0.0
	if (l <= -1.18e-17)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	elseif (l <= -4.8e-301)
		tmp = Float64(Float64(0.125 * Float64(Float64(Float64(D * D) * M_m) * Float64(M_m / d))) * t_0);
	elseif (l <= 7e+48)
		tmp = Float64(Float64(-0.125 * Float64(Float64(Float64(M_m * Float64(D * M_m)) * D) / d)) * t_0);
	else
		tmp = Float64(Float64(1.0 * d) / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = sqrt((h / ((l * l) * l)));
	tmp = 0.0;
	if (l <= -1.18e-17)
		tmp = -sqrt((1.0 / (l * h))) * d;
	elseif (l <= -4.8e-301)
		tmp = (0.125 * (((D * D) * M_m) * (M_m / d))) * t_0;
	elseif (l <= 7e+48)
		tmp = (-0.125 * (((M_m * (D * M_m)) * D) / d)) * t_0;
	else
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.18e-17], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], If[LessEqual[l, -4.8e-301], N[(N[(0.125 * N[(N[(N[(D * D), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[l, 7e+48], N[(N[(-0.125 * N[(N[(N[(M$95$m * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(1.0 * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.18000000000000004e-17

   1. Initial program 60.8%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \cdot \color{blue}{d} \]

   4. Applied rewrites 1.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}, -0.125, \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d} \]

   5. Taylor expanded in l around -inf

      \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d \]
   6. Step-by-step derivation

      1. sqrt-pow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot d \]

      2. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{1}\right) \cdot d \]

      3. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right) \cdot d \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      5. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot d \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      7. *-commutative N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      9. lift-/.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      10. lift-*.f64 49.6

          \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   7. Applied rewrites 49.6%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if -1.18000000000000004e-17 < l < -4.79999999999999982e-301

   1. Initial program 72.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   3. Applied rewrites 76.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      15. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      17. lift-/.f64 76.2

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   6. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

   8. Applied rewrites 46.2%

      \[\leadsto \color{blue}{\left(\frac{\left(\left(M \cdot \left(D \cdot M\right)\right) \cdot D\right) \cdot -1}{d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot -0.125} \]

   9. Taylor expanded in d around 0

      \[\leadsto \frac{1}{8} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       3. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       4. associate-/l* N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       5. unpow2 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left({D}^{2} \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       6. associate-*r/ N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left({D}^{2} \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       7. associate-*r* N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left({D}^{2} \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left({D}^{2} \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left({D}^{2} \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       10. pow2 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       11. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       12. lift-/.f64 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       13. pow3 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

       14. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

       15. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

       16. lift-/.f64 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

       17. lift-sqrt.f64 44.5

           \[\leadsto \left(0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   11. Applied rewrites 44.5%

       \[\leadsto \left(0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \color{blue}{\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}} \]

   if -4.79999999999999982e-301 < l < 6.9999999999999995e48

   1. Initial program 74.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8} + \color{blue}{d} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \color{blue}{\frac{-1}{8}}, d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), -0.125, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \]

   5. Taylor expanded in d around 0

      \[\leadsto \frac{-1}{8} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

   7. Applied rewrites 46.6%

      \[\leadsto \left(-0.125 \cdot \frac{\left(M \cdot \left(D \cdot M\right)\right) \cdot D}{d}\right) \cdot \color{blue}{\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}} \]

   if 6.9999999999999995e48 < l

   1. Initial program 58.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      6. lower-*.f64 52.3

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      4. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lift-*.f64 52.2

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 52.2%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot \color{blue}{d} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      5. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{\ell \cdot h}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{1 \cdot d}{\sqrt{\ell \cdot h}} \]

      11. *-commutative N/A

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      12. lower-*.f64 52.3

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

   8. Applied rewrites 52.3%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\color{blue}{\ell}}} \]

      6. lower-sqrt.f64 63.9

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}} \]

   10. Applied rewrites 63.9%

       \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 50.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left(\left(D \cdot D\right) \cdot M\_m\right) \cdot \frac{M\_m}{d}\\ t_1 := \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{if}\;\ell \leq -1.18 \cdot 10^{-17}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;\ell \leq -4.8 \cdot 10^{-301}:\\ \;\;\;\;\left(0.125 \cdot t\_0\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+48}:\\ \;\;\;\;\left(-0.125 \cdot t\_0\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (* (* D D) M_m) (/ M_m d))) (t_1 (sqrt (/ h (* (* l l) l)))))
   (if (<= l -1.18e-17)
     (* (- (sqrt (/ 1.0 (* l h)))) d)
     (if (<= l -4.8e-301)
       (* (* 0.125 t_0) t_1)
       (if (<= l 4.5e+48)
         (* (* -0.125 t_0) t_1)
         (/ (* 1.0 d) (* (sqrt h) (sqrt l))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = ((D * D) * M_m) * (M_m / d);
	double t_1 = sqrt((h / ((l * l) * l)));
	double tmp;
	if (l <= -1.18e-17) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else if (l <= -4.8e-301) {
		tmp = (0.125 * t_0) * t_1;
	} else if (l <= 4.5e+48) {
		tmp = (-0.125 * t_0) * t_1;
	} else {
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: d, h, l, M_m, and D 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_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_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((d_1 * d_1) * m_m) * (m_m / d)
    t_1 = sqrt((h / ((l * l) * l)))
    if (l <= (-1.18d-17)) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else if (l <= (-4.8d-301)) then
        tmp = (0.125d0 * t_0) * t_1
    else if (l <= 4.5d+48) then
        tmp = ((-0.125d0) * t_0) * t_1
    else
        tmp = (1.0d0 * d) / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = ((D * D) * M_m) * (M_m / d);
	double t_1 = Math.sqrt((h / ((l * l) * l)));
	double tmp;
	if (l <= -1.18e-17) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else if (l <= -4.8e-301) {
		tmp = (0.125 * t_0) * t_1;
	} else if (l <= 4.5e+48) {
		tmp = (-0.125 * t_0) * t_1;
	} else {
		tmp = (1.0 * d) / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = ((D * D) * M_m) * (M_m / d)
	t_1 = math.sqrt((h / ((l * l) * l)))
	tmp = 0
	if l <= -1.18e-17:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	elif l <= -4.8e-301:
		tmp = (0.125 * t_0) * t_1
	elif l <= 4.5e+48:
		tmp = (-0.125 * t_0) * t_1
	else:
		tmp = (1.0 * d) / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64(Float64(D * D) * M_m) * Float64(M_m / d))
	t_1 = sqrt(Float64(h / Float64(Float64(l * l) * l)))
	tmp = 0.0
	if (l <= -1.18e-17)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	elseif (l <= -4.8e-301)
		tmp = Float64(Float64(0.125 * t_0) * t_1);
	elseif (l <= 4.5e+48)
		tmp = Float64(Float64(-0.125 * t_0) * t_1);
	else
		tmp = Float64(Float64(1.0 * d) / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = ((D * D) * M_m) * (M_m / d);
	t_1 = sqrt((h / ((l * l) * l)));
	tmp = 0.0;
	if (l <= -1.18e-17)
		tmp = -sqrt((1.0 / (l * h))) * d;
	elseif (l <= -4.8e-301)
		tmp = (0.125 * t_0) * t_1;
	elseif (l <= 4.5e+48)
		tmp = (-0.125 * t_0) * t_1;
	else
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(N[(D * D), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.18e-17], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], If[LessEqual[l, -4.8e-301], N[(N[(0.125 * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 4.5e+48], N[(N[(-0.125 * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(1.0 * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.18000000000000004e-17

   1. Initial program 60.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \cdot \color{blue}{d} \]

   4. Applied rewrites 1.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}, -0.125, \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d} \]

   5. Taylor expanded in l around -inf

      \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d \]
   6. Step-by-step derivation

      1. sqrt-pow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot d \]

      2. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{1}\right) \cdot d \]

      3. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right) \cdot d \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      5. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot d \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      7. *-commutative N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      9. lift-/.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      10. lift-*.f64 49.6

          \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   7. Applied rewrites 49.6%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if -1.18000000000000004e-17 < l < -4.79999999999999982e-301

   1. Initial program 72.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   3. Applied rewrites 76.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      15. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      17. lift-/.f64 76.2

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   6. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

   8. Applied rewrites 46.2%

      \[\leadsto \color{blue}{\left(\frac{\left(\left(M \cdot \left(D \cdot M\right)\right) \cdot D\right) \cdot -1}{d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot -0.125} \]

   9. Taylor expanded in d around 0

      \[\leadsto \frac{1}{8} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       3. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       4. associate-/l* N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       5. unpow2 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left({D}^{2} \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       6. associate-*r/ N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left({D}^{2} \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       7. associate-*r* N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left({D}^{2} \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left({D}^{2} \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left({D}^{2} \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       10. pow2 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       11. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       12. lift-/.f64 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       13. pow3 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

       14. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

       15. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

       16. lift-/.f64 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

       17. lift-sqrt.f64 44.5

           \[\leadsto \left(0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   11. Applied rewrites 44.5%

       \[\leadsto \left(0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \color{blue}{\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}} \]

   if -4.79999999999999982e-301 < l < 4.49999999999999995e48

   1. Initial program 74.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8} + \color{blue}{d} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \color{blue}{\frac{-1}{8}}, d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), -0.125, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), \frac{-1}{8}, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), \frac{-1}{8}, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right), \frac{-1}{8}, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right), \frac{-1}{8}, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \]

      5. lower-/.f64 46.1

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right), -0.125, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \]

   6. Applied rewrites 46.1%

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right), -0.125, \sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \]

   7. Taylor expanded in d around 0

      \[\leadsto \frac{-1}{8} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      4. associate-/l* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      6. associate-*r/ N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      7. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      10. pow2 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      12. lift-/.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      13. pow3 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      16. lift-/.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   9. Applied rewrites 43.8%

      \[\leadsto \left(-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \color{blue}{\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}} \]

   if 4.49999999999999995e48 < l

   1. Initial program 58.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      6. lower-*.f64 52.2

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      4. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lift-*.f64 52.1

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 52.1%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot \color{blue}{d} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      5. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{\ell \cdot h}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{1 \cdot d}{\sqrt{\ell \cdot h}} \]

      11. *-commutative N/A

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      12. lower-*.f64 52.2

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

   8. Applied rewrites 52.2%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\color{blue}{\ell}}} \]

      6. lower-sqrt.f64 63.8

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}} \]

   10. Applied rewrites 63.8%

       \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 49.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{-33}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\_m\right) \cdot \frac{M\_m}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -9.5e-33)
   (* (- (sqrt (/ 1.0 (* l h)))) d)
   (if (<= d -5e-310)
     (* (* 0.125 (* (* (* D D) M_m) (/ M_m d))) (sqrt (/ h (* (* l l) l))))
     (/ (* 1.0 d) (* (sqrt h) (sqrt l))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -9.5e-33) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else if (d <= -5e-310) {
		tmp = (0.125 * (((D * D) * M_m) * (M_m / d))) * sqrt((h / ((l * l) * l)));
	} else {
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: d, h, l, M_m, and D 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_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_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-9.5d-33)) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else if (d <= (-5d-310)) then
        tmp = (0.125d0 * (((d_1 * d_1) * m_m) * (m_m / d))) * sqrt((h / ((l * l) * l)))
    else
        tmp = (1.0d0 * d) / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -9.5e-33) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else if (d <= -5e-310) {
		tmp = (0.125 * (((D * D) * M_m) * (M_m / d))) * Math.sqrt((h / ((l * l) * l)));
	} else {
		tmp = (1.0 * d) / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= -9.5e-33:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	elif d <= -5e-310:
		tmp = (0.125 * (((D * D) * M_m) * (M_m / d))) * math.sqrt((h / ((l * l) * l)))
	else:
		tmp = (1.0 * d) / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -9.5e-33)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	elseif (d <= -5e-310)
		tmp = Float64(Float64(0.125 * Float64(Float64(Float64(D * D) * M_m) * Float64(M_m / d))) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	else
		tmp = Float64(Float64(1.0 * d) / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= -9.5e-33)
		tmp = -sqrt((1.0 / (l * h))) * d;
	elseif (d <= -5e-310)
		tmp = (0.125 * (((D * D) * M_m) * (M_m / d))) * sqrt((h / ((l * l) * l)));
	else
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -9.5e-33], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(0.125 * N[(N[(N[(D * D), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.50000000000000019e-33

   1. Initial program 77.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \cdot \color{blue}{d} \]

   4. Applied rewrites 0.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}, -0.125, \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d} \]

   5. Taylor expanded in l around -inf

      \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d \]
   6. Step-by-step derivation

      1. sqrt-pow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot d \]

      2. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{1}\right) \cdot d \]

      3. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right) \cdot d \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      5. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot d \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      7. *-commutative N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      9. lift-/.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      10. lift-*.f64 57.4

          \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   7. Applied rewrites 57.4%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if -9.50000000000000019e-33 < d < -4.999999999999985e-310

   1. Initial program 52.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   3. Applied rewrites 51.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      15. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      17. lift-/.f64 51.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   6. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

   8. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\left(\frac{\left(\left(M \cdot \left(D \cdot M\right)\right) \cdot D\right) \cdot -1}{d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot -0.125} \]

   9. Taylor expanded in d around 0

      \[\leadsto \frac{1}{8} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       3. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       4. associate-/l* N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       5. unpow2 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left({D}^{2} \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       6. associate-*r/ N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left({D}^{2} \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       7. associate-*r* N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left({D}^{2} \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left({D}^{2} \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left({D}^{2} \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       10. pow2 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       11. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       12. lift-/.f64 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

       13. pow3 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

       14. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

       15. lift-*.f64 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

       16. lift-/.f64 N/A

           \[\leadsto \left(\frac{1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

       17. lift-sqrt.f64 41.0

           \[\leadsto \left(0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   11. Applied rewrites 41.0%

       \[\leadsto \left(0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d}\right)\right) \cdot \color{blue}{\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}} \]

   if -4.999999999999985e-310 < d

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      6. lower-*.f64 43.7

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      4. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lift-*.f64 43.9

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 43.9%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot \color{blue}{d} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      5. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{\ell \cdot h}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{1 \cdot d}{\sqrt{\ell \cdot h}} \]

      11. *-commutative N/A

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      12. lower-*.f64 43.9

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

   8. Applied rewrites 43.9%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\color{blue}{\ell}}} \]

      6. lower-sqrt.f64 51.6

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}} \]

   10. Applied rewrites 51.6%

       \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 49.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-90}:\\ \;\;\;\;-\left(-\left(-\sqrt{\frac{1}{\left(\left(h \cdot h\right) \cdot h\right) \cdot \ell}}\right) \cdot d\right) \cdot h\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -2e-90)
     (- (* (- (* (- (sqrt (/ 1.0 (* (* (* h h) h) l)))) d)) h))
     (if (<= t_0 INFINITY)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (* (- (sqrt (/ 1.0 (* l h)))) d)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -2e-90) {
		tmp = -(-(-sqrt((1.0 / (((h * h) * h) * l))) * d) * h);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = -sqrt((1.0 / (l * h))) * d;
	}
	return tmp;
}

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (Math.pow((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) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -2e-90) {
		tmp = -(-(-Math.sqrt((1.0 / (((h * h) * h) * l))) * d) * h);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = (math.pow((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) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -2e-90:
		tmp = -(-(-math.sqrt((1.0 / (((h * h) * h) * l))) * d) * h)
	elif t_0 <= math.inf:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64((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) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -2e-90)
		tmp = Float64(-Float64(Float64(-Float64(Float64(-sqrt(Float64(1.0 / Float64(Float64(Float64(h * h) * h) * l)))) * d)) * h));
	elseif (t_0 <= Inf)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -2e-90)
		tmp = -(-(-sqrt((1.0 / (((h * h) * h) * l))) * d) * h);
	elseif (t_0 <= Inf)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = -sqrt((1.0 / (l * h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(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), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-90], (-N[((-N[((-N[Sqrt[N[(1.0 / N[(N[(N[(h * h), $MachinePrecision] * h), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision]) * h), $MachinePrecision]), If[LessEqual[t$95$0, Infinity], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.99999999999999999e-90

   1. Initial program 86.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 43.5

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 43.5%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(h \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right) + \frac{1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right)\right)\right)} \]

   6. Applied rewrites 35.5%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot D\right) \cdot \sqrt{\frac{1}{\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot h}}}{d}, 0.125, -\sqrt{\frac{1}{\left(\left(h \cdot h\right) \cdot h\right) \cdot \ell}} \cdot d\right) \cdot h} \]

   7. Taylor expanded in l around -inf

      \[\leadsto -\left(-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right)\right) \cdot h \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto -\left(\mathsf{neg}\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right)\right) \cdot h \]

      2. lower-neg.f64 N/A

         \[\leadsto -\left(-\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right) \cdot h \]

      3. sqrt-pow2 N/A

         \[\leadsto -\left(-\left(d \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right) \cdot h \]

      4. metadata-eval N/A

         \[\leadsto -\left(-\left(d \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right) \cdot h \]

      5. metadata-eval N/A

         \[\leadsto -\left(-\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right) \cdot h \]

      6. *-commutative N/A

         \[\leadsto -\left(-\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right) \cdot h \]

      7. associate-*r* N/A

         \[\leadsto -\left(--1 \cdot \left(d \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right)\right) \cdot h \]

      8. mul-1-neg N/A

         \[\leadsto -\left(-\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right)\right)\right) \cdot h \]

      9. pow3 N/A

         \[\leadsto -\left(-\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{\left(\left(h \cdot h\right) \cdot h\right) \cdot \ell}}\right)\right)\right) \cdot h \]

      10. lift-*.f64 N/A

          \[\leadsto -\left(-\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{\left(\left(h \cdot h\right) \cdot h\right) \cdot \ell}}\right)\right)\right) \cdot h \]

      11. lift-*.f64 N/A

          \[\leadsto -\left(-\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{\left(\left(h \cdot h\right) \cdot h\right) \cdot \ell}}\right)\right)\right) \cdot h \]

      12. lift-*.f64 N/A

          \[\leadsto -\left(-\left(\mathsf{neg}\left(d \cdot \sqrt{\frac{1}{\left(\left(h \cdot h\right) \cdot h\right) \cdot \ell}}\right)\right)\right) \cdot h \]

   9. Applied rewrites 30.7%

      \[\leadsto -\left(-\left(-\sqrt{\frac{1}{\left(\left(h \cdot h\right) \cdot h\right) \cdot \ell}}\right) \cdot d\right) \cdot h \]

   if -1.99999999999999999e-90 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 78.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 77.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-/.f64 76.6

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   6. Applied rewrites 76.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \cdot \color{blue}{d} \]

   4. Applied rewrites 10.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}, -0.125, \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d} \]

   5. Taylor expanded in l around -inf

      \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d \]
   6. Step-by-step derivation

      1. sqrt-pow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot d \]

      2. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{1}\right) \cdot d \]

      3. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right) \cdot d \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      5. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot d \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      7. *-commutative N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      9. lift-/.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      10. lift-*.f64 13.7

          \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   7. Applied rewrites 13.7%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 47.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.4 \cdot 10^{-283}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l 4.4e-283)
   (* (- (sqrt (/ 1.0 (* l h)))) d)
   (/ (* 1.0 d) (* (sqrt h) (sqrt l)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 4.4e-283) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: d, h, l, M_m, and D 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_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_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 4.4d-283) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else
        tmp = (1.0d0 * d) / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 4.4e-283) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = (1.0 * d) / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= 4.4e-283:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	else:
		tmp = (1.0 * d) / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= 4.4e-283)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	else
		tmp = Float64(Float64(1.0 * d) / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= 4.4e-283)
		tmp = -sqrt((1.0 / (l * h))) * d;
	else
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, 4.4e-283], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], N[(N[(1.0 * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.3999999999999996e-283

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \cdot \color{blue}{d} \]

   4. Applied rewrites 1.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}, -0.125, \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d} \]

   5. Taylor expanded in l around -inf

      \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d \]
   6. Step-by-step derivation

      1. sqrt-pow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot d \]

      2. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{1}\right) \cdot d \]

      3. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right) \cdot d \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      5. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot d \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      7. *-commutative N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      9. lift-/.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      10. lift-*.f64 42.4

          \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   7. Applied rewrites 42.4%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if 4.3999999999999996e-283 < l

   1. Initial program 67.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      6. lower-*.f64 44.5

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 44.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      4. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lift-*.f64 44.7

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 44.7%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot \color{blue}{d} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      5. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{\ell \cdot h}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{1 \cdot d}{\sqrt{\ell \cdot h}} \]

      11. *-commutative N/A

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      12. lower-*.f64 44.8

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

   8. Applied rewrites 44.8%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\color{blue}{\ell}}} \]

      6. lower-sqrt.f64 52.3

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}} \]

   10. Applied rewrites 52.3%

       \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 47.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.4 \cdot 10^{-283}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l 4.4e-283)
   (* (- (sqrt (/ 1.0 (* l h)))) d)
   (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 4.4e-283) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: d, h, l, M_m, and D 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_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_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 4.4d-283) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 4.4e-283) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= 4.4e-283:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= 4.4e-283)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= 4.4e-283)
		tmp = -sqrt((1.0 / (l * h))) * d;
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, 4.4e-283], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.3999999999999996e-283

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \cdot \color{blue}{d} \]

   4. Applied rewrites 1.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}, -0.125, \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d} \]

   5. Taylor expanded in l around -inf

      \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d \]
   6. Step-by-step derivation

      1. sqrt-pow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot d \]

      2. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{1}\right) \cdot d \]

      3. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right) \cdot d \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      5. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot d \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      7. *-commutative N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      9. lift-/.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      10. lift-*.f64 42.4

          \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   7. Applied rewrites 42.4%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if 4.3999999999999996e-283 < l

   1. Initial program 67.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      6. lower-*.f64 44.5

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 44.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      4. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lift-*.f64 44.7

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 44.7%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lower-sqrt.f64 52.3

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   8. Applied rewrites 52.3%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 43.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.3 \cdot 10^{-208}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l 7.3e-208) (* (- (sqrt (/ 1.0 (* l h)))) d) (/ d (sqrt (* h l)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 7.3e-208) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = d / sqrt((h * l));
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: d, h, l, M_m, and D 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_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_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 7.3d-208) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else
        tmp = d / sqrt((h * l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 7.3e-208) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = d / Math.sqrt((h * l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= 7.3e-208:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	else:
		tmp = d / math.sqrt((h * l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= 7.3e-208)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	else
		tmp = Float64(d / sqrt(Float64(h * l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= 7.3e-208)
		tmp = -sqrt((1.0 / (l * h))) * d;
	else
		tmp = d / sqrt((h * l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, 7.3e-208], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.30000000000000002e-208

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} + \frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \cdot \color{blue}{d} \]

   4. Applied rewrites 3.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}, -0.125, \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d} \]

   5. Taylor expanded in l around -inf

      \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d \]
   6. Step-by-step derivation

      1. sqrt-pow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot d \]

      2. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{1}\right) \cdot d \]

      3. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right) \cdot d \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      5. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \cdot d \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      7. *-commutative N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      9. lift-/.f64 N/A

         \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

      10. lift-*.f64 40.3

          \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   7. Applied rewrites 40.3%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if 7.30000000000000002e-208 < l

   1. Initial program 66.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      6. lower-*.f64 46.8

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 46.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      4. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lift-*.f64 46.9

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 46.9%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot \color{blue}{d} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      5. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{\ell \cdot h}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{1 \cdot d}{\sqrt{\ell \cdot h}} \]

      11. *-commutative N/A

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      12. lower-*.f64 47.0

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

   8. Applied rewrites 47.0%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      2. *-lft-identity 47.0

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   10. Applied rewrites 47.0%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 26.3% accurate, 7.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (* (sqrt (/ 1.0 (* l h))) d))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return sqrt((1.0 / (l * h))) * d;
}

Fortran

M_m =     private
NOTE: d, h, l, M_m, and D 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_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_m
    real(8), intent (in) :: d_1
    code = sqrt((1.0d0 / (l * h))) * d
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return Math.sqrt((1.0 / (l * h))) * d;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return math.sqrt((1.0 / (l * h))) * d

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(sqrt(Float64(1.0 / Float64(l * h))) * d)
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = sqrt((1.0 / (l * h))) * d;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 66.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   6. lower-*.f64 26.3

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

4. Applied rewrites 26.3%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Add Preprocessing

Alternative 19: 26.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (/ d (sqrt (* h l))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d / sqrt((h * l));
}

Fortran

M_m =     private
NOTE: d, h, l, M_m, and D 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_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_m
    real(8), intent (in) :: d_1
    code = d / sqrt((h * l))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d / Math.sqrt((h * l));
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d / math.sqrt((h * l))

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d / sqrt((h * l));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   6. lower-*.f64 26.3

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

4. Applied rewrites 26.3%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   3. lift-/.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. sqrt-div N/A

      \[\leadsto \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

   8. *-commutative N/A

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   10. lift-*.f64 26.2

       \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

6. Applied rewrites 26.2%

   \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot \color{blue}{d} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   5. associate-*l/ N/A

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   6. *-commutative N/A

      \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{1 \cdot d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. *-commutative N/A

      \[\leadsto \frac{1 \cdot d}{\sqrt{\ell \cdot h}} \]

   10. lift-sqrt.f64 N/A

       \[\leadsto \frac{1 \cdot d}{\sqrt{\ell \cdot h}} \]

   11. *-commutative N/A

       \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

   12. lower-*.f64 26.2

       \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

8. Applied rewrites 26.2%

   \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1 \cdot d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   2. *-lft-identity 26.2

      \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

10. Applied rewrites 26.2%

    \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025112 
(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)))))