Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.6% → 73.7%

Time: 9.4s

Alternatives: 24

Speedup: 1.8×

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: 66.6% 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: 73.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ t_1 := M \cdot \frac{D}{d + d}\\ t_2 := 1 - \frac{\left(\left(t\_1 \cdot t\_1\right) \cdot 0.5\right) \cdot h}{\ell}\\ \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+53}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{t\_1 \cdot \left(\frac{\left(M \cdot D\right) \cdot 0.5}{d + d} \cdot h\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(-t\_0\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* d (- (sqrt (/ 1.0 (* l h))))))
        (t_1 (* M (/ D (+ d d))))
        (t_2 (- 1.0 (/ (* (* (* t_1 t_1) 0.5) h) l))))
   (if (<= l -4.8e+53)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (/ (* t_1 (* (/ (* (* M D) 0.5) (+ d d)) h)) l)))
     (if (<= l -5e-310) (* t_0 t_2) (* (- t_0) t_2)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = d * -sqrt((1.0 / (l * h)));
	double t_1 = M * (D / (d + d));
	double t_2 = 1.0 - ((((t_1 * t_1) * 0.5) * h) / l);
	double tmp;
	if (l <= -4.8e+53) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((t_1 * ((((M * D) * 0.5) / (d + d)) * h)) / l));
	} else if (l <= -5e-310) {
		tmp = t_0 * t_2;
	} else {
		tmp = -t_0 * t_2;
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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, 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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = d * -sqrt((1.0d0 / (l * h)))
    t_1 = m * (d_1 / (d + d))
    t_2 = 1.0d0 - ((((t_1 * t_1) * 0.5d0) * h) / l)
    if (l <= (-4.8d+53)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((t_1 * ((((m * d_1) * 0.5d0) / (d + d)) * h)) / l))
    else if (l <= (-5d-310)) then
        tmp = t_0 * t_2
    else
        tmp = -t_0 * t_2
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d * -Math.sqrt((1.0 / (l * h)));
	double t_1 = M * (D / (d + d));
	double t_2 = 1.0 - ((((t_1 * t_1) * 0.5) * h) / l);
	double tmp;
	if (l <= -4.8e+53) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((t_1 * ((((M * D) * 0.5) / (d + d)) * h)) / l));
	} else if (l <= -5e-310) {
		tmp = t_0 * t_2;
	} else {
		tmp = -t_0 * t_2;
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = d * -math.sqrt((1.0 / (l * h)))
	t_1 = M * (D / (d + d))
	t_2 = 1.0 - ((((t_1 * t_1) * 0.5) * h) / l)
	tmp = 0
	if l <= -4.8e+53:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((t_1 * ((((M * D) * 0.5) / (d + d)) * h)) / l))
	elif l <= -5e-310:
		tmp = t_0 * t_2
	else:
		tmp = -t_0 * t_2
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(l * h)))))
	t_1 = Float64(M * Float64(D / Float64(d + d)))
	t_2 = Float64(1.0 - Float64(Float64(Float64(Float64(t_1 * t_1) * 0.5) * h) / l))
	tmp = 0.0
	if (l <= -4.8e+53)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(t_1 * Float64(Float64(Float64(Float64(M * D) * 0.5) / Float64(d + d)) * h)) / l)));
	elseif (l <= -5e-310)
		tmp = Float64(t_0 * t_2);
	else
		tmp = Float64(Float64(-t_0) * t_2);
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = d * -sqrt((1.0 / (l * h)));
	t_1 = M * (D / (d + d));
	t_2 = 1.0 - ((((t_1 * t_1) * 0.5) * h) / l);
	tmp = 0.0;
	if (l <= -4.8e+53)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((t_1 * ((((M * D) * 0.5) / (d + d)) * h)) / l));
	elseif (l <= -5e-310)
		tmp = t_0 * t_2;
	else
		tmp = -t_0 * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$1 = N[(M * N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.8e+53], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$1 * N[(N[(N[(N[(M * D), $MachinePrecision] * 0.5), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(t$95$0 * t$95$2), $MachinePrecision], N[((-t$95$0) * t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.8e53

   1. Initial program 56.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 55.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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 55.9

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 55.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-+.f64 55.9

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

   7. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 56.5

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

   9. Applied rewrites 56.5%

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

   if -4.8e53 < l < -4.999999999999985e-310

   1. Initial program 72.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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 75.8%

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

   4. 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. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-*.f64 84.8

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

   6. Applied rewrites 84.8%

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

   if -4.999999999999985e-310 < l

   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. 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 68.1%

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

   4. Taylor expanded in d around -inf

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

      1. metadata-eval N/A

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

      2. pow1/2 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. sqrt-pow2 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. *-commutative N/A

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

   6. Applied rewrites 73.8%

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

Alternative 2: 73.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ t_1 := M \cdot \frac{D}{d + d}\\ t_2 := 1 - \frac{\left(\left(t\_1 \cdot t\_1\right) \cdot 0.5\right) \cdot h}{\ell}\\ \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+53}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot M}{d + d} \cdot \left(0.25 \cdot \frac{M \cdot D}{d}\right)\right) \cdot h}{\ell}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(-t\_0\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* d (- (sqrt (/ 1.0 (* l h))))))
        (t_1 (* M (/ D (+ d d))))
        (t_2 (- 1.0 (/ (* (* (* t_1 t_1) 0.5) h) l))))
   (if (<= l -4.8e+53)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (/ (* (* (/ (* D M) (+ d d)) (* 0.25 (/ (* M D) d))) h) l)))
     (if (<= l -5e-310) (* t_0 t_2) (* (- t_0) t_2)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = d * -sqrt((1.0 / (l * h)));
	double t_1 = M * (D / (d + d));
	double t_2 = 1.0 - ((((t_1 * t_1) * 0.5) * h) / l);
	double tmp;
	if (l <= -4.8e+53) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((((D * M) / (d + d)) * (0.25 * ((M * D) / d))) * h) / l));
	} else if (l <= -5e-310) {
		tmp = t_0 * t_2;
	} else {
		tmp = -t_0 * t_2;
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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, 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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = d * -sqrt((1.0d0 / (l * h)))
    t_1 = m * (d_1 / (d + d))
    t_2 = 1.0d0 - ((((t_1 * t_1) * 0.5d0) * h) / l)
    if (l <= (-4.8d+53)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (((((d_1 * m) / (d + d)) * (0.25d0 * ((m * d_1) / d))) * h) / l))
    else if (l <= (-5d-310)) then
        tmp = t_0 * t_2
    else
        tmp = -t_0 * t_2
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d * -Math.sqrt((1.0 / (l * h)));
	double t_1 = M * (D / (d + d));
	double t_2 = 1.0 - ((((t_1 * t_1) * 0.5) * h) / l);
	double tmp;
	if (l <= -4.8e+53) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (((((D * M) / (d + d)) * (0.25 * ((M * D) / d))) * h) / l));
	} else if (l <= -5e-310) {
		tmp = t_0 * t_2;
	} else {
		tmp = -t_0 * t_2;
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = d * -math.sqrt((1.0 / (l * h)))
	t_1 = M * (D / (d + d))
	t_2 = 1.0 - ((((t_1 * t_1) * 0.5) * h) / l)
	tmp = 0
	if l <= -4.8e+53:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (((((D * M) / (d + d)) * (0.25 * ((M * D) / d))) * h) / l))
	elif l <= -5e-310:
		tmp = t_0 * t_2
	else:
		tmp = -t_0 * t_2
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(l * h)))))
	t_1 = Float64(M * Float64(D / Float64(d + d)))
	t_2 = Float64(1.0 - Float64(Float64(Float64(Float64(t_1 * t_1) * 0.5) * h) / l))
	tmp = 0.0
	if (l <= -4.8e+53)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D * M) / Float64(d + d)) * Float64(0.25 * Float64(Float64(M * D) / d))) * h) / l)));
	elseif (l <= -5e-310)
		tmp = Float64(t_0 * t_2);
	else
		tmp = Float64(Float64(-t_0) * t_2);
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = d * -sqrt((1.0 / (l * h)));
	t_1 = M * (D / (d + d));
	t_2 = 1.0 - ((((t_1 * t_1) * 0.5) * h) / l);
	tmp = 0.0;
	if (l <= -4.8e+53)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((((D * M) / (d + d)) * (0.25 * ((M * D) / d))) * h) / l));
	elseif (l <= -5e-310)
		tmp = t_0 * t_2;
	else
		tmp = -t_0 * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$1 = N[(M * N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.8e+53], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(D * M), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(t$95$0 * t$95$2), $MachinePrecision], N[((-t$95$0) * t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.8e53

   1. Initial program 56.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 55.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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 55.9

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 55.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-+.f64 55.9

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

   7. Applied rewrites 55.9%

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

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot M}{d + d} \cdot \left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)\right) \cdot h}{\ell}\right) \]

      4. lower-*.f64 55.9

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot M}{d + d} \cdot \left(0.25 \cdot \frac{M \cdot D}{d}\right)\right) \cdot h}{\ell}\right) \]

   10. Applied rewrites 55.9%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot M}{d + d} \cdot \color{blue}{\left(0.25 \cdot \frac{M \cdot D}{d}\right)}\right) \cdot h}{\ell}\right) \]

   if -4.8e53 < l < -4.999999999999985e-310

   1. Initial program 72.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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 75.8%

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

   4. 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. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-*.f64 84.8

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

   6. Applied rewrites 84.8%

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

   if -4.999999999999985e-310 < l

   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. 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 68.1%

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

   4. Taylor expanded in d around -inf

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

      1. metadata-eval N/A

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

      2. pow1/2 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. sqrt-pow2 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. *-commutative N/A

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

   6. Applied rewrites 73.8%

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

Alternative 3: 73.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \frac{D \cdot M}{d + d}\\ t_1 := M \cdot \frac{D}{d + d}\\ \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+53}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(t\_0 \cdot \left(0.25 \cdot \frac{M \cdot D}{d}\right)\right) \cdot h}{\ell}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\right) \cdot \left(1 - \frac{\left(\left(t\_1 \cdot t\_1\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \cdot \left(1 - \frac{\left(t\_0 \cdot \left(t\_0 \cdot 0.5\right)\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* D M) (+ d d))) (t_1 (* M (/ D (+ d d)))))
   (if (<= l -4.8e+53)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (/ (* (* t_0 (* 0.25 (/ (* M D) d))) h) l)))
     (if (<= l -5e-310)
       (*
        (* d (- (sqrt (/ 1.0 (* l h)))))
        (- 1.0 (/ (* (* (* t_1 t_1) 0.5) h) l)))
       (*
        (* (sqrt (/ 1.0 (* h l))) d)
        (- 1.0 (/ (* (* t_0 (* t_0 0.5)) h) l)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * M) / (d + d);
	double t_1 = M * (D / (d + d));
	double tmp;
	if (l <= -4.8e+53) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((t_0 * (0.25 * ((M * D) / d))) * h) / l));
	} else if (l <= -5e-310) {
		tmp = (d * -sqrt((1.0 / (l * h)))) * (1.0 - ((((t_1 * t_1) * 0.5) * h) / l));
	} else {
		tmp = (sqrt((1.0 / (h * l))) * d) * (1.0 - (((t_0 * (t_0 * 0.5)) * h) / l));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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, 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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (d_1 * m) / (d + d)
    t_1 = m * (d_1 / (d + d))
    if (l <= (-4.8d+53)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (((t_0 * (0.25d0 * ((m * d_1) / d))) * h) / l))
    else if (l <= (-5d-310)) then
        tmp = (d * -sqrt((1.0d0 / (l * h)))) * (1.0d0 - ((((t_1 * t_1) * 0.5d0) * h) / l))
    else
        tmp = (sqrt((1.0d0 / (h * l))) * d) * (1.0d0 - (((t_0 * (t_0 * 0.5d0)) * h) / l))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * M) / (d + d);
	double t_1 = M * (D / (d + d));
	double tmp;
	if (l <= -4.8e+53) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (((t_0 * (0.25 * ((M * D) / d))) * h) / l));
	} else if (l <= -5e-310) {
		tmp = (d * -Math.sqrt((1.0 / (l * h)))) * (1.0 - ((((t_1 * t_1) * 0.5) * h) / l));
	} else {
		tmp = (Math.sqrt((1.0 / (h * l))) * d) * (1.0 - (((t_0 * (t_0 * 0.5)) * h) / l));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = (D * M) / (d + d)
	t_1 = M * (D / (d + d))
	tmp = 0
	if l <= -4.8e+53:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (((t_0 * (0.25 * ((M * D) / d))) * h) / l))
	elif l <= -5e-310:
		tmp = (d * -math.sqrt((1.0 / (l * h)))) * (1.0 - ((((t_1 * t_1) * 0.5) * h) / l))
	else:
		tmp = (math.sqrt((1.0 / (h * l))) * d) * (1.0 - (((t_0 * (t_0 * 0.5)) * h) / l))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * M) / Float64(d + d))
	t_1 = Float64(M * Float64(D / Float64(d + d)))
	tmp = 0.0
	if (l <= -4.8e+53)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(t_0 * Float64(0.25 * Float64(Float64(M * D) / d))) * h) / l)));
	elseif (l <= -5e-310)
		tmp = Float64(Float64(d * Float64(-sqrt(Float64(1.0 / Float64(l * h))))) * Float64(1.0 - Float64(Float64(Float64(Float64(t_1 * t_1) * 0.5) * h) / l)));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(h * l))) * d) * Float64(1.0 - Float64(Float64(Float64(t_0 * Float64(t_0 * 0.5)) * h) / l)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (D * M) / (d + d);
	t_1 = M * (D / (d + d));
	tmp = 0.0;
	if (l <= -4.8e+53)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((t_0 * (0.25 * ((M * D) / d))) * h) / l));
	elseif (l <= -5e-310)
		tmp = (d * -sqrt((1.0 / (l * h)))) * (1.0 - ((((t_1 * t_1) * 0.5) * h) / l));
	else
		tmp = (sqrt((1.0 / (h * l))) * d) * (1.0 - (((t_0 * (t_0 * 0.5)) * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D * M), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(M * N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.8e+53], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(t$95$0 * N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(d * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(t$95$0 * N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.8e53

   1. Initial program 56.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 55.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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 55.9

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 55.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-+.f64 55.9

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

   7. Applied rewrites 55.9%

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

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot M}{d + d} \cdot \left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)\right) \cdot h}{\ell}\right) \]

      4. lower-*.f64 55.9

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot M}{d + d} \cdot \left(0.25 \cdot \frac{M \cdot D}{d}\right)\right) \cdot h}{\ell}\right) \]

   10. Applied rewrites 55.9%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot M}{d + d} \cdot \color{blue}{\left(0.25 \cdot \frac{M \cdot D}{d}\right)}\right) \cdot h}{\ell}\right) \]

   if -4.8e53 < l < -4.999999999999985e-310

   1. Initial program 72.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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 75.8%

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

   4. 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. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-*.f64 84.8

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

   6. Applied rewrites 84.8%

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

   if -4.999999999999985e-310 < l

   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. 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 68.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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 68.1

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

   5. Applied rewrites 68.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 68.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-+.f64 68.8

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

   7. Applied rewrites 68.8%

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

   8. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 74.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 74.4

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

   10. Applied rewrites 74.4%

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

Alternative 4: 71.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \frac{D \cdot M}{d + d}\\ \mathbf{if}\;h \leq 1.65 \cdot 10^{-294}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(t\_0 \cdot \left(0.25 \cdot \frac{M \cdot D}{d}\right)\right) \cdot h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \cdot \left(1 - \frac{\left(t\_0 \cdot \left(t\_0 \cdot 0.5\right)\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* D M) (+ d d))))
   (if (<= h 1.65e-294)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (/ (* (* t_0 (* 0.25 (/ (* M D) d))) h) l)))
     (*
      (* (sqrt (/ 1.0 (* h l))) d)
      (- 1.0 (/ (* (* t_0 (* t_0 0.5)) h) l))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * M) / (d + d);
	double tmp;
	if (h <= 1.65e-294) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((t_0 * (0.25 * ((M * D) / d))) * h) / l));
	} else {
		tmp = (sqrt((1.0 / (h * l))) * d) * (1.0 - (((t_0 * (t_0 * 0.5)) * h) / l));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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, 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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_1 * m) / (d + d)
    if (h <= 1.65d-294) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (((t_0 * (0.25d0 * ((m * d_1) / d))) * h) / l))
    else
        tmp = (sqrt((1.0d0 / (h * l))) * d) * (1.0d0 - (((t_0 * (t_0 * 0.5d0)) * h) / l))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * M) / (d + d);
	double tmp;
	if (h <= 1.65e-294) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (((t_0 * (0.25 * ((M * D) / d))) * h) / l));
	} else {
		tmp = (Math.sqrt((1.0 / (h * l))) * d) * (1.0 - (((t_0 * (t_0 * 0.5)) * h) / l));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = (D * M) / (d + d)
	tmp = 0
	if h <= 1.65e-294:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (((t_0 * (0.25 * ((M * D) / d))) * h) / l))
	else:
		tmp = (math.sqrt((1.0 / (h * l))) * d) * (1.0 - (((t_0 * (t_0 * 0.5)) * h) / l))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * M) / Float64(d + d))
	tmp = 0.0
	if (h <= 1.65e-294)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(t_0 * Float64(0.25 * Float64(Float64(M * D) / d))) * h) / l)));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(h * l))) * d) * Float64(1.0 - Float64(Float64(Float64(t_0 * Float64(t_0 * 0.5)) * h) / l)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (D * M) / (d + d);
	tmp = 0.0;
	if (h <= 1.65e-294)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((t_0 * (0.25 * ((M * D) / d))) * h) / l));
	else
		tmp = (sqrt((1.0 / (h * l))) * d) * (1.0 - (((t_0 * (t_0 * 0.5)) * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D * M), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, 1.65e-294], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(t$95$0 * N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] * N[(1.0 - N[(N[(N[(t$95$0 * N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < 1.65e-294

   1. Initial program 65.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 67.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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 67.5

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

   5. Applied rewrites 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 67.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-+.f64 67.9

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

   7. Applied rewrites 67.9%

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

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot M}{d + d} \cdot \left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)\right) \cdot h}{\ell}\right) \]

      4. lower-*.f64 67.9

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot M}{d + d} \cdot \left(0.25 \cdot \frac{M \cdot D}{d}\right)\right) \cdot h}{\ell}\right) \]

   10. Applied rewrites 67.9%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot M}{d + d} \cdot \color{blue}{\left(0.25 \cdot \frac{M \cdot D}{d}\right)}\right) \cdot h}{\ell}\right) \]

   if 1.65e-294 < h

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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 68.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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 68.3

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

   5. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 68.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-+.f64 69.0

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

   7. Applied rewrites 69.0%

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

   8. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 74.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 74.5

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

   10. Applied rewrites 74.5%

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

Alternative 5: 69.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{-111}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot M}{d + d} \cdot \left(0.25 \cdot \frac{M \cdot D}{d}\right)\right) \cdot h}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \sqrt{\ell \cdot h}}{d}, -0.125, \sqrt{\frac{\left(\ell \cdot \ell\right) \cdot \ell}{h}} \cdot d\right)}{\ell \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l 2.2e-111)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (/ (* (* (/ (* D M) (+ d d)) (* 0.25 (/ (* M D) d))) h) l)))
   (if (<= l 5.4e+102)
     (/
      (fma
       (/ (* (* (* D M) (* D M)) (sqrt (* l h))) d)
       -0.125
       (* (sqrt (/ (* (* l l) l) h)) d))
      (* l l))
     (* (/ 1.0 (* (sqrt l) (sqrt h))) d))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2.2e-111) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((((D * M) / (d + d)) * (0.25 * ((M * D) / d))) * h) / l));
	} else if (l <= 5.4e+102) {
		tmp = fma(((((D * M) * (D * M)) * sqrt((l * h))) / d), -0.125, (sqrt((((l * l) * l) / h)) * d)) / (l * l);
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 2.2e-111)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D * M) / Float64(d + d)) * Float64(0.25 * Float64(Float64(M * D) / d))) * h) / l)));
	elseif (l <= 5.4e+102)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(D * M) * Float64(D * M)) * sqrt(Float64(l * h))) / d), -0.125, Float64(sqrt(Float64(Float64(Float64(l * l) * l) / h)) * d)) / Float64(l * l));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, 2.2e-111], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(D * M), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.4e+102], N[(N[(N[(N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125 + N[(N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.2e-111

   1. Initial program 67.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 69.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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 69.9

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

   5. Applied rewrites 69.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 69.6

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-+.f64 70.4

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

   7. Applied rewrites 70.4%

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

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot M}{d + d} \cdot \left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)\right) \cdot h}{\ell}\right) \]

      4. lower-*.f64 70.4

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot M}{d + d} \cdot \left(0.25 \cdot \frac{M \cdot D}{d}\right)\right) \cdot h}{\ell}\right) \]

   10. Applied rewrites 70.4%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot M}{d + d} \cdot \color{blue}{\left(0.25 \cdot \frac{M \cdot D}{d}\right)}\right) \cdot h}{\ell}\right) \]

   if 2.2e-111 < l < 5.4000000000000002e102

   1. Initial program 72.3%

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

   2. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unswap-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 72.0

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

   6. Applied rewrites 72.0%

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

   if 5.4000000000000002e102 < l

   1. Initial program 56.0%

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

   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 49.3

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

   4. Applied rewrites 49.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   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. metadata-eval N/A

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

      6. *-commutative 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. lift-sqrt.f64 N/A

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

      10. lift-*.f64 49.2

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

   6. Applied rewrites 49.2%

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

      1. lift-*.f64 N/A

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

      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 63.4

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

   8. Applied rewrites 63.4%

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

Alternative 6: 67.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \frac{D \cdot M}{d + d}\\ \mathbf{if}\;h \leq 1.3 \cdot 10^{-156}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(t\_0 \cdot \left(0.25 \cdot \frac{M \cdot D}{d}\right)\right) \cdot h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(\left(\left(t\_0 \cdot \frac{D}{d + d}\right) \cdot M\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* D M) (+ d d))))
   (if (<= h 1.3e-156)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (/ (* (* t_0 (* 0.25 (/ (* M D) d))) h) l)))
     (*
      (- 1.0 (* (* (* (* t_0 (/ D (+ d d))) M) 0.5) (/ h l)))
      (* (sqrt (/ 1.0 (* h l))) d)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * M) / (d + d);
	double tmp;
	if (h <= 1.3e-156) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((t_0 * (0.25 * ((M * D) / d))) * h) / l));
	} else {
		tmp = (1.0 - ((((t_0 * (D / (d + d))) * M) * 0.5) * (h / l))) * (sqrt((1.0 / (h * l))) * d);
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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, 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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_1 * m) / (d + d)
    if (h <= 1.3d-156) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (((t_0 * (0.25d0 * ((m * d_1) / d))) * h) / l))
    else
        tmp = (1.0d0 - ((((t_0 * (d_1 / (d + d))) * m) * 0.5d0) * (h / l))) * (sqrt((1.0d0 / (h * l))) * d)
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * M) / (d + d);
	double tmp;
	if (h <= 1.3e-156) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (((t_0 * (0.25 * ((M * D) / d))) * h) / l));
	} else {
		tmp = (1.0 - ((((t_0 * (D / (d + d))) * M) * 0.5) * (h / l))) * (Math.sqrt((1.0 / (h * l))) * d);
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = (D * M) / (d + d)
	tmp = 0
	if h <= 1.3e-156:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (((t_0 * (0.25 * ((M * D) / d))) * h) / l))
	else:
		tmp = (1.0 - ((((t_0 * (D / (d + d))) * M) * 0.5) * (h / l))) * (math.sqrt((1.0 / (h * l))) * d)
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * M) / Float64(d + d))
	tmp = 0.0
	if (h <= 1.3e-156)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(t_0 * Float64(0.25 * Float64(Float64(M * D) / d))) * h) / l)));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * Float64(D / Float64(d + d))) * M) * 0.5) * Float64(h / l))) * Float64(sqrt(Float64(1.0 / Float64(h * l))) * d));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (D * M) / (d + d);
	tmp = 0.0;
	if (h <= 1.3e-156)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((t_0 * (0.25 * ((M * D) / d))) * h) / l));
	else
		tmp = (1.0 - ((((t_0 * (D / (d + d))) * M) * 0.5) * (h / l))) * (sqrt((1.0 / (h * l))) * d);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D * M), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, 1.3e-156], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(t$95$0 * N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(N[(t$95$0 * N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * 0.5), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < 1.3e-156

   1. Initial program 66.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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 67.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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 67.6

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

   5. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 67.3

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-+.f64 68.1

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

   7. Applied rewrites 68.1%

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

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot M}{d + d} \cdot \left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)\right) \cdot h}{\ell}\right) \]

      4. lower-*.f64 68.1

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot M}{d + d} \cdot \left(0.25 \cdot \frac{M \cdot D}{d}\right)\right) \cdot h}{\ell}\right) \]

   10. Applied rewrites 68.1%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D \cdot M}{d + d} \cdot \color{blue}{\left(0.25 \cdot \frac{M \cdot D}{d}\right)}\right) \cdot h}{\ell}\right) \]

   if 1.3e-156 < h

   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. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 68.9

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

   4. Applied rewrites 68.9%

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

   6. Applied rewrites 67.4%

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

   8. Applied rewrites 67.2%

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

Alternative 7: 62.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(\ell \cdot \ell\right) \cdot \ell\\ t_1 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{d \cdot d}\right) \cdot 0.125\right) \cdot h}{\ell}\right)\\ \mathbf{if}\;d \leq -6.6 \cdot 10^{+154}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-302}:\\ \;\;\;\;\left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{t\_0}}\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{-84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \sqrt{\ell \cdot h}}{d}, -0.125, \sqrt{\frac{t\_0}{h}} \cdot d\right)}{\ell \cdot \ell}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+145}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* l l) l))
        (t_1
         (*
          (* (sqrt (/ d l)) (sqrt (/ d h)))
          (- 1.0 (/ (* (* (* (* (* M M) D) (/ D (* d d))) 0.125) h) l)))))
   (if (<= d -6.6e+154)
     (* (- (sqrt (/ 1.0 (* l h)))) d)
     (if (<= d -8.5e-157)
       t_1
       (if (<= d -2.6e-302)
         (* (* 0.125 (* (* D D) (* M (/ M d)))) (sqrt (/ h t_0)))
         (if (<= d 1.02e-84)
           (/
            (fma
             (/ (* (* (* D M) (* D M)) (sqrt (* l h))) d)
             -0.125
             (* (sqrt (/ t_0 h)) d))
            (* l l))
           (if (<= d 3.8e+145) t_1 (/ (* 1.0 d) (* (sqrt h) (sqrt l))))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (l * l) * l;
	double t_1 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((((M * M) * D) * (D / (d * d))) * 0.125) * h) / l));
	double tmp;
	if (d <= -6.6e+154) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else if (d <= -8.5e-157) {
		tmp = t_1;
	} else if (d <= -2.6e-302) {
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * sqrt((h / t_0));
	} else if (d <= 1.02e-84) {
		tmp = fma(((((D * M) * (D * M)) * sqrt((l * h))) / d), -0.125, (sqrt((t_0 / h)) * d)) / (l * l);
	} else if (d <= 3.8e+145) {
		tmp = t_1;
	} else {
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(l * l) * l)
	t_1 = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * M) * D) * Float64(D / Float64(d * d))) * 0.125) * h) / l)))
	tmp = 0.0
	if (d <= -6.6e+154)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	elseif (d <= -8.5e-157)
		tmp = t_1;
	elseif (d <= -2.6e-302)
		tmp = Float64(Float64(0.125 * Float64(Float64(D * D) * Float64(M * Float64(M / d)))) * sqrt(Float64(h / t_0)));
	elseif (d <= 1.02e-84)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(D * M) * Float64(D * M)) * sqrt(Float64(l * h))) / d), -0.125, Float64(sqrt(Float64(t_0 / h)) * d)) / Float64(l * l));
	elseif (d <= 3.8e+145)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 * d) / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * D), $MachinePrecision] * N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.6e+154], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], If[LessEqual[d, -8.5e-157], t$95$1, If[LessEqual[d, -2.6e-302], N[(N[(0.125 * N[(N[(D * D), $MachinePrecision] * N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.02e-84], N[(N[(N[(N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125 + N[(N[Sqrt[N[(t$95$0 / h), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.8e+145], t$95$1, N[(N[(1.0 * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -6.6e154

   1. Initial program 68.4%

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

   2. 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 6.5

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

   4. Applied rewrites 6.5%

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

   5. Taylor expanded in h 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 70.7

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

   7. Applied rewrites 70.7%

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

   if -6.6e154 < d < -8.49999999999999976e-157 or 1.02000000000000004e-84 < d < 3.80000000000000012e145

   1. Initial program 77.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 79.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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 79.0

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

   5. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 79.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-+.f64 79.7

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

   7. Applied rewrites 79.7%

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

   8. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot h}{\ell}\right) \]
   9. 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. *-commutative N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 69.1

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

   10. Applied rewrites 69.1%

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

   if -8.49999999999999976e-157 < d < -2.60000000000000011e-302

   1. Initial program 42.9%

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

   2. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 38.6%

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

   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

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

      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. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow3 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-sqrt.f64 38.6

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

   7. Applied rewrites 38.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 42.4

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

   9. Applied rewrites 42.4%

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

   if -2.60000000000000011e-302 < d < 1.02000000000000004e-84

   1. Initial program 49.3%

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

   2. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 32.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unswap-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 40.7

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

   6. Applied rewrites 40.7%

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

   if 3.80000000000000012e145 < d

   1. Initial program 74.0%

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

   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 68.5

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

   4. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   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. metadata-eval N/A

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

      6. *-commutative 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. lift-sqrt.f64 N/A

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

      10. lift-*.f64 68.5

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

   6. Applied rewrites 68.5%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   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 68.5

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

   8. Applied rewrites 68.5%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
   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 77.2

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

   10. Applied rewrites 77.2%

       \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 8: 62.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(\ell \cdot \ell\right) \cdot \ell\\ t_1 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{d \cdot d}\right) \cdot 0.125\right) \cdot h}{\ell}\right)\\ \mathbf{if}\;d \leq -6.6 \cdot 10^{+154}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-302}:\\ \;\;\;\;\left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{t\_0}}\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{-152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(M \cdot \left(M \cdot \left(D \cdot D\right)\right)\right) \cdot \sqrt{\ell \cdot h}}{d}, -0.125, \sqrt{\frac{t\_0}{h}} \cdot d\right)}{\ell \cdot \ell}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+145}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* l l) l))
        (t_1
         (*
          (* (sqrt (/ d l)) (sqrt (/ d h)))
          (- 1.0 (/ (* (* (* (* (* M M) D) (/ D (* d d))) 0.125) h) l)))))
   (if (<= d -6.6e+154)
     (* (- (sqrt (/ 1.0 (* l h)))) d)
     (if (<= d -8.5e-157)
       t_1
       (if (<= d -2.6e-302)
         (* (* 0.125 (* (* D D) (* M (/ M d)))) (sqrt (/ h t_0)))
         (if (<= d 7.6e-152)
           (/
            (fma
             (/ (* (* M (* M (* D D))) (sqrt (* l h))) d)
             -0.125
             (* (sqrt (/ t_0 h)) d))
            (* l l))
           (if (<= d 3.8e+145) t_1 (/ (* 1.0 d) (* (sqrt h) (sqrt l))))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (l * l) * l;
	double t_1 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((((M * M) * D) * (D / (d * d))) * 0.125) * h) / l));
	double tmp;
	if (d <= -6.6e+154) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else if (d <= -8.5e-157) {
		tmp = t_1;
	} else if (d <= -2.6e-302) {
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * sqrt((h / t_0));
	} else if (d <= 7.6e-152) {
		tmp = fma((((M * (M * (D * D))) * sqrt((l * h))) / d), -0.125, (sqrt((t_0 / h)) * d)) / (l * l);
	} else if (d <= 3.8e+145) {
		tmp = t_1;
	} else {
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(l * l) * l)
	t_1 = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * M) * D) * Float64(D / Float64(d * d))) * 0.125) * h) / l)))
	tmp = 0.0
	if (d <= -6.6e+154)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	elseif (d <= -8.5e-157)
		tmp = t_1;
	elseif (d <= -2.6e-302)
		tmp = Float64(Float64(0.125 * Float64(Float64(D * D) * Float64(M * Float64(M / d)))) * sqrt(Float64(h / t_0)));
	elseif (d <= 7.6e-152)
		tmp = Float64(fma(Float64(Float64(Float64(M * Float64(M * Float64(D * D))) * sqrt(Float64(l * h))) / d), -0.125, Float64(sqrt(Float64(t_0 / h)) * d)) / Float64(l * l));
	elseif (d <= 3.8e+145)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 * d) / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * D), $MachinePrecision] * N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.6e+154], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], If[LessEqual[d, -8.5e-157], t$95$1, If[LessEqual[d, -2.6e-302], N[(N[(0.125 * N[(N[(D * D), $MachinePrecision] * N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.6e-152], N[(N[(N[(N[(N[(M * N[(M * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125 + N[(N[Sqrt[N[(t$95$0 / h), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.8e+145], t$95$1, N[(N[(1.0 * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -6.6e154

   1. Initial program 68.4%

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

   2. 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 6.5

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

   4. Applied rewrites 6.5%

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

   5. Taylor expanded in h 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 70.7

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

   7. Applied rewrites 70.7%

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

   if -6.6e154 < d < -8.49999999999999976e-157 or 7.60000000000000024e-152 < d < 3.80000000000000012e145

   1. Initial program 75.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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 77.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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 77.2

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

   5. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 77.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-+.f64 78.0

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

   7. Applied rewrites 78.0%

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

   8. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot h}{\ell}\right) \]
   9. 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. *-commutative N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 67.6

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

   10. Applied rewrites 67.6%

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

   if -8.49999999999999976e-157 < d < -2.60000000000000011e-302

   1. Initial program 42.9%

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

   2. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 38.6%

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

   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

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

      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. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow3 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-sqrt.f64 38.6

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

   7. Applied rewrites 38.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 42.4

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

   9. Applied rewrites 42.4%

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

   if -2.60000000000000011e-302 < d < 7.60000000000000024e-152

   1. Initial program 43.1%

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

   2. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 31.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 34.8

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

   6. Applied rewrites 34.8%

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

   if 3.80000000000000012e145 < d

   1. Initial program 74.0%

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

   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 68.5

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

   4. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   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. metadata-eval N/A

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

      6. *-commutative 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. lift-sqrt.f64 N/A

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

      10. lift-*.f64 68.5

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

   6. Applied rewrites 68.5%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   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 68.5

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

   8. Applied rewrites 68.5%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
   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 77.2

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

   10. Applied rewrites 77.2%

       \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 62.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{d \cdot d}\right) \cdot 0.125\right) \cdot h}{\ell}\right)\\ t_1 := \sqrt{\frac{1}{\ell \cdot h}}\\ t_2 := \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{if}\;d \leq -6.6 \cdot 10^{+154}:\\ \;\;\;\;\left(-t\_1\right) \cdot d\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-157}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-281}:\\ \;\;\;\;\left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot t\_2\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-84}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot D}{d \cdot d} \cdot t\_2, -0.125, t\_1\right) \cdot d\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+145}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (sqrt (/ d l)) (sqrt (/ d h)))
          (- 1.0 (/ (* (* (* (* (* M M) D) (/ D (* d d))) 0.125) h) l))))
        (t_1 (sqrt (/ 1.0 (* l h))))
        (t_2 (sqrt (/ h (* (* l l) l)))))
   (if (<= d -6.6e+154)
     (* (- t_1) d)
     (if (<= d -8.5e-157)
       t_0
       (if (<= d 9.5e-281)
         (* (* 0.125 (* (* D D) (* M (/ M d)))) t_2)
         (if (<= d 1.05e-84)
           (* (fma (* (/ (* (* M (* M D)) D) (* d d)) t_2) -0.125 t_1) d)
           (if (<= d 3.8e+145) t_0 (/ (* 1.0 d) (* (sqrt h) (sqrt l))))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((((M * M) * D) * (D / (d * d))) * 0.125) * h) / l));
	double t_1 = sqrt((1.0 / (l * h)));
	double t_2 = sqrt((h / ((l * l) * l)));
	double tmp;
	if (d <= -6.6e+154) {
		tmp = -t_1 * d;
	} else if (d <= -8.5e-157) {
		tmp = t_0;
	} else if (d <= 9.5e-281) {
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * t_2;
	} else if (d <= 1.05e-84) {
		tmp = fma(((((M * (M * D)) * D) / (d * d)) * t_2), -0.125, t_1) * d;
	} else if (d <= 3.8e+145) {
		tmp = t_0;
	} else {
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * M) * D) * Float64(D / Float64(d * d))) * 0.125) * h) / l)))
	t_1 = sqrt(Float64(1.0 / Float64(l * h)))
	t_2 = sqrt(Float64(h / Float64(Float64(l * l) * l)))
	tmp = 0.0
	if (d <= -6.6e+154)
		tmp = Float64(Float64(-t_1) * d);
	elseif (d <= -8.5e-157)
		tmp = t_0;
	elseif (d <= 9.5e-281)
		tmp = Float64(Float64(0.125 * Float64(Float64(D * D) * Float64(M * Float64(M / d)))) * t_2);
	elseif (d <= 1.05e-84)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(M * Float64(M * D)) * D) / Float64(d * d)) * t_2), -0.125, t_1) * d);
	elseif (d <= 3.8e+145)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 * d) / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * D), $MachinePrecision] * N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -6.6e+154], N[((-t$95$1) * d), $MachinePrecision], If[LessEqual[d, -8.5e-157], t$95$0, If[LessEqual[d, 9.5e-281], N[(N[(0.125 * N[(N[(D * D), $MachinePrecision] * N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[d, 1.05e-84], N[(N[(N[(N[(N[(N[(M * N[(M * D), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * -0.125 + t$95$1), $MachinePrecision] * d), $MachinePrecision], If[LessEqual[d, 3.8e+145], t$95$0, N[(N[(1.0 * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -6.6e154

   1. Initial program 68.4%

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

   2. 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 6.5

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

   4. Applied rewrites 6.5%

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

   5. Taylor expanded in h 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 70.7

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

   7. Applied rewrites 70.7%

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

   if -6.6e154 < d < -8.49999999999999976e-157 or 1.04999999999999999e-84 < d < 3.80000000000000012e145

   1. Initial program 77.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 79.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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 79.0

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

   5. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 79.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-+.f64 79.7

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

   7. Applied rewrites 79.7%

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

   8. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot h}{\ell}\right) \]
   9. 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. *-commutative N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 69.1

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

   10. Applied rewrites 69.1%

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

   if -8.49999999999999976e-157 < d < 9.5000000000000003e-281

   1. Initial program 40.0%

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

   2. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 33.5%

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

   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

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

      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. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow3 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-sqrt.f64 33.5

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

   7. Applied rewrites 33.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 36.8

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

   9. Applied rewrites 36.8%

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

   if 9.5000000000000003e-281 < d < 1.04999999999999999e-84

   1. Initial program 53.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 52.3%

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

   4. Taylor expanded in d around inf

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

   6. Applied rewrites 38.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 41.6

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

   8. Applied rewrites 41.6%

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

   if 3.80000000000000012e145 < d

   1. Initial program 74.0%

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

   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 68.5

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

   4. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   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. metadata-eval N/A

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

      6. *-commutative 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. lift-sqrt.f64 N/A

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

      10. lift-*.f64 68.5

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

   6. Applied rewrites 68.5%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   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 68.5

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

   8. Applied rewrites 68.5%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
   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 77.2

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

   10. Applied rewrites 77.2%

       \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 10: 61.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{d \cdot d}\right) \cdot 0.125\right) \cdot h}{\ell}\right)\\ t_1 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;d \leq -6.6 \cdot 10^{+154}:\\ \;\;\;\;\left(-t\_1\right) \cdot d\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-157}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 9.4 \cdot 10^{-279}:\\ \;\;\;\;\left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{elif}\;d \leq 9000000000:\\ \;\;\;\;\left(1 - \left(0.5 \cdot \left(M \cdot \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot M\right)}{d \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \cdot \left(t\_1 \cdot d\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+145}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (sqrt (/ d l)) (sqrt (/ d h)))
          (- 1.0 (/ (* (* (* (* (* M M) D) (/ D (* d d))) 0.125) h) l))))
        (t_1 (sqrt (/ 1.0 (* l h)))))
   (if (<= d -6.6e+154)
     (* (- t_1) d)
     (if (<= d -8.5e-157)
       t_0
       (if (<= d 9.4e-279)
         (* (* 0.125 (* (* D D) (* M (/ M d)))) (sqrt (/ h (* (* l l) l))))
         (if (<= d 9000000000.0)
           (*
            (-
             1.0
             (* (* 0.5 (* M (/ (* 0.25 (* (* D D) M)) (* d d)))) (/ h l)))
            (* t_1 d))
           (if (<= d 3.8e+145) t_0 (/ (* 1.0 d) (* (sqrt h) (sqrt l))))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((((M * M) * D) * (D / (d * d))) * 0.125) * h) / l));
	double t_1 = sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -6.6e+154) {
		tmp = -t_1 * d;
	} else if (d <= -8.5e-157) {
		tmp = t_0;
	} else if (d <= 9.4e-279) {
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * sqrt((h / ((l * l) * l)));
	} else if (d <= 9000000000.0) {
		tmp = (1.0 - ((0.5 * (M * ((0.25 * ((D * D) * M)) / (d * d)))) * (h / l))) * (t_1 * d);
	} else if (d <= 3.8e+145) {
		tmp = t_0;
	} else {
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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, 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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((((((m * m) * d_1) * (d_1 / (d * d))) * 0.125d0) * h) / l))
    t_1 = sqrt((1.0d0 / (l * h)))
    if (d <= (-6.6d+154)) then
        tmp = -t_1 * d
    else if (d <= (-8.5d-157)) then
        tmp = t_0
    else if (d <= 9.4d-279) then
        tmp = (0.125d0 * ((d_1 * d_1) * (m * (m / d)))) * sqrt((h / ((l * l) * l)))
    else if (d <= 9000000000.0d0) then
        tmp = (1.0d0 - ((0.5d0 * (m * ((0.25d0 * ((d_1 * d_1) * m)) / (d * d)))) * (h / l))) * (t_1 * d)
    else if (d <= 3.8d+145) then
        tmp = t_0
    else
        tmp = (1.0d0 * d) / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((((((M * M) * D) * (D / (d * d))) * 0.125) * h) / l));
	double t_1 = Math.sqrt((1.0 / (l * h)));
	double tmp;
	if (d <= -6.6e+154) {
		tmp = -t_1 * d;
	} else if (d <= -8.5e-157) {
		tmp = t_0;
	} else if (d <= 9.4e-279) {
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * Math.sqrt((h / ((l * l) * l)));
	} else if (d <= 9000000000.0) {
		tmp = (1.0 - ((0.5 * (M * ((0.25 * ((D * D) * M)) / (d * d)))) * (h / l))) * (t_1 * d);
	} else if (d <= 3.8e+145) {
		tmp = t_0;
	} else {
		tmp = (1.0 * d) / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((((((M * M) * D) * (D / (d * d))) * 0.125) * h) / l))
	t_1 = math.sqrt((1.0 / (l * h)))
	tmp = 0
	if d <= -6.6e+154:
		tmp = -t_1 * d
	elif d <= -8.5e-157:
		tmp = t_0
	elif d <= 9.4e-279:
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * math.sqrt((h / ((l * l) * l)))
	elif d <= 9000000000.0:
		tmp = (1.0 - ((0.5 * (M * ((0.25 * ((D * D) * M)) / (d * d)))) * (h / l))) * (t_1 * d)
	elif d <= 3.8e+145:
		tmp = t_0
	else:
		tmp = (1.0 * d) / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * M) * D) * Float64(D / Float64(d * d))) * 0.125) * h) / l)))
	t_1 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (d <= -6.6e+154)
		tmp = Float64(Float64(-t_1) * d);
	elseif (d <= -8.5e-157)
		tmp = t_0;
	elseif (d <= 9.4e-279)
		tmp = Float64(Float64(0.125 * Float64(Float64(D * D) * Float64(M * Float64(M / d)))) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	elseif (d <= 9000000000.0)
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 * Float64(M * Float64(Float64(0.25 * Float64(Float64(D * D) * M)) / Float64(d * d)))) * Float64(h / l))) * Float64(t_1 * d));
	elseif (d <= 3.8e+145)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 * d) / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((((M * M) * D) * (D / (d * d))) * 0.125) * h) / l));
	t_1 = sqrt((1.0 / (l * h)));
	tmp = 0.0;
	if (d <= -6.6e+154)
		tmp = -t_1 * d;
	elseif (d <= -8.5e-157)
		tmp = t_0;
	elseif (d <= 9.4e-279)
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * sqrt((h / ((l * l) * l)));
	elseif (d <= 9000000000.0)
		tmp = (1.0 - ((0.5 * (M * ((0.25 * ((D * D) * M)) / (d * d)))) * (h / l))) * (t_1 * d);
	elseif (d <= 3.8e+145)
		tmp = t_0;
	else
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * D), $MachinePrecision] * N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -6.6e+154], N[((-t$95$1) * d), $MachinePrecision], If[LessEqual[d, -8.5e-157], t$95$0, If[LessEqual[d, 9.4e-279], N[(N[(0.125 * N[(N[(D * D), $MachinePrecision] * N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9000000000.0], N[(N[(1.0 - N[(N[(0.5 * N[(M * N[(N[(0.25 * N[(N[(D * D), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.8e+145], t$95$0, N[(N[(1.0 * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -6.6e154

   1. Initial program 68.4%

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

   2. 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 6.5

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

   4. Applied rewrites 6.5%

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

   5. Taylor expanded in h 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 70.7

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

   7. Applied rewrites 70.7%

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

   if -6.6e154 < d < -8.49999999999999976e-157 or 9e9 < d < 3.80000000000000012e145

   1. Initial program 77.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 78.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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 78.8

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

   5. Applied rewrites 78.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 78.8

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-+.f64 79.5

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

   7. Applied rewrites 79.5%

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

   8. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot h}{\ell}\right) \]
   9. 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. *-commutative N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 68.7

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

   10. Applied rewrites 68.7%

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

   if -8.49999999999999976e-157 < d < 9.3999999999999997e-279

   1. Initial program 39.9%

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

   2. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 33.4%

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

   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

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

      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. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow3 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-sqrt.f64 33.4

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

   7. Applied rewrites 33.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 36.7

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

   9. Applied rewrites 36.7%

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

   if 9.3999999999999997e-279 < d < 9e9

   1. Initial program 61.2%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 66.9

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

   4. Applied rewrites 66.9%

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

   6. Applied rewrites 64.1%

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

   7. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 51.2

         \[\leadsto \left(1 - \left(0.5 \cdot \left(M \cdot \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot M\right)}{d \cdot \color{blue}{d}}\right)\right) \cdot \frac{h}{\ell}\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \]

   9. Applied rewrites 51.2%

      \[\leadsto \left(1 - \left(0.5 \cdot \left(M \cdot \color{blue}{\frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot M\right)}{d \cdot d}}\right)\right) \cdot \frac{h}{\ell}\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \]

   if 3.80000000000000012e145 < d

   1. Initial program 74.0%

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

   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 68.5

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

   4. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   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. metadata-eval N/A

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

      6. *-commutative 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. lift-sqrt.f64 N/A

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

      10. lift-*.f64 68.5

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

   6. Applied rewrites 68.5%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   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 68.5

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

   8. Applied rewrites 68.5%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
   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 77.2

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

   10. Applied rewrites 77.2%

       \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 11: 61.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{d \cdot d}\right) \cdot 0.125\right) \cdot h}{\ell}\right)\\ t_1 := \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{if}\;d \leq -6.6 \cdot 10^{+154}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-157}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-302}:\\ \;\;\;\;\left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot t\_1\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{-152}:\\ \;\;\;\;\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot t\_1\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+145}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (sqrt (/ d l)) (sqrt (/ d h)))
          (- 1.0 (/ (* (* (* (* (* M M) D) (/ D (* d d))) 0.125) h) l))))
        (t_1 (sqrt (/ h (* (* l l) l)))))
   (if (<= d -6.6e+154)
     (* (- (sqrt (/ 1.0 (* l h)))) d)
     (if (<= d -8.5e-157)
       t_0
       (if (<= d -2.6e-302)
         (* (* 0.125 (* (* D D) (* M (/ M d)))) t_1)
         (if (<= d 7.6e-152)
           (* (* -0.125 (* (* D D) (/ (* M M) d))) t_1)
           (if (<= d 3.8e+145) t_0 (/ (* 1.0 d) (* (sqrt h) (sqrt l))))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((((M * M) * D) * (D / (d * d))) * 0.125) * h) / l));
	double t_1 = sqrt((h / ((l * l) * l)));
	double tmp;
	if (d <= -6.6e+154) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else if (d <= -8.5e-157) {
		tmp = t_0;
	} else if (d <= -2.6e-302) {
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * t_1;
	} else if (d <= 7.6e-152) {
		tmp = (-0.125 * ((D * D) * ((M * M) / d))) * t_1;
	} else if (d <= 3.8e+145) {
		tmp = t_0;
	} else {
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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, 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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((((((m * m) * d_1) * (d_1 / (d * d))) * 0.125d0) * h) / l))
    t_1 = sqrt((h / ((l * l) * l)))
    if (d <= (-6.6d+154)) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else if (d <= (-8.5d-157)) then
        tmp = t_0
    else if (d <= (-2.6d-302)) then
        tmp = (0.125d0 * ((d_1 * d_1) * (m * (m / d)))) * t_1
    else if (d <= 7.6d-152) then
        tmp = ((-0.125d0) * ((d_1 * d_1) * ((m * m) / d))) * t_1
    else if (d <= 3.8d+145) then
        tmp = t_0
    else
        tmp = (1.0d0 * d) / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((((((M * M) * D) * (D / (d * d))) * 0.125) * h) / l));
	double t_1 = Math.sqrt((h / ((l * l) * l)));
	double tmp;
	if (d <= -6.6e+154) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else if (d <= -8.5e-157) {
		tmp = t_0;
	} else if (d <= -2.6e-302) {
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * t_1;
	} else if (d <= 7.6e-152) {
		tmp = (-0.125 * ((D * D) * ((M * M) / d))) * t_1;
	} else if (d <= 3.8e+145) {
		tmp = t_0;
	} else {
		tmp = (1.0 * d) / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((((((M * M) * D) * (D / (d * d))) * 0.125) * h) / l))
	t_1 = math.sqrt((h / ((l * l) * l)))
	tmp = 0
	if d <= -6.6e+154:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	elif d <= -8.5e-157:
		tmp = t_0
	elif d <= -2.6e-302:
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * t_1
	elif d <= 7.6e-152:
		tmp = (-0.125 * ((D * D) * ((M * M) / d))) * t_1
	elif d <= 3.8e+145:
		tmp = t_0
	else:
		tmp = (1.0 * d) / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * M) * D) * Float64(D / Float64(d * d))) * 0.125) * h) / l)))
	t_1 = sqrt(Float64(h / Float64(Float64(l * l) * l)))
	tmp = 0.0
	if (d <= -6.6e+154)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	elseif (d <= -8.5e-157)
		tmp = t_0;
	elseif (d <= -2.6e-302)
		tmp = Float64(Float64(0.125 * Float64(Float64(D * D) * Float64(M * Float64(M / d)))) * t_1);
	elseif (d <= 7.6e-152)
		tmp = Float64(Float64(-0.125 * Float64(Float64(D * D) * Float64(Float64(M * M) / d))) * t_1);
	elseif (d <= 3.8e+145)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 * d) / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((((M * M) * D) * (D / (d * d))) * 0.125) * h) / l));
	t_1 = sqrt((h / ((l * l) * l)));
	tmp = 0.0;
	if (d <= -6.6e+154)
		tmp = -sqrt((1.0 / (l * h))) * d;
	elseif (d <= -8.5e-157)
		tmp = t_0;
	elseif (d <= -2.6e-302)
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * t_1;
	elseif (d <= 7.6e-152)
		tmp = (-0.125 * ((D * D) * ((M * M) / d))) * t_1;
	elseif (d <= 3.8e+145)
		tmp = t_0;
	else
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * D), $MachinePrecision] * N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -6.6e+154], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], If[LessEqual[d, -8.5e-157], t$95$0, If[LessEqual[d, -2.6e-302], N[(N[(0.125 * N[(N[(D * D), $MachinePrecision] * N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 7.6e-152], N[(N[(-0.125 * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 3.8e+145], t$95$0, N[(N[(1.0 * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -6.6e154

   1. Initial program 68.4%

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

   2. 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 6.5

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

   4. Applied rewrites 6.5%

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

   5. Taylor expanded in h 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 70.7

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

   7. Applied rewrites 70.7%

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

   if -6.6e154 < d < -8.49999999999999976e-157 or 7.60000000000000024e-152 < d < 3.80000000000000012e145

   1. Initial program 75.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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 77.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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 77.2

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

   5. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 77.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-+.f64 78.0

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

   7. Applied rewrites 78.0%

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

   8. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot h}{\ell}\right) \]
   9. 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. *-commutative N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 67.6

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

   10. Applied rewrites 67.6%

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

   if -8.49999999999999976e-157 < d < -2.60000000000000011e-302

   1. Initial program 42.9%

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

   2. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 38.6%

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

   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

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

      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. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow3 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-sqrt.f64 38.6

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

   7. Applied rewrites 38.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 42.4

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

   9. Applied rewrites 42.4%

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

   if -2.60000000000000011e-302 < d < 7.60000000000000024e-152

   1. Initial program 43.1%

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

   2. Taylor expanded in d around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 39.8

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

   4. Applied rewrites 39.8%

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

   if 3.80000000000000012e145 < d

   1. Initial program 74.0%

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

   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 68.5

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

   4. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   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. metadata-eval N/A

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

      6. *-commutative 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. lift-sqrt.f64 N/A

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

      10. lift-*.f64 68.5

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

   6. Applied rewrites 68.5%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   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 68.5

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

   8. Applied rewrites 68.5%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
   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 77.2

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

   10. Applied rewrites 77.2%

       \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 12: 60.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.125\right) \cdot h}{\ell}\right)\\ t_1 := \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{if}\;d \leq -5.8 \cdot 10^{+153}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-157}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-302}:\\ \;\;\;\;\left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot t\_1\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-152}:\\ \;\;\;\;\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot t\_1\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+20}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (sqrt (/ d l)) (sqrt (/ d h)))
          (- 1.0 (/ (* (* (/ (* (* M M) (* D D)) (* d d)) 0.125) h) l))))
        (t_1 (sqrt (/ h (* (* l l) l)))))
   (if (<= d -5.8e+153)
     (* (- (sqrt (/ 1.0 (* l h)))) d)
     (if (<= d -8.5e-157)
       t_0
       (if (<= d -2.6e-302)
         (* (* 0.125 (* (* D D) (* M (/ M d)))) t_1)
         (if (<= d 7.5e-152)
           (* (* -0.125 (* (* D D) (/ (* M M) d))) t_1)
           (if (<= d 1.7e+20) t_0 (/ (* 1.0 d) (* (sqrt h) (sqrt l))))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((((M * M) * (D * D)) / (d * d)) * 0.125) * h) / l));
	double t_1 = sqrt((h / ((l * l) * l)));
	double tmp;
	if (d <= -5.8e+153) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else if (d <= -8.5e-157) {
		tmp = t_0;
	} else if (d <= -2.6e-302) {
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * t_1;
	} else if (d <= 7.5e-152) {
		tmp = (-0.125 * ((D * D) * ((M * M) / d))) * t_1;
	} else if (d <= 1.7e+20) {
		tmp = t_0;
	} else {
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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, 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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((((((m * m) * (d_1 * d_1)) / (d * d)) * 0.125d0) * h) / l))
    t_1 = sqrt((h / ((l * l) * l)))
    if (d <= (-5.8d+153)) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else if (d <= (-8.5d-157)) then
        tmp = t_0
    else if (d <= (-2.6d-302)) then
        tmp = (0.125d0 * ((d_1 * d_1) * (m * (m / d)))) * t_1
    else if (d <= 7.5d-152) then
        tmp = ((-0.125d0) * ((d_1 * d_1) * ((m * m) / d))) * t_1
    else if (d <= 1.7d+20) then
        tmp = t_0
    else
        tmp = (1.0d0 * d) / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((((((M * M) * (D * D)) / (d * d)) * 0.125) * h) / l));
	double t_1 = Math.sqrt((h / ((l * l) * l)));
	double tmp;
	if (d <= -5.8e+153) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else if (d <= -8.5e-157) {
		tmp = t_0;
	} else if (d <= -2.6e-302) {
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * t_1;
	} else if (d <= 7.5e-152) {
		tmp = (-0.125 * ((D * D) * ((M * M) / d))) * t_1;
	} else if (d <= 1.7e+20) {
		tmp = t_0;
	} else {
		tmp = (1.0 * d) / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((((((M * M) * (D * D)) / (d * d)) * 0.125) * h) / l))
	t_1 = math.sqrt((h / ((l * l) * l)))
	tmp = 0
	if d <= -5.8e+153:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	elif d <= -8.5e-157:
		tmp = t_0
	elif d <= -2.6e-302:
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * t_1
	elif d <= 7.5e-152:
		tmp = (-0.125 * ((D * D) * ((M * M) / d))) * t_1
	elif d <= 1.7e+20:
		tmp = t_0
	else:
		tmp = (1.0 * d) / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * M) * Float64(D * D)) / Float64(d * d)) * 0.125) * h) / l)))
	t_1 = sqrt(Float64(h / Float64(Float64(l * l) * l)))
	tmp = 0.0
	if (d <= -5.8e+153)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	elseif (d <= -8.5e-157)
		tmp = t_0;
	elseif (d <= -2.6e-302)
		tmp = Float64(Float64(0.125 * Float64(Float64(D * D) * Float64(M * Float64(M / d)))) * t_1);
	elseif (d <= 7.5e-152)
		tmp = Float64(Float64(-0.125 * Float64(Float64(D * D) * Float64(Float64(M * M) / d))) * t_1);
	elseif (d <= 1.7e+20)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 * d) / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((((M * M) * (D * D)) / (d * d)) * 0.125) * h) / l));
	t_1 = sqrt((h / ((l * l) * l)));
	tmp = 0.0;
	if (d <= -5.8e+153)
		tmp = -sqrt((1.0 / (l * h))) * d;
	elseif (d <= -8.5e-157)
		tmp = t_0;
	elseif (d <= -2.6e-302)
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * t_1;
	elseif (d <= 7.5e-152)
		tmp = (-0.125 * ((D * D) * ((M * M) / d))) * t_1;
	elseif (d <= 1.7e+20)
		tmp = t_0;
	else
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -5.8e+153], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], If[LessEqual[d, -8.5e-157], t$95$0, If[LessEqual[d, -2.6e-302], N[(N[(0.125 * N[(N[(D * D), $MachinePrecision] * N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 7.5e-152], N[(N[(-0.125 * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 1.7e+20], t$95$0, N[(N[(1.0 * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -5.80000000000000004e153

   1. Initial program 68.4%

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

   2. 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 6.5

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

   4. Applied rewrites 6.5%

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

   5. Taylor expanded in h 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 70.8

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

   7. Applied rewrites 70.8%

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

   if -5.80000000000000004e153 < d < -8.49999999999999976e-157 or 7.5e-152 < d < 1.7e20

   1. Initial program 75.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 76.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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. 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 76.4

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

   5. Applied rewrites 76.4%

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

   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

   8. Applied rewrites 63.2%

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

   if -8.49999999999999976e-157 < d < -2.60000000000000011e-302

   1. Initial program 42.9%

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

   2. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 38.6%

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

   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

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

      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. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow3 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-sqrt.f64 38.6

          \[\leadsto \left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   7. Applied rewrites 38.6%

      \[\leadsto \left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \color{blue}{\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      3. associate-/l* N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      5. lower-/.f64 42.4

         \[\leadsto \left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   9. Applied rewrites 42.4%

      \[\leadsto \left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   if -2.60000000000000011e-302 < d < 7.5e-152

   1. Initial program 43.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \]

      4. associate-/l* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{\color{blue}{3}}}} \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      13. unpow3 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      14. unpow2 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{2} \cdot \ell}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{2} \cdot \ell}} \]

      16. unpow2 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      17. lower-*.f64 39.8

          \[\leadsto \left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   4. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}} \]

   if 1.7e20 < d

   1. Initial program 75.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 58.3

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   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. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      6. *-commutative 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. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lift-*.f64 58.5

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 58.5%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   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 58.6

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

   8. Applied rewrites 58.6%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
   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 68.5

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}} \]

   10. Applied rewrites 68.5%

       \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 13: 56.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{+149}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-49}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-302}:\\ \;\;\;\;\left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{\sqrt{\ell \cdot h} \cdot \left(\left(\left(M \cdot M\right) \cdot D\right) \cdot D\right)}{d} \cdot -0.125}{\ell \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -8.5e+149)
   (* (- (sqrt (/ 1.0 (* l h)))) d)
   (if (<= d -3.5e-49)
     (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
     (if (<= d -2.6e-302)
       (* (* 0.125 (* (* D D) (* M (/ M d)))) (sqrt (/ h (* (* l l) l))))
       (if (<= d 2.2e-133)
         (/ (* (/ (* (sqrt (* l h)) (* (* (* M M) D) D)) d) -0.125) (* l l))
         (/ (* 1.0 d) (* (sqrt h) (sqrt l))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -8.5e+149) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else if (d <= -3.5e-49) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else if (d <= -2.6e-302) {
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * sqrt((h / ((l * l) * l)));
	} else if (d <= 2.2e-133) {
		tmp = (((sqrt((l * h)) * (((M * M) * D) * D)) / d) * -0.125) / (l * l);
	} else {
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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, 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
    real(8) :: tmp
    if (d <= (-8.5d+149)) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else if (d <= (-3.5d-49)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    else if (d <= (-2.6d-302)) then
        tmp = (0.125d0 * ((d_1 * d_1) * (m * (m / d)))) * sqrt((h / ((l * l) * l)))
    else if (d <= 2.2d-133) then
        tmp = (((sqrt((l * h)) * (((m * m) * d_1) * d_1)) / d) * (-0.125d0)) / (l * l)
    else
        tmp = (1.0d0 * d) / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -8.5e+149) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else if (d <= -3.5e-49) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else if (d <= -2.6e-302) {
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * Math.sqrt((h / ((l * l) * l)));
	} else if (d <= 2.2e-133) {
		tmp = (((Math.sqrt((l * h)) * (((M * M) * D) * D)) / d) * -0.125) / (l * l);
	} else {
		tmp = (1.0 * d) / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -8.5e+149:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	elif d <= -3.5e-49:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	elif d <= -2.6e-302:
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * math.sqrt((h / ((l * l) * l)))
	elif d <= 2.2e-133:
		tmp = (((math.sqrt((l * h)) * (((M * M) * D) * D)) / d) * -0.125) / (l * l)
	else:
		tmp = (1.0 * d) / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -8.5e+149)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	elseif (d <= -3.5e-49)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	elseif (d <= -2.6e-302)
		tmp = Float64(Float64(0.125 * Float64(Float64(D * D) * Float64(M * Float64(M / d)))) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	elseif (d <= 2.2e-133)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(l * h)) * Float64(Float64(Float64(M * M) * D) * D)) / d) * -0.125) / Float64(l * l));
	else
		tmp = Float64(Float64(1.0 * d) / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -8.5e+149)
		tmp = -sqrt((1.0 / (l * h))) * d;
	elseif (d <= -3.5e-49)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	elseif (d <= -2.6e-302)
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * sqrt((h / ((l * l) * l)));
	elseif (d <= 2.2e-133)
		tmp = (((sqrt((l * h)) * (((M * M) * D) * D)) / d) * -0.125) / (l * l);
	else
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -8.5e+149], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], If[LessEqual[d, -3.5e-49], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[d, -2.6e-302], N[(N[(0.125 * N[(N[(D * D), $MachinePrecision] * N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.2e-133], N[(N[(N[(N[(N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * d), $MachinePrecision] / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -8.49999999999999956e149

   1. Initial program 68.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 6.5

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 6.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   5. Taylor expanded in h 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 70.9

          \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   7. Applied rewrites 70.9%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if -8.49999999999999956e149 < d < -3.50000000000000006e-49

   1. Initial program 81.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 83.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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. 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 83.4

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied rewrites 83.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.4%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if -3.50000000000000006e-49 < d < -2.60000000000000011e-302

   1. Initial program 53.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \color{blue}{\frac{-1}{8}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \color{blue}{\frac{-1}{8}} \]

   4. Applied rewrites 35.7%

      \[\leadsto \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot -0.125} \]

   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

         \[\leadsto \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      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. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      6. pow2 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      9. pow2 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      11. pow3 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      15. lift-sqrt.f64 35.7

          \[\leadsto \left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   7. Applied rewrites 35.7%

      \[\leadsto \left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \color{blue}{\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      3. associate-/l* N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      5. lower-/.f64 38.4

         \[\leadsto \left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   9. Applied rewrites 38.4%

      \[\leadsto \left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   if -2.60000000000000011e-302 < d < 2.2000000000000001e-133

   1. Initial program 44.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{\color{blue}{{\ell}^{2}}} \]

   4. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot D\right)\right) \cdot \sqrt{\ell \cdot h}}{d}, -0.125, \sqrt{\frac{\left(\ell \cdot \ell\right) \cdot \ell}{h}} \cdot d\right)}{\ell \cdot \ell}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right)}{\color{blue}{\ell} \cdot \ell} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) \cdot \frac{-1}{8}}{\ell \cdot \ell} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{h \cdot \ell}}{d} \cdot \frac{-1}{8}}{\ell \cdot \ell} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{\left({M}^{2} \cdot {D}^{2}\right) \cdot \sqrt{h \cdot \ell}}{d} \cdot \frac{-1}{8}}{\ell \cdot \ell} \]

      4. pow2 N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot {D}^{2}\right) \cdot \sqrt{h \cdot \ell}}{d} \cdot \frac{-1}{8}}{\ell \cdot \ell} \]

      5. pow2 N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot D\right)\right) \cdot \sqrt{h \cdot \ell}}{d} \cdot \frac{-1}{8}}{\ell \cdot \ell} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot D\right)\right) \cdot \sqrt{\ell \cdot h}}{d} \cdot \frac{-1}{8}}{\ell \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(M \cdot M\right) \cdot \left(D \cdot D\right)\right) \cdot \sqrt{\ell \cdot h}}{d} \cdot \frac{-1}{8}}{\ell \cdot \ell} \]

   7. Applied rewrites 38.4%

      \[\leadsto \frac{\frac{\sqrt{\ell \cdot h} \cdot \left(\left(\left(M \cdot M\right) \cdot D\right) \cdot D\right)}{d} \cdot -0.125}{\color{blue}{\ell} \cdot \ell} \]

   if 2.2000000000000001e-133 < d

   1. Initial program 75.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 49.6

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 49.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   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. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      6. *-commutative 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. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lift-*.f64 49.8

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 49.8%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   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 49.8

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

   8. Applied rewrites 49.8%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
   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 59.5

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}} \]

   10. Applied rewrites 59.5%

       \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 14: 51.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{if}\;d \leq -8.5 \cdot 10^{+149}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-49}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-302}:\\ \;\;\;\;\left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-206}:\\ \;\;\;\;\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h (* (* l l) l)))))
   (if (<= d -8.5e+149)
     (* (- (sqrt (/ 1.0 (* l h)))) d)
     (if (<= d -3.5e-49)
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
       (if (<= d -2.6e-302)
         (* (* 0.125 (* (* D D) (* M (/ M d)))) t_0)
         (if (<= d 4e-206)
           (* (* -0.125 (* (* D D) (/ (* M M) d))) t_0)
           (/ (* 1.0 d) (* (sqrt h) (sqrt l)))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / ((l * l) * l)));
	double tmp;
	if (d <= -8.5e+149) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else if (d <= -3.5e-49) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else if (d <= -2.6e-302) {
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * t_0;
	} else if (d <= 4e-206) {
		tmp = (-0.125 * ((D * D) * ((M * M) / d))) * t_0;
	} else {
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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, 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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((h / ((l * l) * l)))
    if (d <= (-8.5d+149)) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else if (d <= (-3.5d-49)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    else if (d <= (-2.6d-302)) then
        tmp = (0.125d0 * ((d_1 * d_1) * (m * (m / d)))) * t_0
    else if (d <= 4d-206) then
        tmp = ((-0.125d0) * ((d_1 * d_1) * ((m * m) / d))) * t_0
    else
        tmp = (1.0d0 * d) / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / ((l * l) * l)));
	double tmp;
	if (d <= -8.5e+149) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else if (d <= -3.5e-49) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else if (d <= -2.6e-302) {
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * t_0;
	} else if (d <= 4e-206) {
		tmp = (-0.125 * ((D * D) * ((M * M) / d))) * t_0;
	} else {
		tmp = (1.0 * d) / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((h / ((l * l) * l)))
	tmp = 0
	if d <= -8.5e+149:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	elif d <= -3.5e-49:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	elif d <= -2.6e-302:
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * t_0
	elif d <= 4e-206:
		tmp = (-0.125 * ((D * D) * ((M * M) / d))) * t_0
	else:
		tmp = (1.0 * d) / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / Float64(Float64(l * l) * l)))
	tmp = 0.0
	if (d <= -8.5e+149)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	elseif (d <= -3.5e-49)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	elseif (d <= -2.6e-302)
		tmp = Float64(Float64(0.125 * Float64(Float64(D * D) * Float64(M * Float64(M / d)))) * t_0);
	elseif (d <= 4e-206)
		tmp = Float64(Float64(-0.125 * Float64(Float64(D * D) * Float64(Float64(M * M) / d))) * t_0);
	else
		tmp = Float64(Float64(1.0 * d) / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / ((l * l) * l)));
	tmp = 0.0;
	if (d <= -8.5e+149)
		tmp = -sqrt((1.0 / (l * h))) * d;
	elseif (d <= -3.5e-49)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	elseif (d <= -2.6e-302)
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * t_0;
	elseif (d <= 4e-206)
		tmp = (-0.125 * ((D * D) * ((M * M) / d))) * t_0;
	else
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -8.5e+149], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], If[LessEqual[d, -3.5e-49], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[d, -2.6e-302], N[(N[(0.125 * N[(N[(D * D), $MachinePrecision] * N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, 4e-206], N[(N[(-0.125 * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $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 5 regimes

2. if d < -8.49999999999999956e149

   1. Initial program 68.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 6.5

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 6.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   5. Taylor expanded in h 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 70.9

          \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   7. Applied rewrites 70.9%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if -8.49999999999999956e149 < d < -3.50000000000000006e-49

   1. Initial program 81.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 83.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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. 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 83.4

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied rewrites 83.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.4%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if -3.50000000000000006e-49 < d < -2.60000000000000011e-302

   1. Initial program 53.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \color{blue}{\frac{-1}{8}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \color{blue}{\frac{-1}{8}} \]

   4. Applied rewrites 35.7%

      \[\leadsto \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot -0.125} \]

   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

         \[\leadsto \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      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. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      6. pow2 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      9. pow2 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      11. pow3 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      15. lift-sqrt.f64 35.7

          \[\leadsto \left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   7. Applied rewrites 35.7%

      \[\leadsto \left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \color{blue}{\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      3. associate-/l* N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      5. lower-/.f64 38.4

         \[\leadsto \left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   9. Applied rewrites 38.4%

      \[\leadsto \left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   if -2.60000000000000011e-302 < d < 4.00000000000000011e-206

   1. Initial program 37.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \]

      4. associate-/l* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{\color{blue}{3}}}} \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      13. unpow3 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      14. unpow2 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{2} \cdot \ell}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{2} \cdot \ell}} \]

      16. unpow2 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      17. lower-*.f64 40.2

          \[\leadsto \left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   4. Applied rewrites 40.2%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}} \]

   if 4.00000000000000011e-206 < d

   1. Initial program 72.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 46.1

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 46.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   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. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      6. *-commutative 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. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lift-*.f64 46.3

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 46.3%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   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 46.3

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

   8. Applied rewrites 46.3%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
   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 54.9

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}} \]

   10. Applied rewrites 54.9%

       \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 15: 51.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-103}:\\ \;\;\;\;t\_1 \cdot \left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-200}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l 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 D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (* (sqrt (/ d l)) (sqrt (/ d h)))))
   (if (<= t_0 -5e-103)
     (* t_1 (* (/ (* (* (* M M) h) (* D D)) (* (* d d) l)) -0.125))
     (if (<= t_0 4e-200) (/ d (sqrt (* h l))) (* t_1 1.0)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double 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 * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d / l)) * sqrt((d / h));
	double tmp;
	if (t_0 <= -5e-103) {
		tmp = t_1 * (((((M * M) * h) * (D * D)) / ((d * d) * l)) * -0.125);
	} else if (t_0 <= 4e-200) {
		tmp = d / sqrt((h * l));
	} else {
		tmp = t_1 * 1.0;
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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, 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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((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)))
    t_1 = sqrt((d / l)) * sqrt((d / h))
    if (t_0 <= (-5d-103)) then
        tmp = t_1 * (((((m * m) * h) * (d_1 * d_1)) / ((d * d) * l)) * (-0.125d0))
    else if (t_0 <= 4d-200) then
        tmp = d / sqrt((h * l))
    else
        tmp = t_1 * 1.0d0
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double 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 * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.sqrt((d / l)) * Math.sqrt((d / h));
	double tmp;
	if (t_0 <= -5e-103) {
		tmp = t_1 * (((((M * M) * h) * (D * D)) / ((d * d) * l)) * -0.125);
	} else if (t_0 <= 4e-200) {
		tmp = d / Math.sqrt((h * l));
	} else {
		tmp = t_1 * 1.0;
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, 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 * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.sqrt((d / l)) * math.sqrt((d / h))
	tmp = 0
	if t_0 <= -5e-103:
		tmp = t_1 * (((((M * M) * h) * (D * D)) / ((d * d) * l)) * -0.125)
	elif t_0 <= 4e-200:
		tmp = d / math.sqrt((h * l))
	else:
		tmp = t_1 * 1.0
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, 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 * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)))
	tmp = 0.0
	if (t_0 <= -5e-103)
		tmp = Float64(t_1 * Float64(Float64(Float64(Float64(Float64(M * M) * h) * Float64(D * D)) / Float64(Float64(d * d) * l)) * -0.125));
	elseif (t_0 <= 4e-200)
		tmp = Float64(d / sqrt(Float64(h * l)));
	else
		tmp = Float64(t_1 * 1.0);
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (((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)));
	t_1 = sqrt((d / l)) * sqrt((d / h));
	tmp = 0.0;
	if (t_0 <= -5e-103)
		tmp = t_1 * (((((M * M) * h) * (D * D)) / ((d * d) * l)) * -0.125);
	elseif (t_0 <= 4e-200)
		tmp = d / sqrt((h * l));
	else
		tmp = t_1 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, 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 * 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[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-103], N[(t$95$1 * N[(N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-200], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.99999999999999966e-103

   1. Initial program 86.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 84.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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 84.9

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.8%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\right)} \]

   if -4.99999999999999966e-103 < (*.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)))) < 3.9999999999999999e-200

   1. Initial program 49.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 45.7

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   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. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      6. *-commutative 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. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lift-*.f64 45.6

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 45.6%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   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 45.7

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

   8. Applied rewrites 45.7%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      2. *-lft-identity 45.7

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   10. Applied rewrites 45.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   if 3.9999999999999999e-200 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 57.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 60.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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 60.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.6%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 49.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{+149}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-49}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-302}:\\ \;\;\;\;\left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -8.5e+149)
   (* (- (sqrt (/ 1.0 (* l h)))) d)
   (if (<= d -3.5e-49)
     (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
     (if (<= d -2.6e-302)
       (* (* 0.125 (* (* D D) (* M (/ M d)))) (sqrt (/ h (* (* l l) l))))
       (/ (* 1.0 d) (* (sqrt h) (sqrt l)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -8.5e+149) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else if (d <= -3.5e-49) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else if (d <= -2.6e-302) {
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * sqrt((h / ((l * l) * l)));
	} else {
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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, 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
    real(8) :: tmp
    if (d <= (-8.5d+149)) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else if (d <= (-3.5d-49)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    else if (d <= (-2.6d-302)) then
        tmp = (0.125d0 * ((d_1 * d_1) * (m * (m / d)))) * sqrt((h / ((l * l) * l)))
    else
        tmp = (1.0d0 * d) / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -8.5e+149) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else if (d <= -3.5e-49) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else if (d <= -2.6e-302) {
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * Math.sqrt((h / ((l * l) * l)));
	} else {
		tmp = (1.0 * d) / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -8.5e+149:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	elif d <= -3.5e-49:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	elif d <= -2.6e-302:
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * math.sqrt((h / ((l * l) * l)))
	else:
		tmp = (1.0 * d) / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -8.5e+149)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	elseif (d <= -3.5e-49)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	elseif (d <= -2.6e-302)
		tmp = Float64(Float64(0.125 * Float64(Float64(D * D) * Float64(M * Float64(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

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -8.5e+149)
		tmp = -sqrt((1.0 / (l * h))) * d;
	elseif (d <= -3.5e-49)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	elseif (d <= -2.6e-302)
		tmp = (0.125 * ((D * D) * (M * (M / d)))) * sqrt((h / ((l * l) * l)));
	else
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -8.5e+149], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], If[LessEqual[d, -3.5e-49], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[d, -2.6e-302], N[(N[(0.125 * N[(N[(D * D), $MachinePrecision] * N[(M * N[(M / d), $MachinePrecision]), $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 4 regimes

2. if d < -8.49999999999999956e149

   1. Initial program 68.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 6.5

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 6.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   5. Taylor expanded in h 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 70.9

          \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   7. Applied rewrites 70.9%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if -8.49999999999999956e149 < d < -3.50000000000000006e-49

   1. Initial program 81.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 83.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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. 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 83.4

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied rewrites 83.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.4%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if -3.50000000000000006e-49 < d < -2.60000000000000011e-302

   1. Initial program 53.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \color{blue}{\frac{-1}{8}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \color{blue}{\frac{-1}{8}} \]

   4. Applied rewrites 35.7%

      \[\leadsto \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot -0.125} \]

   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

         \[\leadsto \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      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. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      6. pow2 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      9. pow2 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      11. pow3 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      15. lift-sqrt.f64 35.7

          \[\leadsto \left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   7. Applied rewrites 35.7%

      \[\leadsto \left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \color{blue}{\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      3. associate-/l* N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      5. lower-/.f64 38.4

         \[\leadsto \left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   9. Applied rewrites 38.4%

      \[\leadsto \left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(M \cdot \frac{M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   if -2.60000000000000011e-302 < d

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 41.8

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   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. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      6. *-commutative 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. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lift-*.f64 41.9

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 41.9%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   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 42.0

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

   8. Applied rewrites 42.0%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      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 49.2

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}} \]

   10. Applied rewrites 49.2%

       \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 17: 49.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+36}:\\ \;\;\;\;\left(0.125 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-200}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l 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 D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -1e+36)
     (* (* 0.125 (* D (* D (/ (* M M) d)))) (sqrt (/ h (* (* l l) l))))
     (if (<= t_0 4e-200)
       (/ d (sqrt (* h l)))
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double 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 * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e+36) {
		tmp = (0.125 * (D * (D * ((M * M) / d)))) * sqrt((h / ((l * l) * l)));
	} else if (t_0 <= 4e-200) {
		tmp = d / sqrt((h * l));
	} else {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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, 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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-1d+36)) then
        tmp = (0.125d0 * (d_1 * (d_1 * ((m * m) / d)))) * sqrt((h / ((l * l) * l)))
    else if (t_0 <= 4d-200) then
        tmp = d / sqrt((h * l))
    else
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double 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 * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e+36) {
		tmp = (0.125 * (D * (D * ((M * M) / d)))) * Math.sqrt((h / ((l * l) * l)));
	} else if (t_0 <= 4e-200) {
		tmp = d / Math.sqrt((h * l));
	} else {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, 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 * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -1e+36:
		tmp = (0.125 * (D * (D * ((M * M) / d)))) * math.sqrt((h / ((l * l) * l)))
	elif t_0 <= 4e-200:
		tmp = d / math.sqrt((h * l))
	else:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, 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 * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -1e+36)
		tmp = Float64(Float64(0.125 * Float64(D * Float64(D * Float64(Float64(M * M) / d)))) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	elseif (t_0 <= 4e-200)
		tmp = Float64(d / sqrt(Float64(h * l)));
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (((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)));
	tmp = 0.0;
	if (t_0 <= -1e+36)
		tmp = (0.125 * (D * (D * ((M * M) / d)))) * sqrt((h / ((l * l) * l)));
	elseif (t_0 <= 4e-200)
		tmp = d / sqrt((h * l));
	else
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, 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 * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+36], N[(N[(0.125 * N[(D * N[(D * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-200], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.00000000000000004e36

   1. Initial program 85.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \color{blue}{\frac{-1}{8}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \color{blue}{\frac{-1}{8}} \]

   4. Applied rewrites 30.8%

      \[\leadsto \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \frac{-1 \cdot \left(M \cdot M\right)}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot -0.125} \]

   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

         \[\leadsto \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      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. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      6. pow2 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      9. pow2 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      11. pow3 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      15. lift-sqrt.f64 30.8

          \[\leadsto \left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   7. Applied rewrites 30.8%

      \[\leadsto \left(0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \color{blue}{\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      5. associate-*l* N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\frac{1}{8} \cdot \left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      9. lift-*.f64 32.5

         \[\leadsto \left(0.125 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   9. Applied rewrites 32.5%

      \[\leadsto \left(0.125 \cdot \left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   if -1.00000000000000004e36 < (*.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)))) < 3.9999999999999999e-200

   1. Initial program 59.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 37.5

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   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. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      6. *-commutative 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. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lift-*.f64 37.5

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 37.5%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   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 37.5

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

   8. Applied rewrites 37.5%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      2. *-lft-identity 37.5

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   10. Applied rewrites 37.5%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   if 3.9999999999999999e-200 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 57.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 60.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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 60.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.6%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 46.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+52}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{-222}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.3e+52)
   (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
   (if (<= l 9.5e-222)
     (* (- (sqrt (/ 1.0 (* l h)))) d)
     (/ (* 1.0 d) (* (sqrt h) (sqrt l))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.3e+52) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else if (l <= 9.5e-222) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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, 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
    real(8) :: tmp
    if (l <= (-1.3d+52)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    else if (l <= 9.5d-222) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else
        tmp = (1.0d0 * d) / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.3e+52) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else if (l <= 9.5e-222) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = (1.0 * d) / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.3e+52:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	elif l <= 9.5e-222:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	else:
		tmp = (1.0 * d) / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.3e+52)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	elseif (l <= 9.5e-222)
		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

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.3e+52)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	elseif (l <= 9.5e-222)
		tmp = -sqrt((1.0 / (l * h))) * d;
	else
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.3e+52], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[l, 9.5e-222], 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 3 regimes

2. if l < -1.3e52

   1. Initial program 56.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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 56.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   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 56.0

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   5. Applied rewrites 56.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.8%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if -1.3e52 < l < 9.5000000000000002e-222

   1. Initial program 72.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 13.7

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 13.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   5. Taylor expanded in h 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 35.7

          \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   7. Applied rewrites 35.7%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if 9.5000000000000002e-222 < l

   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

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 45.3

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 45.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   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. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      6. *-commutative 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. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lift-*.f64 45.4

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 45.4%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   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 45.5

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

   8. Applied rewrites 45.5%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
   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 53.0

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}} \]

   10. Applied rewrites 53.0%

       \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 46.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{-222}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l 9.5e-222)
   (* (- (sqrt (/ 1.0 (* l h)))) d)
   (/ (* 1.0 d) (* (sqrt h) (sqrt l)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 9.5e-222) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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, 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
    real(8) :: tmp
    if (l <= 9.5d-222) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else
        tmp = (1.0d0 * d) / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 9.5e-222) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = (1.0 * d) / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= 9.5e-222:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	else:
		tmp = (1.0 * d) / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 9.5e-222)
		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

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 9.5e-222)
		tmp = -sqrt((1.0 / (l * h))) * d;
	else
		tmp = (1.0 * d) / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, 9.5e-222], 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 < 9.5000000000000002e-222

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 11.3

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 11.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   5. Taylor expanded in h 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 41.5

          \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   7. Applied rewrites 41.5%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if 9.5000000000000002e-222 < l

   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

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 45.3

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 45.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   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. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      6. *-commutative 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. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lift-*.f64 45.4

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 45.4%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   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 45.5

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

   8. Applied rewrites 45.5%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
   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 53.0

         \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \sqrt{\ell}} \]

   10. Applied rewrites 53.0%

       \[\leadsto \frac{1 \cdot d}{\sqrt{h} \cdot \color{blue}{\sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 45.2% accurate, 5.2× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{-222}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l 9.5e-222)
   (* (- (sqrt (/ 1.0 (* l h)))) d)
   (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 9.5e-222) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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, 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
    real(8) :: tmp
    if (l <= 9.5d-222) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 9.5e-222) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= 9.5e-222:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 9.5e-222)
		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

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 9.5e-222)
		tmp = -sqrt((1.0 / (l * h))) * d;
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, 9.5e-222], 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 < 9.5000000000000002e-222

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 11.3

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 11.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   5. Taylor expanded in h 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 41.5

          \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   7. Applied rewrites 41.5%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if 9.5000000000000002e-222 < l

   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

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 45.3

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 45.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   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. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      6. *-commutative 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. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lift-*.f64 45.4

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 45.4%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   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 52.9

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   8. Applied rewrites 52.9%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 43.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9.8 \cdot 10^{-222}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l 9.8e-222)
   (* (- (sqrt (/ 1.0 (* l h)))) d)
   (* (sqrt (/ (/ 1.0 l) h)) d)))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 9.8e-222) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = sqrt(((1.0 / l) / h)) * d;
	}
	return tmp;
}

Fortran

NOTE: d, h, l, 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, 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
    real(8) :: tmp
    if (l <= 9.8d-222) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else
        tmp = sqrt(((1.0d0 / l) / h)) * d
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 9.8e-222) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = Math.sqrt(((1.0 / l) / h)) * d;
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= 9.8e-222:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	else:
		tmp = math.sqrt(((1.0 / l) / h)) * d
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 9.8e-222)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	else
		tmp = Float64(sqrt(Float64(Float64(1.0 / l) / h)) * d);
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 9.8e-222)
		tmp = -sqrt((1.0 / (l * h))) * d;
	else
		tmp = sqrt(((1.0 / l) / h)) * d;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, 9.8e-222], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.7999999999999999e-222

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 11.3

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 11.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   5. Taylor expanded in h 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 41.5

          \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   7. Applied rewrites 41.5%

      \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if 9.7999999999999999e-222 < l

   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

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 45.3

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 45.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-*.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. associate-/r* N/A

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

      5. lower-/.f64 45.6

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

   6. Applied rewrites 45.6%

      \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 26.1% accurate, 7.4× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (* (sqrt (/ (/ 1.0 l) h)) d))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	return sqrt(((1.0 / l) / h)) * d;
}

Fortran

NOTE: d, h, l, 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, 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 = sqrt(((1.0d0 / l) / h)) * d
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	return Math.sqrt(((1.0 / l) / h)) * d;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	return math.sqrt(((1.0 / l) / h)) * d

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	return Float64(sqrt(Float64(Float64(1.0 / l) / h)) * d)
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = sqrt(((1.0 / l) / h)) * d;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 66.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
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 26.0

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

4. Applied rewrites 26.0%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Step-by-step derivation

   1. lift-*.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. associate-/r* N/A

      \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

   5. lower-/.f64 26.1

      \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

6. Applied rewrites 26.1%

   \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]
7. Add Preprocessing

Alternative 23: 26.0% accurate, 7.7× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (* (sqrt (/ 1.0 (* l h))) d))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	return sqrt((1.0 / (l * h))) * d;
}

Fortran

NOTE: d, h, l, 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, 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 = sqrt((1.0d0 / (l * h))) * d
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	return Math.sqrt((1.0 / (l * h))) * d;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	return math.sqrt((1.0 / (l * h))) * d

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	return Float64(sqrt(Float64(1.0 / Float64(l * h))) * d)
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = sqrt((1.0 / (l * h))) * d;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 66.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
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 26.0

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

4. Applied rewrites 26.0%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Add Preprocessing

Alternative 24: 25.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* h l))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	return d / sqrt((h * l));
}

Fortran

NOTE: d, h, l, 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, 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 / sqrt((h * l))
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((h * l));
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	return d / math.sqrt((h * l))

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((h * l));
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
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 26.0

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

4. Applied rewrites 26.0%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
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. metadata-eval N/A

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. *-commutative 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. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   10. lift-*.f64 25.8

       \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

6. Applied rewrites 25.8%

   \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
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 25.9

       \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

8. Applied rewrites 25.9%

   \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1 \cdot d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   2. *-lft-identity 25.9

      \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

10. Applied rewrites 25.9%

    \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(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)))))