Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.6% → 75.8%

Time: 11.7s

Alternatives: 18

Speedup: N/A×

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: 75.8% accurate, N/A× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := 1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_1 := {\left(\frac{h}{\ell}\right)}^{0.75}\\ t_2 := {\left(\frac{d}{h}\right)}^{0.25}\\ t_3 := {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\\ t_4 := \frac{M}{2} \cdot \frac{D}{d}\\ t_5 := 1 - \frac{\left(\left(t\_4 \cdot t\_4\right) \cdot 0.5\right) \cdot h}{\ell}\\ t_6 := {\left(\ell \cdot h\right)}^{-1}\\ t_7 := {t\_6}^{0.25}\\ \mathbf{if}\;d \leq -9.5 \cdot 10^{+133}:\\ \;\;\;\;\left(\left(-1 \cdot d\right) \cdot \left(t\_7 \cdot t\_7\right)\right) \cdot t\_5\\ \mathbf{elif}\;d \leq -5.6 \cdot 10^{-140}:\\ \;\;\;\;\left(\left(t\_2 \cdot t\_2\right) \cdot t\_3\right) \cdot t\_5\\ \mathbf{elif}\;d \leq -1.02 \cdot 10^{-207}:\\ \;\;\;\;\left(\left(-1 \cdot d\right) \cdot {t\_6}^{0.5}\right) \cdot t\_0\\ \mathbf{elif}\;d \leq 6 \cdot 10^{-202}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, t\_1 \cdot t\_1, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t\_3\right) \cdot t\_0\\ \end{array} \end{array} \]

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
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
        (t_1 (pow (/ h l) 0.75))
        (t_2 (pow (/ d h) 0.25))
        (t_3 (pow (/ d l) (/ 1.0 2.0)))
        (t_4 (* (/ M 2.0) (/ D d)))
        (t_5 (- 1.0 (/ (* (* (* t_4 t_4) 0.5) h) l)))
        (t_6 (pow (* l h) -1.0))
        (t_7 (pow t_6 0.25)))
   (if (<= d -9.5e+133)
     (* (* (* -1.0 d) (* t_7 t_7)) t_5)
     (if (<= d -5.6e-140)
       (* (* (* t_2 t_2) t_3) t_5)
       (if (<= d -1.02e-207)
         (* (* (* -1.0 d) (pow t_6 0.5)) t_0)
         (if (<= d 6e-202)
           (/
            (fma
             (* -0.125 (/ (pow (* D M) 2.0) d))
             (* t_1 t_1)
             (* (pow (/ h l) 0.5) d))
            h)
           (* (* (/ (sqrt d) (sqrt h)) t_3) t_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 = 1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l));
	double t_1 = pow((h / l), 0.75);
	double t_2 = pow((d / h), 0.25);
	double t_3 = pow((d / l), (1.0 / 2.0));
	double t_4 = (M / 2.0) * (D / d);
	double t_5 = 1.0 - ((((t_4 * t_4) * 0.5) * h) / l);
	double t_6 = pow((l * h), -1.0);
	double t_7 = pow(t_6, 0.25);
	double tmp;
	if (d <= -9.5e+133) {
		tmp = ((-1.0 * d) * (t_7 * t_7)) * t_5;
	} else if (d <= -5.6e-140) {
		tmp = ((t_2 * t_2) * t_3) * t_5;
	} else if (d <= -1.02e-207) {
		tmp = ((-1.0 * d) * pow(t_6, 0.5)) * t_0;
	} else if (d <= 6e-202) {
		tmp = fma((-0.125 * (pow((D * M), 2.0) / d)), (t_1 * t_1), (pow((h / l), 0.5) * d)) / h;
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * t_3) * t_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(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(h / l) ^ 0.75
	t_2 = Float64(d / h) ^ 0.25
	t_3 = Float64(d / l) ^ Float64(1.0 / 2.0)
	t_4 = Float64(Float64(M / 2.0) * Float64(D / d))
	t_5 = Float64(1.0 - Float64(Float64(Float64(Float64(t_4 * t_4) * 0.5) * h) / l))
	t_6 = Float64(l * h) ^ -1.0
	t_7 = t_6 ^ 0.25
	tmp = 0.0
	if (d <= -9.5e+133)
		tmp = Float64(Float64(Float64(-1.0 * d) * Float64(t_7 * t_7)) * t_5);
	elseif (d <= -5.6e-140)
		tmp = Float64(Float64(Float64(t_2 * t_2) * t_3) * t_5);
	elseif (d <= -1.02e-207)
		tmp = Float64(Float64(Float64(-1.0 * d) * (t_6 ^ 0.5)) * t_0);
	elseif (d <= 6e-202)
		tmp = Float64(fma(Float64(-0.125 * Float64((Float64(D * M) ^ 2.0) / d)), Float64(t_1 * t_1), Float64((Float64(h / l) ^ 0.5) * d)) / h);
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_3) * t_0);
	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[(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]}, Block[{t$95$1 = N[Power[N[(h / l), $MachinePrecision], 0.75], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(d / h), $MachinePrecision], 0.25], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(1.0 - N[(N[(N[(N[(t$95$4 * t$95$4), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$7 = N[Power[t$95$6, 0.25], $MachinePrecision]}, If[LessEqual[d, -9.5e+133], N[(N[(N[(-1.0 * d), $MachinePrecision] * N[(t$95$7 * t$95$7), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[d, -5.6e-140], N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[d, -1.02e-207], N[(N[(N[(-1.0 * d), $MachinePrecision] * N[Power[t$95$6, 0.5], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, 6e-202], N[(N[(N[(-0.125 * N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision] + N[(N[Power[N[(h / l), $MachinePrecision], 0.5], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -9.49999999999999996e133

   1. Initial program 59.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-sqrt.f64 0.0

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. inv-pow N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. lift-pow.f64 N/A

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      19. lift-pow.f64 N/A

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

      20. lift-*.f64 85.2

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot \color{blue}{{\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   9. Applied rewrites 85.2%

      \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right)\right)} \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if -9.49999999999999996e133 < d < -5.6000000000000005e-140

   1. Initial program 87.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 87.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. sqr-pow N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-/.f64 87.4

          \[\leadsto \left(\left({\left(\frac{d}{h}\right)}^{0.25} \cdot {\color{blue}{\left(\frac{d}{h}\right)}}^{0.25}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   6. Applied rewrites 87.4%

      \[\leadsto \left(\color{blue}{\left({\left(\frac{d}{h}\right)}^{0.25} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if -5.6000000000000005e-140 < d < -1.02e-207

   1. Initial program 50.4%

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

   3. Taylor expanded in h around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 81.5

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

      15. lift-/.f64 N/A

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

      16. metadata-eval 81.5

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

   5. Applied rewrites 81.5%

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

   if -1.02e-207 < d < 6.00000000000000022e-202

   1. Initial program 36.3%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 45.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 54.2

         \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   7. Applied rewrites 54.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{\frac{3}{4}} \cdot {\left(\frac{h}{\ell}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}, {\left(\frac{h}{\ell}\right)}^{\frac{1}{2}} \cdot d\right)}{h} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{\frac{3}{4}} \cdot {\left(\frac{h}{\ell}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}, {\left(\frac{h}{\ell}\right)}^{\frac{1}{2}} \cdot d\right)}{h} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{\frac{3}{4}} \cdot {\left(\frac{h}{\ell}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}, {\left(\frac{h}{\ell}\right)}^{\frac{1}{2}} \cdot d\right)}{h} \]

      10. metadata-eval 54.2

          \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{0.75} \cdot {\left(\frac{h}{\ell}\right)}^{0.75}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   9. Applied rewrites 54.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{0.75} \cdot {\left(\frac{h}{\ell}\right)}^{0.75}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   if 6.00000000000000022e-202 < d

   1. Initial program 74.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 83.1

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

   4. Applied rewrites 83.1%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 2: 75.7% accurate, N/A× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{h}{\ell}\right)}^{0.75}\\ t_1 := {\left(\frac{d}{h}\right)}^{0.25}\\ t_2 := {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\\ t_3 := \frac{M}{2} \cdot \frac{D}{d}\\ t_4 := 1 - \frac{\left(\left(t\_3 \cdot t\_3\right) \cdot 0.5\right) \cdot h}{\ell}\\ t_5 := {\left(\ell \cdot h\right)}^{-1}\\ t_6 := {t\_5}^{0.25}\\ \mathbf{if}\;d \leq -9.5 \cdot 10^{+133}:\\ \;\;\;\;\left(\left(-1 \cdot d\right) \cdot \left(t\_6 \cdot t\_6\right)\right) \cdot t\_4\\ \mathbf{elif}\;d \leq -5.6 \cdot 10^{-140}:\\ \;\;\;\;\left(\left(t\_1 \cdot t\_1\right) \cdot t\_2\right) \cdot t\_4\\ \mathbf{elif}\;d \leq -1.02 \cdot 10^{-207}:\\ \;\;\;\;\left(\left(-1 \cdot d\right) \cdot {t\_5}^{0.5}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-202}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, t\_0 \cdot t\_0, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{e^{\log d \cdot 0.5}}{\sqrt{h}} \cdot t\_2\right) \cdot t\_4\\ \end{array} \end{array} \]

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 (/ h l) 0.75))
        (t_1 (pow (/ d h) 0.25))
        (t_2 (pow (/ d l) (/ 1.0 2.0)))
        (t_3 (* (/ M 2.0) (/ D d)))
        (t_4 (- 1.0 (/ (* (* (* t_3 t_3) 0.5) h) l)))
        (t_5 (pow (* l h) -1.0))
        (t_6 (pow t_5 0.25)))
   (if (<= d -9.5e+133)
     (* (* (* -1.0 d) (* t_6 t_6)) t_4)
     (if (<= d -5.6e-140)
       (* (* (* t_1 t_1) t_2) t_4)
       (if (<= d -1.02e-207)
         (*
          (* (* -1.0 d) (pow t_5 0.5))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
         (if (<= d 6.2e-202)
           (/
            (fma
             (* -0.125 (/ (pow (* D M) 2.0) d))
             (* t_0 t_0)
             (* (pow (/ h l) 0.5) d))
            h)
           (* (* (/ (exp (* (log d) 0.5)) (sqrt h)) t_2) t_4)))))))

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((h / l), 0.75);
	double t_1 = pow((d / h), 0.25);
	double t_2 = pow((d / l), (1.0 / 2.0));
	double t_3 = (M / 2.0) * (D / d);
	double t_4 = 1.0 - ((((t_3 * t_3) * 0.5) * h) / l);
	double t_5 = pow((l * h), -1.0);
	double t_6 = pow(t_5, 0.25);
	double tmp;
	if (d <= -9.5e+133) {
		tmp = ((-1.0 * d) * (t_6 * t_6)) * t_4;
	} else if (d <= -5.6e-140) {
		tmp = ((t_1 * t_1) * t_2) * t_4;
	} else if (d <= -1.02e-207) {
		tmp = ((-1.0 * d) * pow(t_5, 0.5)) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	} else if (d <= 6.2e-202) {
		tmp = fma((-0.125 * (pow((D * M), 2.0) / d)), (t_0 * t_0), (pow((h / l), 0.5) * d)) / h;
	} else {
		tmp = ((exp((log(d) * 0.5)) / sqrt(h)) * t_2) * t_4;
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(h / l) ^ 0.75
	t_1 = Float64(d / h) ^ 0.25
	t_2 = Float64(d / l) ^ Float64(1.0 / 2.0)
	t_3 = Float64(Float64(M / 2.0) * Float64(D / d))
	t_4 = Float64(1.0 - Float64(Float64(Float64(Float64(t_3 * t_3) * 0.5) * h) / l))
	t_5 = Float64(l * h) ^ -1.0
	t_6 = t_5 ^ 0.25
	tmp = 0.0
	if (d <= -9.5e+133)
		tmp = Float64(Float64(Float64(-1.0 * d) * Float64(t_6 * t_6)) * t_4);
	elseif (d <= -5.6e-140)
		tmp = Float64(Float64(Float64(t_1 * t_1) * t_2) * t_4);
	elseif (d <= -1.02e-207)
		tmp = Float64(Float64(Float64(-1.0 * d) * (t_5 ^ 0.5)) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	elseif (d <= 6.2e-202)
		tmp = Float64(fma(Float64(-0.125 * Float64((Float64(D * M) ^ 2.0) / d)), Float64(t_0 * t_0), Float64((Float64(h / l) ^ 0.5) * d)) / h);
	else
		tmp = Float64(Float64(Float64(exp(Float64(log(d) * 0.5)) / sqrt(h)) * t_2) * t_4);
	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[Power[N[(h / l), $MachinePrecision], 0.75], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / h), $MachinePrecision], 0.25], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - N[(N[(N[(N[(t$95$3 * t$95$3), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$6 = N[Power[t$95$5, 0.25], $MachinePrecision]}, If[LessEqual[d, -9.5e+133], N[(N[(N[(-1.0 * d), $MachinePrecision] * N[(t$95$6 * t$95$6), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[d, -5.6e-140], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[d, -1.02e-207], N[(N[(N[(-1.0 * d), $MachinePrecision] * N[Power[t$95$5, 0.5], $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[d, 6.2e-202], N[(N[(N[(-0.125 * N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[Power[N[(h / l), $MachinePrecision], 0.5], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[Exp[N[(N[Log[d], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$4), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -9.49999999999999996e133

   1. Initial program 59.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-sqrt.f64 0.0

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. inv-pow N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. lift-pow.f64 N/A

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      19. lift-pow.f64 N/A

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

      20. lift-*.f64 85.2

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot \color{blue}{{\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   9. Applied rewrites 85.2%

      \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right)\right)} \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if -9.49999999999999996e133 < d < -5.6000000000000005e-140

   1. Initial program 87.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 87.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. sqr-pow N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-/.f64 87.4

          \[\leadsto \left(\left({\left(\frac{d}{h}\right)}^{0.25} \cdot {\color{blue}{\left(\frac{d}{h}\right)}}^{0.25}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   6. Applied rewrites 87.4%

      \[\leadsto \left(\color{blue}{\left({\left(\frac{d}{h}\right)}^{0.25} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if -5.6000000000000005e-140 < d < -1.02e-207

   1. Initial program 50.4%

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

   3. Taylor expanded in h around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 81.5

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

      15. lift-/.f64 N/A

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

      16. metadata-eval 81.5

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

   5. Applied rewrites 81.5%

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

   if -1.02e-207 < d < 6.2e-202

   1. Initial program 36.3%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 45.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 54.2

         \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   7. Applied rewrites 54.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{\frac{3}{4}} \cdot {\left(\frac{h}{\ell}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}, {\left(\frac{h}{\ell}\right)}^{\frac{1}{2}} \cdot d\right)}{h} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{\frac{3}{4}} \cdot {\left(\frac{h}{\ell}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}, {\left(\frac{h}{\ell}\right)}^{\frac{1}{2}} \cdot d\right)}{h} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{\frac{3}{4}} \cdot {\left(\frac{h}{\ell}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}, {\left(\frac{h}{\ell}\right)}^{\frac{1}{2}} \cdot d\right)}{h} \]

      10. metadata-eval 54.2

          \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{0.75} \cdot {\left(\frac{h}{\ell}\right)}^{0.75}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   9. Applied rewrites 54.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{0.75} \cdot {\left(\frac{h}{\ell}\right)}^{0.75}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   if 6.2e-202 < d

   1. Initial program 74.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 74.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-sqrt.f64 83.8

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

   6. Applied rewrites 83.8%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 80.9

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

   8. Applied rewrites 80.9%

      \[\leadsto \left(\frac{\color{blue}{e^{\log d \cdot 0.5}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.2% accurate, N/A× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\\ t_1 := 1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_2 := {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\\ t_3 := t\_2 \cdot t\_2\\ t_4 := \frac{M}{2} \cdot \frac{D}{d}\\ t_5 := 1 - \frac{\left(\left(t\_4 \cdot t\_4\right) \cdot 0.5\right) \cdot h}{\ell}\\ \mathbf{if}\;h \leq -3.25 \cdot 10^{+156}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot t\_0\right) \cdot t\_1\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-1 \cdot d\right) \cdot t\_3\right) \cdot t\_5\\ \mathbf{elif}\;h \leq 4.4 \cdot 10^{+186}:\\ \;\;\;\;\left(t\_3 \cdot d\right) \cdot t\_5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t\_0\right) \cdot t\_1\\ \end{array} \end{array} \]

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 l) (/ 1.0 2.0)))
        (t_1
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
        (t_2 (pow (pow (* l h) -1.0) 0.25))
        (t_3 (* t_2 t_2))
        (t_4 (* (/ M 2.0) (/ D d)))
        (t_5 (- 1.0 (/ (* (* (* t_4 t_4) 0.5) h) l))))
   (if (<= h -3.25e+156)
     (* (* (pow (/ d h) (/ 1.0 2.0)) t_0) t_1)
     (if (<= h -2e-310)
       (* (* (* -1.0 d) t_3) t_5)
       (if (<= h 4.4e+186)
         (* (* t_3 d) t_5)
         (* (* (/ (sqrt d) (sqrt h)) t_0) t_1))))))

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 / l), (1.0 / 2.0));
	double t_1 = 1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l));
	double t_2 = pow(pow((l * h), -1.0), 0.25);
	double t_3 = t_2 * t_2;
	double t_4 = (M / 2.0) * (D / d);
	double t_5 = 1.0 - ((((t_4 * t_4) * 0.5) * h) / l);
	double tmp;
	if (h <= -3.25e+156) {
		tmp = (pow((d / h), (1.0 / 2.0)) * t_0) * t_1;
	} else if (h <= -2e-310) {
		tmp = ((-1.0 * d) * t_3) * t_5;
	} else if (h <= 4.4e+186) {
		tmp = (t_3 * d) * t_5;
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * t_0) * t_1;
	}
	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) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = (d / l) ** (1.0d0 / 2.0d0)
    t_1 = 1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l))
    t_2 = ((l * h) ** (-1.0d0)) ** 0.25d0
    t_3 = t_2 * t_2
    t_4 = (m / 2.0d0) * (d_1 / d)
    t_5 = 1.0d0 - ((((t_4 * t_4) * 0.5d0) * h) / l)
    if (h <= (-3.25d+156)) then
        tmp = (((d / h) ** (1.0d0 / 2.0d0)) * t_0) * t_1
    else if (h <= (-2d-310)) then
        tmp = (((-1.0d0) * d) * t_3) * t_5
    else if (h <= 4.4d+186) then
        tmp = (t_3 * d) * t_5
    else
        tmp = ((sqrt(d) / sqrt(h)) * t_0) * t_1
    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 / l), (1.0 / 2.0));
	double t_1 = 1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l));
	double t_2 = Math.pow(Math.pow((l * h), -1.0), 0.25);
	double t_3 = t_2 * t_2;
	double t_4 = (M / 2.0) * (D / d);
	double t_5 = 1.0 - ((((t_4 * t_4) * 0.5) * h) / l);
	double tmp;
	if (h <= -3.25e+156) {
		tmp = (Math.pow((d / h), (1.0 / 2.0)) * t_0) * t_1;
	} else if (h <= -2e-310) {
		tmp = ((-1.0 * d) * t_3) * t_5;
	} else if (h <= 4.4e+186) {
		tmp = (t_3 * d) * t_5;
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * t_0) * t_1;
	}
	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 / l), (1.0 / 2.0))
	t_1 = 1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l))
	t_2 = math.pow(math.pow((l * h), -1.0), 0.25)
	t_3 = t_2 * t_2
	t_4 = (M / 2.0) * (D / d)
	t_5 = 1.0 - ((((t_4 * t_4) * 0.5) * h) / l)
	tmp = 0
	if h <= -3.25e+156:
		tmp = (math.pow((d / h), (1.0 / 2.0)) * t_0) * t_1
	elif h <= -2e-310:
		tmp = ((-1.0 * d) * t_3) * t_5
	elif h <= 4.4e+186:
		tmp = (t_3 * d) * t_5
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * t_0) * t_1
	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 / l) ^ Float64(1.0 / 2.0)
	t_1 = Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))
	t_2 = (Float64(l * h) ^ -1.0) ^ 0.25
	t_3 = Float64(t_2 * t_2)
	t_4 = Float64(Float64(M / 2.0) * Float64(D / d))
	t_5 = Float64(1.0 - Float64(Float64(Float64(Float64(t_4 * t_4) * 0.5) * h) / l))
	tmp = 0.0
	if (h <= -3.25e+156)
		tmp = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * t_0) * t_1);
	elseif (h <= -2e-310)
		tmp = Float64(Float64(Float64(-1.0 * d) * t_3) * t_5);
	elseif (h <= 4.4e+186)
		tmp = Float64(Float64(t_3 * d) * t_5);
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_0) * t_1);
	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 / l) ^ (1.0 / 2.0);
	t_1 = 1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l));
	t_2 = ((l * h) ^ -1.0) ^ 0.25;
	t_3 = t_2 * t_2;
	t_4 = (M / 2.0) * (D / d);
	t_5 = 1.0 - ((((t_4 * t_4) * 0.5) * h) / l);
	tmp = 0.0;
	if (h <= -3.25e+156)
		tmp = (((d / h) ^ (1.0 / 2.0)) * t_0) * t_1;
	elseif (h <= -2e-310)
		tmp = ((-1.0 * d) * t_3) * t_5;
	elseif (h <= 4.4e+186)
		tmp = (t_3 * d) * t_5;
	else
		tmp = ((sqrt(d) / sqrt(h)) * t_0) * t_1;
	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[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[Power[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision], 0.25], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(1.0 - N[(N[(N[(N[(t$95$4 * t$95$4), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -3.25e+156], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[h, -2e-310], N[(N[(N[(-1.0 * d), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[h, 4.4e+186], N[(N[(t$95$3 * d), $MachinePrecision] * t$95$5), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -3.25000000000000014e156

   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. Add Preprocessing

   if -3.25000000000000014e156 < h < -1.999999999999994e-310

   1. Initial program 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 70.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-sqrt.f64 0.0

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. inv-pow N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. lift-pow.f64 N/A

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      19. lift-pow.f64 N/A

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

      20. lift-*.f64 81.9

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot \color{blue}{{\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   9. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right)\right)} \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if -1.999999999999994e-310 < h < 4.3999999999999997e186

   1. Initial program 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 69.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-sqrt.f64 74.4

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

   6. Applied rewrites 74.4%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

         \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      13. lift-pow.f64 N/A

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      14. lift-*.f64 79.7

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   9. Applied rewrites 79.7%

      \[\leadsto \color{blue}{\left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right) \cdot d\right)} \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if 4.3999999999999997e186 < h

   1. Initial program 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 76.1

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

   4. Applied rewrites 76.1%

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

Alternative 4: 74.5% accurate, N/A× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \frac{M}{2} \cdot \frac{D}{d}\\ t_1 := {\left(\frac{d}{\ell}\right)}^{0.25}\\ t_2 := 1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_3 := {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\\ t_4 := t\_3 \cdot t\_3\\ t_5 := 1 - \frac{\left(\left(t\_0 \cdot t\_0\right) \cdot 0.5\right) \cdot h}{\ell}\\ \mathbf{if}\;h \leq -4.4 \cdot 10^{+156}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(t\_1 \cdot t\_1\right)\right) \cdot t\_2\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-1 \cdot d\right) \cdot t\_4\right) \cdot t\_5\\ \mathbf{elif}\;h \leq 4.4 \cdot 10^{+186}:\\ \;\;\;\;\left(t\_4 \cdot d\right) \cdot t\_5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot t\_2\\ \end{array} \end{array} \]

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 (* (/ M 2.0) (/ D d)))
        (t_1 (pow (/ d l) 0.25))
        (t_2
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
        (t_3 (pow (pow (* l h) -1.0) 0.25))
        (t_4 (* t_3 t_3))
        (t_5 (- 1.0 (/ (* (* (* t_0 t_0) 0.5) h) l))))
   (if (<= h -4.4e+156)
     (* (* (pow (/ d h) (/ 1.0 2.0)) (* t_1 t_1)) t_2)
     (if (<= h -2e-310)
       (* (* (* -1.0 d) t_4) t_5)
       (if (<= h 4.4e+186)
         (* (* t_4 d) t_5)
         (* (* (/ (sqrt d) (sqrt h)) (pow (/ d l) (/ 1.0 2.0))) 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 = (M / 2.0) * (D / d);
	double t_1 = pow((d / l), 0.25);
	double t_2 = 1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l));
	double t_3 = pow(pow((l * h), -1.0), 0.25);
	double t_4 = t_3 * t_3;
	double t_5 = 1.0 - ((((t_0 * t_0) * 0.5) * h) / l);
	double tmp;
	if (h <= -4.4e+156) {
		tmp = (pow((d / h), (1.0 / 2.0)) * (t_1 * t_1)) * t_2;
	} else if (h <= -2e-310) {
		tmp = ((-1.0 * d) * t_4) * t_5;
	} else if (h <= 4.4e+186) {
		tmp = (t_4 * d) * t_5;
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), (1.0 / 2.0))) * 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) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = (m / 2.0d0) * (d_1 / d)
    t_1 = (d / l) ** 0.25d0
    t_2 = 1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l))
    t_3 = ((l * h) ** (-1.0d0)) ** 0.25d0
    t_4 = t_3 * t_3
    t_5 = 1.0d0 - ((((t_0 * t_0) * 0.5d0) * h) / l)
    if (h <= (-4.4d+156)) then
        tmp = (((d / h) ** (1.0d0 / 2.0d0)) * (t_1 * t_1)) * t_2
    else if (h <= (-2d-310)) then
        tmp = (((-1.0d0) * d) * t_4) * t_5
    else if (h <= 4.4d+186) then
        tmp = (t_4 * d) * t_5
    else
        tmp = ((sqrt(d) / sqrt(h)) * ((d / l) ** (1.0d0 / 2.0d0))) * 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 = (M / 2.0) * (D / d);
	double t_1 = Math.pow((d / l), 0.25);
	double t_2 = 1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l));
	double t_3 = Math.pow(Math.pow((l * h), -1.0), 0.25);
	double t_4 = t_3 * t_3;
	double t_5 = 1.0 - ((((t_0 * t_0) * 0.5) * h) / l);
	double tmp;
	if (h <= -4.4e+156) {
		tmp = (Math.pow((d / h), (1.0 / 2.0)) * (t_1 * t_1)) * t_2;
	} else if (h <= -2e-310) {
		tmp = ((-1.0 * d) * t_4) * t_5;
	} else if (h <= 4.4e+186) {
		tmp = (t_4 * d) * t_5;
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * Math.pow((d / l), (1.0 / 2.0))) * 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 = (M / 2.0) * (D / d)
	t_1 = math.pow((d / l), 0.25)
	t_2 = 1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l))
	t_3 = math.pow(math.pow((l * h), -1.0), 0.25)
	t_4 = t_3 * t_3
	t_5 = 1.0 - ((((t_0 * t_0) * 0.5) * h) / l)
	tmp = 0
	if h <= -4.4e+156:
		tmp = (math.pow((d / h), (1.0 / 2.0)) * (t_1 * t_1)) * t_2
	elif h <= -2e-310:
		tmp = ((-1.0 * d) * t_4) * t_5
	elif h <= 4.4e+186:
		tmp = (t_4 * d) * t_5
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * math.pow((d / l), (1.0 / 2.0))) * 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(Float64(M / 2.0) * Float64(D / d))
	t_1 = Float64(d / l) ^ 0.25
	t_2 = Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))
	t_3 = (Float64(l * h) ^ -1.0) ^ 0.25
	t_4 = Float64(t_3 * t_3)
	t_5 = Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) * 0.5) * h) / l))
	tmp = 0.0
	if (h <= -4.4e+156)
		tmp = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * Float64(t_1 * t_1)) * t_2);
	elseif (h <= -2e-310)
		tmp = Float64(Float64(Float64(-1.0 * d) * t_4) * t_5);
	elseif (h <= 4.4e+186)
		tmp = Float64(Float64(t_4 * d) * t_5);
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * 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 = (M / 2.0) * (D / d);
	t_1 = (d / l) ^ 0.25;
	t_2 = 1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l));
	t_3 = ((l * h) ^ -1.0) ^ 0.25;
	t_4 = t_3 * t_3;
	t_5 = 1.0 - ((((t_0 * t_0) * 0.5) * h) / l);
	tmp = 0.0;
	if (h <= -4.4e+156)
		tmp = (((d / h) ^ (1.0 / 2.0)) * (t_1 * t_1)) * t_2;
	elseif (h <= -2e-310)
		tmp = ((-1.0 * d) * t_4) * t_5;
	elseif (h <= 4.4e+186)
		tmp = (t_4 * d) * t_5;
	else
		tmp = ((sqrt(d) / sqrt(h)) * ((d / l) ^ (1.0 / 2.0))) * 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[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / l), $MachinePrecision], 0.25], $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[Power[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision], 0.25], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -4.4e+156], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[h, -2e-310], N[(N[(N[(-1.0 * d), $MachinePrecision] * t$95$4), $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[h, 4.4e+186], N[(N[(t$95$4 * d), $MachinePrecision] * t$95$5), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -4.40000000000000008e156

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-/.f64 66.4

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 66.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   if -4.40000000000000008e156 < h < -1.999999999999994e-310

   1. Initial program 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 70.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-sqrt.f64 0.0

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. inv-pow N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. lift-pow.f64 N/A

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      19. lift-pow.f64 N/A

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

      20. lift-*.f64 81.9

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot \color{blue}{{\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   9. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right)\right)} \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if -1.999999999999994e-310 < h < 4.3999999999999997e186

   1. Initial program 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 69.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-sqrt.f64 74.4

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

   6. Applied rewrites 74.4%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

         \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      13. lift-pow.f64 N/A

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      14. lift-*.f64 79.7

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   9. Applied rewrites 79.7%

      \[\leadsto \color{blue}{\left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right) \cdot d\right)} \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if 4.3999999999999997e186 < h

   1. Initial program 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 76.1

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

   4. Applied rewrites 76.1%

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

Alternative 5: 74.5% accurate, N/A× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \frac{M}{2} \cdot \frac{D}{d}\\ t_1 := {\left(\frac{d}{\ell}\right)}^{0.125}\\ t_2 := 1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_3 := {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\\ t_4 := t\_3 \cdot t\_3\\ t_5 := 1 - \frac{\left(\left(t\_0 \cdot t\_0\right) \cdot 0.5\right) \cdot h}{\ell}\\ \mathbf{if}\;h \leq -3.25 \cdot 10^{+156}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot \left(t\_1 \cdot t\_1\right)\right)\right) \cdot t\_2\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-1 \cdot d\right) \cdot t\_4\right) \cdot t\_5\\ \mathbf{elif}\;h \leq 4.4 \cdot 10^{+186}:\\ \;\;\;\;\left(t\_4 \cdot d\right) \cdot t\_5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot t\_2\\ \end{array} \end{array} \]

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 (* (/ M 2.0) (/ D d)))
        (t_1 (pow (/ d l) 0.125))
        (t_2
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
        (t_3 (pow (pow (* l h) -1.0) 0.25))
        (t_4 (* t_3 t_3))
        (t_5 (- 1.0 (/ (* (* (* t_0 t_0) 0.5) h) l))))
   (if (<= h -3.25e+156)
     (* (* (pow (/ d h) (/ 1.0 2.0)) (* (pow (/ d l) 0.25) (* t_1 t_1))) t_2)
     (if (<= h -2e-310)
       (* (* (* -1.0 d) t_4) t_5)
       (if (<= h 4.4e+186)
         (* (* t_4 d) t_5)
         (* (* (/ (sqrt d) (sqrt h)) (pow (/ d l) (/ 1.0 2.0))) 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 = (M / 2.0) * (D / d);
	double t_1 = pow((d / l), 0.125);
	double t_2 = 1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l));
	double t_3 = pow(pow((l * h), -1.0), 0.25);
	double t_4 = t_3 * t_3;
	double t_5 = 1.0 - ((((t_0 * t_0) * 0.5) * h) / l);
	double tmp;
	if (h <= -3.25e+156) {
		tmp = (pow((d / h), (1.0 / 2.0)) * (pow((d / l), 0.25) * (t_1 * t_1))) * t_2;
	} else if (h <= -2e-310) {
		tmp = ((-1.0 * d) * t_4) * t_5;
	} else if (h <= 4.4e+186) {
		tmp = (t_4 * d) * t_5;
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), (1.0 / 2.0))) * 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) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = (m / 2.0d0) * (d_1 / d)
    t_1 = (d / l) ** 0.125d0
    t_2 = 1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l))
    t_3 = ((l * h) ** (-1.0d0)) ** 0.25d0
    t_4 = t_3 * t_3
    t_5 = 1.0d0 - ((((t_0 * t_0) * 0.5d0) * h) / l)
    if (h <= (-3.25d+156)) then
        tmp = (((d / h) ** (1.0d0 / 2.0d0)) * (((d / l) ** 0.25d0) * (t_1 * t_1))) * t_2
    else if (h <= (-2d-310)) then
        tmp = (((-1.0d0) * d) * t_4) * t_5
    else if (h <= 4.4d+186) then
        tmp = (t_4 * d) * t_5
    else
        tmp = ((sqrt(d) / sqrt(h)) * ((d / l) ** (1.0d0 / 2.0d0))) * 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 = (M / 2.0) * (D / d);
	double t_1 = Math.pow((d / l), 0.125);
	double t_2 = 1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l));
	double t_3 = Math.pow(Math.pow((l * h), -1.0), 0.25);
	double t_4 = t_3 * t_3;
	double t_5 = 1.0 - ((((t_0 * t_0) * 0.5) * h) / l);
	double tmp;
	if (h <= -3.25e+156) {
		tmp = (Math.pow((d / h), (1.0 / 2.0)) * (Math.pow((d / l), 0.25) * (t_1 * t_1))) * t_2;
	} else if (h <= -2e-310) {
		tmp = ((-1.0 * d) * t_4) * t_5;
	} else if (h <= 4.4e+186) {
		tmp = (t_4 * d) * t_5;
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * Math.pow((d / l), (1.0 / 2.0))) * 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 = (M / 2.0) * (D / d)
	t_1 = math.pow((d / l), 0.125)
	t_2 = 1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l))
	t_3 = math.pow(math.pow((l * h), -1.0), 0.25)
	t_4 = t_3 * t_3
	t_5 = 1.0 - ((((t_0 * t_0) * 0.5) * h) / l)
	tmp = 0
	if h <= -3.25e+156:
		tmp = (math.pow((d / h), (1.0 / 2.0)) * (math.pow((d / l), 0.25) * (t_1 * t_1))) * t_2
	elif h <= -2e-310:
		tmp = ((-1.0 * d) * t_4) * t_5
	elif h <= 4.4e+186:
		tmp = (t_4 * d) * t_5
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * math.pow((d / l), (1.0 / 2.0))) * 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(Float64(M / 2.0) * Float64(D / d))
	t_1 = Float64(d / l) ^ 0.125
	t_2 = Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))
	t_3 = (Float64(l * h) ^ -1.0) ^ 0.25
	t_4 = Float64(t_3 * t_3)
	t_5 = Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) * 0.5) * h) / l))
	tmp = 0.0
	if (h <= -3.25e+156)
		tmp = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * Float64((Float64(d / l) ^ 0.25) * Float64(t_1 * t_1))) * t_2);
	elseif (h <= -2e-310)
		tmp = Float64(Float64(Float64(-1.0 * d) * t_4) * t_5);
	elseif (h <= 4.4e+186)
		tmp = Float64(Float64(t_4 * d) * t_5);
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * 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 = (M / 2.0) * (D / d);
	t_1 = (d / l) ^ 0.125;
	t_2 = 1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l));
	t_3 = ((l * h) ^ -1.0) ^ 0.25;
	t_4 = t_3 * t_3;
	t_5 = 1.0 - ((((t_0 * t_0) * 0.5) * h) / l);
	tmp = 0.0;
	if (h <= -3.25e+156)
		tmp = (((d / h) ^ (1.0 / 2.0)) * (((d / l) ^ 0.25) * (t_1 * t_1))) * t_2;
	elseif (h <= -2e-310)
		tmp = ((-1.0 * d) * t_4) * t_5;
	elseif (h <= 4.4e+186)
		tmp = (t_4 * d) * t_5;
	else
		tmp = ((sqrt(d) / sqrt(h)) * ((d / l) ^ (1.0 / 2.0))) * 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[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / l), $MachinePrecision], 0.125], $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[Power[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision], 0.25], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -3.25e+156], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], 0.25], $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[h, -2e-310], N[(N[(N[(-1.0 * d), $MachinePrecision] * t$95$4), $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[h, 4.4e+186], N[(N[(t$95$4 * d), $MachinePrecision] * t$95$5), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -3.25000000000000014e156

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-/.f64 66.4

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 66.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{4}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{8}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{4}} \cdot \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{8}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{4}} \cdot \left({\color{blue}{\left(\frac{d}{\ell}\right)}}^{\frac{1}{8}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{4}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{8}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{8}}}\right)\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{4}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{8}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{8}}}\right)\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 66.3

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.125} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{0.125}\right)\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 66.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{0.125} \cdot {\left(\frac{d}{\ell}\right)}^{0.125}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   if -3.25000000000000014e156 < h < -1.999999999999994e-310

   1. Initial program 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 70.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-sqrt.f64 0.0

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. inv-pow N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. lift-pow.f64 N/A

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      19. lift-pow.f64 N/A

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

      20. lift-*.f64 81.9

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot \color{blue}{{\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   9. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right)\right)} \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if -1.999999999999994e-310 < h < 4.3999999999999997e186

   1. Initial program 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 69.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-sqrt.f64 74.4

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

   6. Applied rewrites 74.4%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

         \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      13. lift-pow.f64 N/A

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      14. lift-*.f64 79.7

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   9. Applied rewrites 79.7%

      \[\leadsto \color{blue}{\left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right) \cdot d\right)} \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if 4.3999999999999997e186 < h

   1. Initial program 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 76.1

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

   4. Applied rewrites 76.1%

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

Alternative 6: 74.5% accurate, N/A× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \frac{M}{2} \cdot \frac{D}{d}\\ t_1 := 1 - \frac{\left(\left(t\_0 \cdot t\_0\right) \cdot 0.5\right) \cdot h}{\ell}\\ t_2 := {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\\ t_3 := t\_2 \cdot t\_2\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-1 \cdot d\right) \cdot t\_3\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+173}:\\ \;\;\;\;\left(t\_3 \cdot d\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, \frac{\sqrt{h}}{\sqrt{\ell}} \cdot d\right)}{h}\\ \end{array} \end{array} \]

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 (* (/ M 2.0) (/ D d)))
        (t_1 (- 1.0 (/ (* (* (* t_0 t_0) 0.5) h) l)))
        (t_2 (pow (pow (* l h) -1.0) 0.25))
        (t_3 (* t_2 t_2)))
   (if (<= l -4e-310)
     (* (* (* -1.0 d) t_3) t_1)
     (if (<= l 4.6e+173)
       (* (* t_3 d) t_1)
       (/
        (fma
         (* -0.125 (/ (pow (* D M) 2.0) d))
         (pow (/ h l) 1.5)
         (* (/ (sqrt h) (sqrt l)) d))
        h)))))

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 = (M / 2.0) * (D / d);
	double t_1 = 1.0 - ((((t_0 * t_0) * 0.5) * h) / l);
	double t_2 = pow(pow((l * h), -1.0), 0.25);
	double t_3 = t_2 * t_2;
	double tmp;
	if (l <= -4e-310) {
		tmp = ((-1.0 * d) * t_3) * t_1;
	} else if (l <= 4.6e+173) {
		tmp = (t_3 * d) * t_1;
	} else {
		tmp = fma((-0.125 * (pow((D * M), 2.0) / d)), pow((h / l), 1.5), ((sqrt(h) / sqrt(l)) * d)) / h;
	}
	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(M / 2.0) * Float64(D / d))
	t_1 = Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) * 0.5) * h) / l))
	t_2 = (Float64(l * h) ^ -1.0) ^ 0.25
	t_3 = Float64(t_2 * t_2)
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(Float64(-1.0 * d) * t_3) * t_1);
	elseif (l <= 4.6e+173)
		tmp = Float64(Float64(t_3 * d) * t_1);
	else
		tmp = Float64(fma(Float64(-0.125 * Float64((Float64(D * M) ^ 2.0) / d)), (Float64(h / l) ^ 1.5), Float64(Float64(sqrt(h) / sqrt(l)) * d)) / h);
	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[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision], 0.25], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$2), $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(N[(-1.0 * d), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 4.6e+173], N[(N[(t$95$3 * d), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(-0.125 * N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] + N[(N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 68.4%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 68.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-sqrt.f64 0.0

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. inv-pow N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. lift-pow.f64 N/A

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      19. lift-pow.f64 N/A

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

      20. lift-*.f64 73.8

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot \color{blue}{{\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   9. Applied rewrites 73.8%

      \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right)\right)} \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if -3.999999999999988e-310 < l < 4.5999999999999999e173

   1. Initial program 70.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 70.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-sqrt.f64 79.5

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

         \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      13. lift-pow.f64 N/A

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      14. lift-*.f64 79.5

          \[\leadsto \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right) \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   9. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right) \cdot d\right)} \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if 4.5999999999999999e173 < l

   1. Initial program 48.5%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 45.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 48.3

         \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   7. Applied rewrites 48.3%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 60.2

         \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, \frac{\sqrt{h}}{\sqrt{\ell}} \cdot d\right)}{h} \]

   9. Applied rewrites 60.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, \frac{\sqrt{h}}{\sqrt{\ell}} \cdot d\right)}{h} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 73.3% accurate, N/A× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \frac{M}{2} \cdot \frac{D}{d}\\ t_1 := {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-1 \cdot d\right) \cdot t\_2\right) \cdot \left(1 - \frac{\left(\left(t\_0 \cdot t\_0\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+173}:\\ \;\;\;\;\left(t\_2 \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, \frac{\sqrt{h}}{\sqrt{\ell}} \cdot d\right)}{h}\\ \end{array} \end{array} \]

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 (* (/ M 2.0) (/ D d)))
        (t_1 (pow (pow (* l h) -1.0) 0.25))
        (t_2 (* t_1 t_1)))
   (if (<= l -4e-310)
     (* (* (* -1.0 d) t_2) (- 1.0 (/ (* (* (* t_0 t_0) 0.5) h) l)))
     (if (<= l 4.6e+173)
       (*
        (* t_2 d)
        (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
       (/
        (fma
         (* -0.125 (/ (pow (* D M) 2.0) d))
         (pow (/ h l) 1.5)
         (* (/ (sqrt h) (sqrt l)) d))
        h)))))

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 = (M / 2.0) * (D / d);
	double t_1 = pow(pow((l * h), -1.0), 0.25);
	double t_2 = t_1 * t_1;
	double tmp;
	if (l <= -4e-310) {
		tmp = ((-1.0 * d) * t_2) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	} else if (l <= 4.6e+173) {
		tmp = (t_2 * d) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	} else {
		tmp = fma((-0.125 * (pow((D * M), 2.0) / d)), pow((h / l), 1.5), ((sqrt(h) / sqrt(l)) * d)) / h;
	}
	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(M / 2.0) * Float64(D / d))
	t_1 = (Float64(l * h) ^ -1.0) ^ 0.25
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(Float64(-1.0 * d) * t_2) * Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) * 0.5) * h) / l)));
	elseif (l <= 4.6e+173)
		tmp = Float64(Float64(t_2 * d) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	else
		tmp = Float64(fma(Float64(-0.125 * Float64((Float64(D * M) ^ 2.0) / d)), (Float64(h / l) ^ 1.5), Float64(Float64(sqrt(h) / sqrt(l)) * d)) / h);
	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[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision], 0.25], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(N[(-1.0 * d), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.6e+173], N[(N[(t$95$2 * d), $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], N[(N[(N[(-0.125 * N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] + N[(N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 68.4%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 68.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-sqrt.f64 0.0

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. inv-pow N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. lift-pow.f64 N/A

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\frac{1}{4}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}\right) \]

      19. lift-pow.f64 N/A

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

      20. lift-*.f64 73.8

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot \color{blue}{{\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}}\right)\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   9. Applied rewrites 73.8%

      \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right)\right)} \cdot \left(1 - \frac{\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if -3.999999999999988e-310 < l < 4.5999999999999999e173

   1. Initial program 70.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-/.f64 70.2

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 70.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{4}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{8}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{4}} \cdot \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{8}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{4}} \cdot \left({\color{blue}{\left(\frac{d}{\ell}\right)}}^{\frac{1}{8}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{4}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{8}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{8}}}\right)\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{4}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{8}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{8}}}\right)\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 70.2

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.125} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{0.125}\right)\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 70.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{0.125} \cdot {\left(\frac{d}{\ell}\right)}^{0.125}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Taylor expanded in d around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. sqr-pow N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. pow-prod-up N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 76.9%

      \[\leadsto \color{blue}{\left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right) \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   if 4.5999999999999999e173 < l

   1. Initial program 48.5%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 45.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 48.3

         \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   7. Applied rewrites 48.3%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 60.2

         \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, \frac{\sqrt{h}}{\sqrt{\ell}} \cdot d\right)}{h} \]

   9. Applied rewrites 60.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, \frac{\sqrt{h}}{\sqrt{\ell}} \cdot d\right)}{h} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.2% accurate, N/A× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{h}{\ell}\right)}^{0.75}\\ t_1 := -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\\ t_2 := {\left(\ell \cdot h\right)}^{-1}\\ t_3 := {t\_2}^{0.25}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+181}:\\ \;\;\;\;\left(-1 \cdot d\right) \cdot {t\_2}^{0.5}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-285}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_0 \cdot t\_0, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}\\ \mathbf{elif}\;d \leq 6 \cdot 10^{+80}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, {\left(\frac{h}{\ell}\right)}^{1.5}, \frac{\sqrt{h}}{\sqrt{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_3 \cdot t\_3\right) \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

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 (/ h l) 0.75))
        (t_1 (* -0.125 (/ (pow (* D M) 2.0) d)))
        (t_2 (pow (* l h) -1.0))
        (t_3 (pow t_2 0.25)))
   (if (<= d -7e+181)
     (* (* -1.0 d) (pow t_2 0.5))
     (if (<= d 4e-285)
       (/ (fma t_1 (* t_0 t_0) (* (pow (/ h l) 0.5) d)) h)
       (if (<= d 6e+80)
         (/ (fma t_1 (pow (/ h l) 1.5) (* (/ (sqrt h) (sqrt l)) d)) h)
         (*
          (* (* t_3 t_3) d)
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ 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 = pow((h / l), 0.75);
	double t_1 = -0.125 * (pow((D * M), 2.0) / d);
	double t_2 = pow((l * h), -1.0);
	double t_3 = pow(t_2, 0.25);
	double tmp;
	if (d <= -7e+181) {
		tmp = (-1.0 * d) * pow(t_2, 0.5);
	} else if (d <= 4e-285) {
		tmp = fma(t_1, (t_0 * t_0), (pow((h / l), 0.5) * d)) / h;
	} else if (d <= 6e+80) {
		tmp = fma(t_1, pow((h / l), 1.5), ((sqrt(h) / sqrt(l)) * d)) / h;
	} else {
		tmp = ((t_3 * t_3) * d) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (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(h / l) ^ 0.75
	t_1 = Float64(-0.125 * Float64((Float64(D * M) ^ 2.0) / d))
	t_2 = Float64(l * h) ^ -1.0
	t_3 = t_2 ^ 0.25
	tmp = 0.0
	if (d <= -7e+181)
		tmp = Float64(Float64(-1.0 * d) * (t_2 ^ 0.5));
	elseif (d <= 4e-285)
		tmp = Float64(fma(t_1, Float64(t_0 * t_0), Float64((Float64(h / l) ^ 0.5) * d)) / h);
	elseif (d <= 6e+80)
		tmp = Float64(fma(t_1, (Float64(h / l) ^ 1.5), Float64(Float64(sqrt(h) / sqrt(l)) * d)) / h);
	else
		tmp = Float64(Float64(Float64(t_3 * t_3) * d) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / 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[Power[N[(h / l), $MachinePrecision], 0.75], $MachinePrecision]}, Block[{t$95$1 = N[(-0.125 * N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 0.25], $MachinePrecision]}, If[LessEqual[d, -7e+181], N[(N[(-1.0 * d), $MachinePrecision] * N[Power[t$95$2, 0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4e-285], N[(N[(t$95$1 * N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[Power[N[(h / l), $MachinePrecision], 0.5], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 6e+80], N[(N[(t$95$1 * N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] + N[(N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(t$95$3 * t$95$3), $MachinePrecision] * d), $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]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7.00000000000000016e181

   1. Initial program 57.0%

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

   3. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 76.2

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

      15. lift-/.f64 N/A

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

      16. metadata-eval 76.2

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

   5. Applied rewrites 76.2%

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

   if -7.00000000000000016e181 < d < 4.0000000000000003e-285

   1. Initial program 69.3%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 67.1

         \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   7. Applied rewrites 67.1%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{\frac{3}{4}} \cdot {\left(\frac{h}{\ell}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}, {\left(\frac{h}{\ell}\right)}^{\frac{1}{2}} \cdot d\right)}{h} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{\frac{3}{4}} \cdot {\left(\frac{h}{\ell}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}, {\left(\frac{h}{\ell}\right)}^{\frac{1}{2}} \cdot d\right)}{h} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{\frac{3}{4}} \cdot {\left(\frac{h}{\ell}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}, {\left(\frac{h}{\ell}\right)}^{\frac{1}{2}} \cdot d\right)}{h} \]

      10. metadata-eval 67.1

          \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{0.75} \cdot {\left(\frac{h}{\ell}\right)}^{0.75}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   9. Applied rewrites 67.1%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{0.75} \cdot {\left(\frac{h}{\ell}\right)}^{0.75}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   if 4.0000000000000003e-285 < d < 5.99999999999999974e80

   1. Initial program 55.9%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 58.1

         \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   7. Applied rewrites 58.1%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 69.0

         \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, \frac{\sqrt{h}}{\sqrt{\ell}} \cdot d\right)}{h} \]

   9. Applied rewrites 69.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, \frac{\sqrt{h}}{\sqrt{\ell}} \cdot d\right)}{h} \]

   if 5.99999999999999974e80 < d

   1. Initial program 86.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-/.f64 86.3

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{0.25}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 86.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{4}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{8}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{4}} \cdot \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{8}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{4}} \cdot \left({\color{blue}{\left(\frac{d}{\ell}\right)}}^{\frac{1}{8}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{4}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{8}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{8}}}\right)\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{4}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{8}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{8}}}\right)\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 86.3

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.125} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{0.125}\right)\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 86.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{0.125} \cdot {\left(\frac{d}{\ell}\right)}^{0.125}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Taylor expanded in d around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. sqr-pow N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. pow-prod-up N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 81.8%

      \[\leadsto \color{blue}{\left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right) \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 65.5% accurate, N/A× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{h}{\ell}\right)}^{0.75}\\ t_1 := -0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+181}:\\ \;\;\;\;\left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-285}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_0 \cdot t\_0, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, {\left(\frac{h}{\ell}\right)}^{1.5}, \frac{\sqrt{h}}{\sqrt{\ell}} \cdot d\right)}{h}\\ \end{array} \end{array} \]

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 (/ h l) 0.75)) (t_1 (* -0.125 (/ (pow (* D M) 2.0) d))))
   (if (<= d -7e+181)
     (* (* -1.0 d) (pow (pow (* l h) -1.0) 0.5))
     (if (<= d 4e-285)
       (/ (fma t_1 (* t_0 t_0) (* (pow (/ h l) 0.5) d)) h)
       (/ (fma t_1 (pow (/ h l) 1.5) (* (/ (sqrt h) (sqrt l)) d)) h)))))

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((h / l), 0.75);
	double t_1 = -0.125 * (pow((D * M), 2.0) / d);
	double tmp;
	if (d <= -7e+181) {
		tmp = (-1.0 * d) * pow(pow((l * h), -1.0), 0.5);
	} else if (d <= 4e-285) {
		tmp = fma(t_1, (t_0 * t_0), (pow((h / l), 0.5) * d)) / h;
	} else {
		tmp = fma(t_1, pow((h / l), 1.5), ((sqrt(h) / sqrt(l)) * d)) / h;
	}
	return tmp;
}

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(h / l) ^ 0.75
	t_1 = Float64(-0.125 * Float64((Float64(D * M) ^ 2.0) / d))
	tmp = 0.0
	if (d <= -7e+181)
		tmp = Float64(Float64(-1.0 * d) * ((Float64(l * h) ^ -1.0) ^ 0.5));
	elseif (d <= 4e-285)
		tmp = Float64(fma(t_1, Float64(t_0 * t_0), Float64((Float64(h / l) ^ 0.5) * d)) / h);
	else
		tmp = Float64(fma(t_1, (Float64(h / l) ^ 1.5), Float64(Float64(sqrt(h) / sqrt(l)) * d)) / h);
	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[Power[N[(h / l), $MachinePrecision], 0.75], $MachinePrecision]}, Block[{t$95$1 = N[(-0.125 * N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -7e+181], N[(N[(-1.0 * d), $MachinePrecision] * N[Power[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4e-285], N[(N[(t$95$1 * N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[Power[N[(h / l), $MachinePrecision], 0.5], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(t$95$1 * N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] + N[(N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.00000000000000016e181

   1. Initial program 57.0%

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

   3. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 76.2

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

      15. lift-/.f64 N/A

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

      16. metadata-eval 76.2

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

   5. Applied rewrites 76.2%

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

   if -7.00000000000000016e181 < d < 4.0000000000000003e-285

   1. Initial program 69.3%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 67.1

         \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   7. Applied rewrites 67.1%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{\frac{3}{4}} \cdot {\left(\frac{h}{\ell}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}, {\left(\frac{h}{\ell}\right)}^{\frac{1}{2}} \cdot d\right)}{h} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{\frac{3}{4}} \cdot {\left(\frac{h}{\ell}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}, {\left(\frac{h}{\ell}\right)}^{\frac{1}{2}} \cdot d\right)}{h} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{\frac{3}{4}} \cdot {\left(\frac{h}{\ell}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}, {\left(\frac{h}{\ell}\right)}^{\frac{1}{2}} \cdot d\right)}{h} \]

      10. metadata-eval 67.1

          \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{0.75} \cdot {\left(\frac{h}{\ell}\right)}^{0.75}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   9. Applied rewrites 67.1%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{0.75} \cdot {\left(\frac{h}{\ell}\right)}^{0.75}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   if 4.0000000000000003e-285 < 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. Add Preprocessing

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 61.1

         \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   7. Applied rewrites 61.1%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 68.9

         \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, \frac{\sqrt{h}}{\sqrt{\ell}} \cdot d\right)}{h} \]

   9. Applied rewrites 68.9%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, \frac{\sqrt{h}}{\sqrt{\ell}} \cdot d\right)}{h} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 61.9% accurate, N/A× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{h}{\ell}\right)}^{0.75}\\ t_1 := {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+181}:\\ \;\;\;\;\left(-1 \cdot d\right) \cdot t\_1\\ \mathbf{elif}\;d \leq 8.2 \cdot 10^{+200}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, t\_0 \cdot t\_0, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot d\\ \end{array} \end{array} \]

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
 (let* ((t_0 (pow (/ h l) 0.75)) (t_1 (pow (pow (* l h) -1.0) 0.5)))
   (if (<= d -7e+181)
     (* (* -1.0 d) t_1)
     (if (<= d 8.2e+200)
       (/
        (fma
         (* -0.125 (/ (pow (* D M) 2.0) d))
         (* t_0 t_0)
         (* (pow (/ h l) 0.5) d))
        h)
       (* t_1 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 = pow((h / l), 0.75);
	double t_1 = pow(pow((l * h), -1.0), 0.5);
	double tmp;
	if (d <= -7e+181) {
		tmp = (-1.0 * d) * t_1;
	} else if (d <= 8.2e+200) {
		tmp = fma((-0.125 * (pow((D * M), 2.0) / d)), (t_0 * t_0), (pow((h / l), 0.5) * d)) / h;
	} else {
		tmp = t_1 * 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(h / l) ^ 0.75
	t_1 = (Float64(l * h) ^ -1.0) ^ 0.5
	tmp = 0.0
	if (d <= -7e+181)
		tmp = Float64(Float64(-1.0 * d) * t_1);
	elseif (d <= 8.2e+200)
		tmp = Float64(fma(Float64(-0.125 * Float64((Float64(D * M) ^ 2.0) / d)), Float64(t_0 * t_0), Float64((Float64(h / l) ^ 0.5) * d)) / h);
	else
		tmp = Float64(t_1 * 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_] := Block[{t$95$0 = N[Power[N[(h / l), $MachinePrecision], 0.75], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]}, If[LessEqual[d, -7e+181], N[(N[(-1.0 * d), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 8.2e+200], N[(N[(N[(-0.125 * N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[Power[N[(h / l), $MachinePrecision], 0.5], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(t$95$1 * d), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.00000000000000016e181

   1. Initial program 57.0%

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

   3. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 76.2

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

      15. lift-/.f64 N/A

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

      16. metadata-eval 76.2

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

   5. Applied rewrites 76.2%

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

   if -7.00000000000000016e181 < d < 8.2000000000000005e200

   1. Initial program 67.5%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 53.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 64.9

         \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   7. Applied rewrites 64.9%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{1.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{\frac{3}{4}} \cdot {\left(\frac{h}{\ell}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}, {\left(\frac{h}{\ell}\right)}^{\frac{1}{2}} \cdot d\right)}{h} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{\frac{3}{4}} \cdot {\left(\frac{h}{\ell}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}, {\left(\frac{h}{\ell}\right)}^{\frac{1}{2}} \cdot d\right)}{h} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{\frac{3}{4}} \cdot {\left(\frac{h}{\ell}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}, {\left(\frac{h}{\ell}\right)}^{\frac{1}{2}} \cdot d\right)}{h} \]

      10. metadata-eval 64.9

          \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{0.75} \cdot {\left(\frac{h}{\ell}\right)}^{0.75}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   9. Applied rewrites 64.9%

      \[\leadsto \frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left(\frac{h}{\ell}\right)}^{0.75} \cdot {\left(\frac{h}{\ell}\right)}^{0.75}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h} \]

   if 8.2000000000000005e200 < d

   1. Initial program 72.9%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. pow1/2 N/A

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

      4. metadata-eval N/A

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

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      6. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      7. inv-pow N/A

         \[\leadsto {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      8. lower-pow.f64 N/A

         \[\leadsto {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      9. *-commutative N/A

         \[\leadsto {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      10. lower-*.f64 77.9

          \[\leadsto {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      11. lift-/.f64 N/A

          \[\leadsto {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      12. metadata-eval 77.9

          \[\leadsto {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5} \cdot d \]

   5. Applied rewrites 77.9%

      \[\leadsto \color{blue}{{\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5} \cdot d} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 52.4% accurate, N/A× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5}\\ t_1 := {\left(D \cdot M\right)}^{1}\\ t_2 := \frac{\frac{\mathsf{fma}\left({\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5} \cdot \left(t\_1 \cdot t\_1\right), -0.125, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot \left(d \cdot d\right)\right)}{d}}{h}\\ \mathbf{if}\;d \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\left(-1 \cdot d\right) \cdot t\_0\\ \mathbf{elif}\;d \leq 9.6 \cdot 10^{-266}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{{\left(\ell \cdot h\right)}^{0.5} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right)}{\ell}}{\ell}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+152}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot d\\ \end{array} \end{array} \]

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
 (let* ((t_0 (pow (pow (* l h) -1.0) 0.5))
        (t_1 (pow (* D M) 1.0))
        (t_2
         (/
          (/
           (fma
            (* (pow (pow (/ h l) 3.0) 0.5) (* t_1 t_1))
            -0.125
            (* (pow (/ h l) 0.5) (* d d)))
           d)
          h)))
   (if (<= d -5e+51)
     (* (* -1.0 d) t_0)
     (if (<= d 9.6e-266)
       t_2
       (if (<= d 7.2e-130)
         (/ (/ (* (pow (* l h) 0.5) (* (/ (pow (* D M) 2.0) d) -0.125)) l) l)
         (if (<= d 2.2e+152) t_2 (* t_0 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 = pow(pow((l * h), -1.0), 0.5);
	double t_1 = pow((D * M), 1.0);
	double t_2 = (fma((pow(pow((h / l), 3.0), 0.5) * (t_1 * t_1)), -0.125, (pow((h / l), 0.5) * (d * d))) / d) / h;
	double tmp;
	if (d <= -5e+51) {
		tmp = (-1.0 * d) * t_0;
	} else if (d <= 9.6e-266) {
		tmp = t_2;
	} else if (d <= 7.2e-130) {
		tmp = ((pow((l * h), 0.5) * ((pow((D * M), 2.0) / d) * -0.125)) / l) / l;
	} else if (d <= 2.2e+152) {
		tmp = t_2;
	} else {
		tmp = t_0 * 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(l * h) ^ -1.0) ^ 0.5
	t_1 = Float64(D * M) ^ 1.0
	t_2 = Float64(Float64(fma(Float64(((Float64(h / l) ^ 3.0) ^ 0.5) * Float64(t_1 * t_1)), -0.125, Float64((Float64(h / l) ^ 0.5) * Float64(d * d))) / d) / h)
	tmp = 0.0
	if (d <= -5e+51)
		tmp = Float64(Float64(-1.0 * d) * t_0);
	elseif (d <= 9.6e-266)
		tmp = t_2;
	elseif (d <= 7.2e-130)
		tmp = Float64(Float64(Float64((Float64(l * h) ^ 0.5) * Float64(Float64((Float64(D * M) ^ 2.0) / d) * -0.125)) / l) / l);
	elseif (d <= 2.2e+152)
		tmp = t_2;
	else
		tmp = Float64(t_0 * 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_] := Block[{t$95$0 = N[Power[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(D * M), $MachinePrecision], 1.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Power[N[Power[N[(h / l), $MachinePrecision], 3.0], $MachinePrecision], 0.5], $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(N[Power[N[(h / l), $MachinePrecision], 0.5], $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[d, -5e+51], N[(N[(-1.0 * d), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, 9.6e-266], t$95$2, If[LessEqual[d, 7.2e-130], N[(N[(N[(N[Power[N[(l * h), $MachinePrecision], 0.5], $MachinePrecision] * N[(N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[d, 2.2e+152], t$95$2, N[(t$95$0 * d), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5e51

   1. Initial program 66.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(d \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto \left(d \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{h \cdot \color{blue}{\ell}}} \]

      4. metadata-eval N/A

         \[\leadsto \left(d \cdot -1\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      7. pow1/2 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{\frac{1}{2}}} \]

      8. metadata-eval N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \]

      11. inv-pow N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

      12. lower-pow.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

      13. *-commutative N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \]

      14. lower-*.f64 61.7

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \]

      15. lift-/.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      16. metadata-eval 61.7

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5} \]

   5. Applied rewrites 61.7%

      \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5}} \]

   if -5e51 < d < 9.5999999999999999e-266 or 7.2000000000000003e-130 < d < 2.1999999999999998e152

   1. Initial program 72.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around 0

      \[\leadsto \frac{\frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + {d}^{2} \cdot \sqrt{\frac{h}{\ell}}}{d}}{h} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + {d}^{2} \cdot \sqrt{\frac{h}{\ell}}}{d}}{h} \]

   8. Applied rewrites 58.5%

      \[\leadsto \frac{\frac{\mathsf{fma}\left({\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5} \cdot \left({\left(D \cdot M\right)}^{1} \cdot {\left(D \cdot M\right)}^{1}\right), -0.125, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot \left(d \cdot d\right)\right)}{d}}{h} \]

   if 9.5999999999999999e-266 < d < 7.2000000000000003e-130

   1. Initial program 26.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{\color{blue}{{\ell}^{2}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{\color{blue}{2}}} \]

   8. Applied rewrites 24.4%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\ell \cdot h\right)}^{0.5} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), -0.125, {\left(\frac{{\ell}^{3}}{h}\right)}^{0.5} \cdot d\right)}{\color{blue}{\ell \cdot \ell}} \]

   9. Taylor expanded in d around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right)}{\ell \cdot \ell} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       3. *-commutative N/A

          \[\leadsto \frac{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       5. unpow-prod-down N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       6. lift-pow.f64 N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       8. lift-/.f64 N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       9. pow1/2 N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(h \cdot \ell\right)}^{\frac{1}{2}}}{\ell \cdot \ell} \]

       10. *-commutative N/A

           \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell \cdot \ell} \]

       11. metadata-eval N/A

           \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\left(\frac{1}{2}\right)}}{\ell \cdot \ell} \]

       12. lift-pow.f64 N/A

           \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\left(\frac{1}{2}\right)}}{\ell \cdot \ell} \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\left(\frac{1}{2}\right)}}{\ell \cdot \ell} \]

       14. metadata-eval 39.3

           \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right) \cdot {\left(\ell \cdot h\right)}^{0.5}}{\ell \cdot \ell} \]

   11. Applied rewrites 39.3%

       \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right) \cdot {\left(\ell \cdot h\right)}^{0.5}}{\ell \cdot \ell} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell \cdot \ell} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell \cdot \color{blue}{\ell}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell}}{\ell} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell}}{\ell} \]

   13. Applied rewrites 49.1%

       \[\leadsto \frac{\frac{{\left(\ell \cdot h\right)}^{0.5} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right)}{\ell}}{\ell} \]

   if 2.1999999999999998e152 < d

   1. Initial program 82.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. pow1/2 N/A

         \[\leadsto {\left(\frac{1}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot d \]

      4. metadata-eval N/A

         \[\leadsto {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      6. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      7. inv-pow N/A

         \[\leadsto {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      8. lower-pow.f64 N/A

         \[\leadsto {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      9. *-commutative N/A

         \[\leadsto {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      10. lower-*.f64 79.5

          \[\leadsto {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      11. lift-/.f64 N/A

          \[\leadsto {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      12. metadata-eval 79.5

          \[\leadsto {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5} \cdot d \]

   5. Applied rewrites 79.5%

      \[\leadsto \color{blue}{{\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5} \cdot d} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 48.4% accurate, N/A× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5}\\ \mathbf{if}\;\ell \leq 3.8 \cdot 10^{-287}:\\ \;\;\;\;\left(-1 \cdot d\right) \cdot t\_0\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125, {\left(\ell \cdot h\right)}^{0.5}, {\left(\frac{{\ell}^{3}}{h}\right)}^{0.5} \cdot d\right)}{\ell}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot d\\ \end{array} \end{array} \]

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
 (let* ((t_0 (pow (pow (* l h) -1.0) 0.5)))
   (if (<= l 3.8e-287)
     (* (* -1.0 d) t_0)
     (if (<= l 3.2e+77)
       (/
        (/
         (fma
          (* (/ (pow (* D M) 2.0) d) -0.125)
          (pow (* l h) 0.5)
          (* (pow (/ (pow l 3.0) h) 0.5) d))
         l)
        l)
       (* t_0 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 = pow(pow((l * h), -1.0), 0.5);
	double tmp;
	if (l <= 3.8e-287) {
		tmp = (-1.0 * d) * t_0;
	} else if (l <= 3.2e+77) {
		tmp = (fma(((pow((D * M), 2.0) / d) * -0.125), pow((l * h), 0.5), (pow((pow(l, 3.0) / h), 0.5) * d)) / l) / l;
	} else {
		tmp = t_0 * 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(l * h) ^ -1.0) ^ 0.5
	tmp = 0.0
	if (l <= 3.8e-287)
		tmp = Float64(Float64(-1.0 * d) * t_0);
	elseif (l <= 3.2e+77)
		tmp = Float64(Float64(fma(Float64(Float64((Float64(D * M) ^ 2.0) / d) * -0.125), (Float64(l * h) ^ 0.5), Float64((Float64((l ^ 3.0) / h) ^ 0.5) * d)) / l) / l);
	else
		tmp = Float64(t_0 * 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_] := Block[{t$95$0 = N[Power[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]}, If[LessEqual[l, 3.8e-287], N[(N[(-1.0 * d), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[l, 3.2e+77], N[(N[(N[(N[(N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision] * N[Power[N[(l * h), $MachinePrecision], 0.5], $MachinePrecision] + N[(N[Power[N[(N[Power[l, 3.0], $MachinePrecision] / h), $MachinePrecision], 0.5], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision], N[(t$95$0 * d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.79999999999999982e-287

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(d \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto \left(d \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{h \cdot \color{blue}{\ell}}} \]

      4. metadata-eval N/A

         \[\leadsto \left(d \cdot -1\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      7. pow1/2 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{\frac{1}{2}}} \]

      8. metadata-eval N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \]

      11. inv-pow N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

      12. lower-pow.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

      13. *-commutative N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \]

      14. lower-*.f64 42.2

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \]

      15. lift-/.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      16. metadata-eval 42.2

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5} \]

   5. Applied rewrites 42.2%

      \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5}} \]

   if 3.79999999999999982e-287 < l < 3.2000000000000002e77

   1. Initial program 69.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{\color{blue}{{\ell}^{2}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{\color{blue}{2}}} \]

   8. Applied rewrites 50.1%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\ell \cdot h\right)}^{0.5} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), -0.125, {\left(\frac{{\ell}^{3}}{h}\right)}^{0.5} \cdot d\right)}{\color{blue}{\ell \cdot \ell}} \]

   10. Applied rewrites 63.2%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125, {\left(\ell \cdot h\right)}^{0.5}, {\left(\frac{{\ell}^{3}}{h}\right)}^{0.5} \cdot d\right)}{\ell}}{\ell} \]

   if 3.2000000000000002e77 < l

   1. Initial program 57.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. 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. pow1/2 N/A

         \[\leadsto {\left(\frac{1}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot d \]

      4. metadata-eval N/A

         \[\leadsto {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      6. lower-pow.f64 N/A

         \[\leadsto {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      7. inv-pow N/A

         \[\leadsto {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      8. lower-pow.f64 N/A

         \[\leadsto {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      9. *-commutative N/A

         \[\leadsto {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      10. lower-*.f64 47.6

          \[\leadsto {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      11. lift-/.f64 N/A

          \[\leadsto {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \cdot d \]

      12. metadata-eval 47.6

          \[\leadsto {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5} \cdot d \]

   5. Applied rewrites 47.6%

      \[\leadsto \color{blue}{{\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5} \cdot d} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 48.3% accurate, N/A× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{-1}\\ t_1 := {t\_0}^{0.25}\\ \mathbf{if}\;\ell \leq 3.8 \cdot 10^{-287}:\\ \;\;\;\;\left(-1 \cdot d\right) \cdot {t\_0}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125, {\left(\ell \cdot h\right)}^{0.5}, {\left(\frac{{\ell}^{3}}{h}\right)}^{0.5} \cdot d\right)}{\ell}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot d\right) \cdot \left(\left(t\_1 \cdot t\_1\right) \cdot -1\right)\\ \end{array} \end{array} \]

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 (pow (* l h) -1.0)) (t_1 (pow t_0 0.25)))
   (if (<= l 3.8e-287)
     (* (* -1.0 d) (pow t_0 0.5))
     (if (<= l 1.8e+79)
       (/
        (/
         (fma
          (* (/ (pow (* D M) 2.0) d) -0.125)
          (pow (* l h) 0.5)
          (* (pow (/ (pow l 3.0) h) 0.5) d))
         l)
        l)
       (* (* -1.0 d) (* (* t_1 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((l * h), -1.0);
	double t_1 = pow(t_0, 0.25);
	double tmp;
	if (l <= 3.8e-287) {
		tmp = (-1.0 * d) * pow(t_0, 0.5);
	} else if (l <= 1.8e+79) {
		tmp = (fma(((pow((D * M), 2.0) / d) * -0.125), pow((l * h), 0.5), (pow((pow(l, 3.0) / h), 0.5) * d)) / l) / l;
	} else {
		tmp = (-1.0 * d) * ((t_1 * 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(l * h) ^ -1.0
	t_1 = t_0 ^ 0.25
	tmp = 0.0
	if (l <= 3.8e-287)
		tmp = Float64(Float64(-1.0 * d) * (t_0 ^ 0.5));
	elseif (l <= 1.8e+79)
		tmp = Float64(Float64(fma(Float64(Float64((Float64(D * M) ^ 2.0) / d) * -0.125), (Float64(l * h) ^ 0.5), Float64((Float64((l ^ 3.0) / h) ^ 0.5) * d)) / l) / l);
	else
		tmp = Float64(Float64(-1.0 * d) * Float64(Float64(t_1 * t_1) * -1.0));
	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[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 0.25], $MachinePrecision]}, If[LessEqual[l, 3.8e-287], N[(N[(-1.0 * d), $MachinePrecision] * N[Power[t$95$0, 0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.8e+79], N[(N[(N[(N[(N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision] * N[Power[N[(l * h), $MachinePrecision], 0.5], $MachinePrecision] + N[(N[Power[N[(N[Power[l, 3.0], $MachinePrecision] / h), $MachinePrecision], 0.5], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision], N[(N[(-1.0 * d), $MachinePrecision] * N[(N[(t$95$1 * t$95$1), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.79999999999999982e-287

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(d \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto \left(d \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{h \cdot \color{blue}{\ell}}} \]

      4. metadata-eval N/A

         \[\leadsto \left(d \cdot -1\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      7. pow1/2 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{\frac{1}{2}}} \]

      8. metadata-eval N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \]

      11. inv-pow N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

      12. lower-pow.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

      13. *-commutative N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \]

      14. lower-*.f64 42.2

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \]

      15. lift-/.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      16. metadata-eval 42.2

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5} \]

   5. Applied rewrites 42.2%

      \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5}} \]

   if 3.79999999999999982e-287 < l < 1.8e79

   1. Initial program 69.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{\color{blue}{{\ell}^{2}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{\color{blue}{2}}} \]

   8. Applied rewrites 50.7%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\ell \cdot h\right)}^{0.5} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), -0.125, {\left(\frac{{\ell}^{3}}{h}\right)}^{0.5} \cdot d\right)}{\color{blue}{\ell \cdot \ell}} \]

   10. Applied rewrites 63.7%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125, {\left(\ell \cdot h\right)}^{0.5}, {\left(\frac{{\ell}^{3}}{h}\right)}^{0.5} \cdot d\right)}{\ell}}{\ell} \]

   if 1.8e79 < l

   1. Initial program 56.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(d \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto \left(d \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{h \cdot \color{blue}{\ell}}} \]

      4. metadata-eval N/A

         \[\leadsto \left(d \cdot -1\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      7. pow1/2 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{\frac{1}{2}}} \]

      8. metadata-eval N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \]

      11. inv-pow N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

      12. lower-pow.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

      13. *-commutative N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \]

      14. lower-*.f64 8.8

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \]

      15. lift-/.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      16. metadata-eval 8.8

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5} \]

   5. Applied rewrites 8.8%

      \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5}} \]

   6. Taylor expanded in h around -inf

      \[\leadsto \left(-1 \cdot d\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. sqrt-pow2 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{\left(\frac{2}{\color{blue}{2}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{1}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right) \]

   8. Applied rewrites 46.4%

      \[\leadsto \left(-1 \cdot d\right) \cdot \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right) \cdot \color{blue}{-1}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 48.2% accurate, N/A× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\\ t_1 := {\left(\ell \cdot h\right)}^{0.5}\\ \mathbf{if}\;\ell \leq 3.8 \cdot 10^{-287}:\\ \;\;\;\;\left(-1 \cdot d\right) \cdot \frac{1}{t\_1}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125, t\_1, {\left(\frac{{\ell}^{3}}{h}\right)}^{0.5} \cdot d\right)}{\ell}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot d\right) \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot -1\right)\\ \end{array} \end{array} \]

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 (pow (pow (* l h) -1.0) 0.25)) (t_1 (pow (* l h) 0.5)))
   (if (<= l 3.8e-287)
     (* (* -1.0 d) (/ 1.0 t_1))
     (if (<= l 1.8e+79)
       (/
        (/
         (fma
          (* (/ (pow (* D M) 2.0) d) -0.125)
          t_1
          (* (pow (/ (pow l 3.0) h) 0.5) d))
         l)
        l)
       (* (* -1.0 d) (* (* t_0 t_0) -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(pow((l * h), -1.0), 0.25);
	double t_1 = pow((l * h), 0.5);
	double tmp;
	if (l <= 3.8e-287) {
		tmp = (-1.0 * d) * (1.0 / t_1);
	} else if (l <= 1.8e+79) {
		tmp = (fma(((pow((D * M), 2.0) / d) * -0.125), t_1, (pow((pow(l, 3.0) / h), 0.5) * d)) / l) / l;
	} else {
		tmp = (-1.0 * d) * ((t_0 * t_0) * -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(l * h) ^ -1.0) ^ 0.25
	t_1 = Float64(l * h) ^ 0.5
	tmp = 0.0
	if (l <= 3.8e-287)
		tmp = Float64(Float64(-1.0 * d) * Float64(1.0 / t_1));
	elseif (l <= 1.8e+79)
		tmp = Float64(Float64(fma(Float64(Float64((Float64(D * M) ^ 2.0) / d) * -0.125), t_1, Float64((Float64((l ^ 3.0) / h) ^ 0.5) * d)) / l) / l);
	else
		tmp = Float64(Float64(-1.0 * d) * Float64(Float64(t_0 * t_0) * -1.0));
	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[Power[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision], 0.25], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(l * h), $MachinePrecision], 0.5], $MachinePrecision]}, If[LessEqual[l, 3.8e-287], N[(N[(-1.0 * d), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.8e+79], N[(N[(N[(N[(N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision] * t$95$1 + N[(N[Power[N[(N[Power[l, 3.0], $MachinePrecision] / h), $MachinePrecision], 0.5], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision], N[(N[(-1.0 * d), $MachinePrecision] * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.79999999999999982e-287

   1. Initial program 68.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(d \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto \left(d \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{h \cdot \color{blue}{\ell}}} \]

      4. metadata-eval N/A

         \[\leadsto \left(d \cdot -1\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      7. pow1/2 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{\frac{1}{2}}} \]

      8. metadata-eval N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \]

      11. inv-pow N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

      12. lower-pow.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

      13. *-commutative N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \]

      14. lower-*.f64 42.2

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \]

      15. lift-/.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      16. metadata-eval 42.2

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5} \]

   5. Applied rewrites 42.2%

      \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \]

      6. inv-pow N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\frac{1}{2}} \]

      8. pow1/2 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. sqrt-div N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      10. metadata-eval N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      12. pow1/2 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot \frac{1}{{\left(h \cdot \ell\right)}^{\color{blue}{\frac{1}{2}}}} \]

      13. *-commutative N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot \frac{1}{{\left(\ell \cdot h\right)}^{\frac{1}{2}}} \]

      14. metadata-eval N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot \frac{1}{{\left(\ell \cdot h\right)}^{\left(\frac{1}{\color{blue}{2}}\right)}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot \frac{1}{{\left(\ell \cdot h\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

      16. lift-*.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot \frac{1}{{\left(\ell \cdot h\right)}^{\left(\frac{\color{blue}{1}}{2}\right)}} \]

      17. metadata-eval 42.1

          \[\leadsto \left(-1 \cdot d\right) \cdot \frac{1}{{\left(\ell \cdot h\right)}^{0.5}} \]

   7. Applied rewrites 42.1%

      \[\leadsto \left(-1 \cdot d\right) \cdot \frac{1}{\color{blue}{{\left(\ell \cdot h\right)}^{0.5}}} \]

   if 3.79999999999999982e-287 < l < 1.8e79

   1. Initial program 69.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{\color{blue}{{\ell}^{2}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{\color{blue}{2}}} \]

   8. Applied rewrites 50.7%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\ell \cdot h\right)}^{0.5} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), -0.125, {\left(\frac{{\ell}^{3}}{h}\right)}^{0.5} \cdot d\right)}{\color{blue}{\ell \cdot \ell}} \]

   10. Applied rewrites 63.7%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125, {\left(\ell \cdot h\right)}^{0.5}, {\left(\frac{{\ell}^{3}}{h}\right)}^{0.5} \cdot d\right)}{\ell}}{\ell} \]

   if 1.8e79 < l

   1. Initial program 56.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(d \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto \left(d \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{h \cdot \color{blue}{\ell}}} \]

      4. metadata-eval N/A

         \[\leadsto \left(d \cdot -1\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      7. pow1/2 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{\frac{1}{2}}} \]

      8. metadata-eval N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \]

      11. inv-pow N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

      12. lower-pow.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

      13. *-commutative N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \]

      14. lower-*.f64 8.8

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \]

      15. lift-/.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      16. metadata-eval 8.8

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5} \]

   5. Applied rewrites 8.8%

      \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5}} \]

   6. Taylor expanded in h around -inf

      \[\leadsto \left(-1 \cdot d\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. sqrt-pow2 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{\left(\frac{2}{\color{blue}{2}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{1}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right) \]

   8. Applied rewrites 46.4%

      \[\leadsto \left(-1 \cdot d\right) \cdot \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right) \cdot \color{blue}{-1}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 36.3% accurate, N/A× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{\frac{{\left(\ell \cdot h\right)}^{0.5} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right)}{\ell}}{\ell}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+149}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(-1 \cdot d\right) \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (pow (pow (* l h) -1.0) 0.25))
        (t_1
         (*
          (* (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_2
         (/ (/ (* (pow (* l h) 0.5) (* (/ (pow (* D M) 2.0) d) -0.125)) l) l)))
   (if (<= t_1 -5e+149)
     t_2
     (if (<= t_1 INFINITY) (* (* -1.0 d) (* (* t_0 t_0) -1.0)) 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 = pow(pow((l * h), -1.0), 0.25);
	double t_1 = (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_2 = ((pow((l * h), 0.5) * ((pow((D * M), 2.0) / d) * -0.125)) / l) / l;
	double tmp;
	if (t_1 <= -5e+149) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (-1.0 * d) * ((t_0 * t_0) * -1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

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(Math.pow((l * h), -1.0), 0.25);
	double t_1 = (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_2 = ((Math.pow((l * h), 0.5) * ((Math.pow((D * M), 2.0) / d) * -0.125)) / l) / l;
	double tmp;
	if (t_1 <= -5e+149) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (-1.0 * d) * ((t_0 * t_0) * -1.0);
	} else {
		tmp = 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 = math.pow(math.pow((l * h), -1.0), 0.25)
	t_1 = (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_2 = ((math.pow((l * h), 0.5) * ((math.pow((D * M), 2.0) / d) * -0.125)) / l) / l
	tmp = 0
	if t_1 <= -5e+149:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (-1.0 * d) * ((t_0 * t_0) * -1.0)
	else:
		tmp = 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(l * h) ^ -1.0) ^ 0.25
	t_1 = 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_2 = Float64(Float64(Float64((Float64(l * h) ^ 0.5) * Float64(Float64((Float64(D * M) ^ 2.0) / d) * -0.125)) / l) / l)
	tmp = 0.0
	if (t_1 <= -5e+149)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(-1.0 * d) * Float64(Float64(t_0 * t_0) * -1.0));
	else
		tmp = 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 = ((l * h) ^ -1.0) ^ 0.25;
	t_1 = (((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_2 = ((((l * h) ^ 0.5) * ((((D * M) ^ 2.0) / d) * -0.125)) / l) / l;
	tmp = 0.0;
	if (t_1 <= -5e+149)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (-1.0 * d) * ((t_0 * t_0) * -1.0);
	else
		tmp = 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[Power[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision], 0.25], $MachinePrecision]}, Block[{t$95$1 = 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$2 = N[(N[(N[(N[Power[N[(l * h), $MachinePrecision], 0.5], $MachinePrecision] * N[(N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+149], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(-1.0 * d), $MachinePrecision] * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.9999999999999999e149 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 53.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{\color{blue}{{\ell}^{2}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{\color{blue}{2}}} \]

   8. Applied rewrites 21.8%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\ell \cdot h\right)}^{0.5} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), -0.125, {\left(\frac{{\ell}^{3}}{h}\right)}^{0.5} \cdot d\right)}{\color{blue}{\ell \cdot \ell}} \]

   9. Taylor expanded in d around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right)}{\ell \cdot \ell} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       3. *-commutative N/A

          \[\leadsto \frac{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       5. unpow-prod-down N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       6. lift-pow.f64 N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       8. lift-/.f64 N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       9. pow1/2 N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(h \cdot \ell\right)}^{\frac{1}{2}}}{\ell \cdot \ell} \]

       10. *-commutative N/A

           \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell \cdot \ell} \]

       11. metadata-eval N/A

           \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\left(\frac{1}{2}\right)}}{\ell \cdot \ell} \]

       12. lift-pow.f64 N/A

           \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\left(\frac{1}{2}\right)}}{\ell \cdot \ell} \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\left(\frac{1}{2}\right)}}{\ell \cdot \ell} \]

       14. metadata-eval 29.2

           \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right) \cdot {\left(\ell \cdot h\right)}^{0.5}}{\ell \cdot \ell} \]

   11. Applied rewrites 29.2%

       \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right) \cdot {\left(\ell \cdot h\right)}^{0.5}}{\ell \cdot \ell} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell \cdot \ell} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell \cdot \color{blue}{\ell}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell}}{\ell} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell}}{\ell} \]

   13. Applied rewrites 34.8%

       \[\leadsto \frac{\frac{{\left(\ell \cdot h\right)}^{0.5} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right)}{\ell}}{\ell} \]

   if -4.9999999999999999e149 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 80.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(d \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto \left(d \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{h \cdot \color{blue}{\ell}}} \]

      4. metadata-eval N/A

         \[\leadsto \left(d \cdot -1\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      7. pow1/2 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{\frac{1}{2}}} \]

      8. metadata-eval N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \]

      11. inv-pow N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

      12. lower-pow.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(h \cdot \ell\right)}^{-1}\right)}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

      13. *-commutative N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \]

      14. lower-*.f64 40.9

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{2}\right)} \]

      15. lift-/.f64 N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{\left(\frac{1}{\color{blue}{2}}\right)} \]

      16. metadata-eval 40.9

          \[\leadsto \left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5} \]

   5. Applied rewrites 40.9%

      \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.5}} \]

   6. Taylor expanded in h around -inf

      \[\leadsto \left(-1 \cdot d\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. sqrt-pow2 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{\left(\frac{2}{\color{blue}{2}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {-1}^{1}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right) \]

   8. Applied rewrites 41.7%

      \[\leadsto \left(-1 \cdot d\right) \cdot \left(\left({\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\ell \cdot h\right)}^{-1}\right)}^{0.25}\right) \cdot \color{blue}{-1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 23.6% accurate, N/A× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{0.5}\\ t_1 := \frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\\ \mathbf{if}\;\ell \leq 7.2 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_1, t\_0, {\left(\frac{{\ell}^{3}}{h}\right)}^{0.5} \cdot d\right)}{\ell}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 \cdot t\_1}{\ell}}{\ell}\\ \end{array} \end{array} \]

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 (pow (* l h) 0.5)) (t_1 (* (/ (pow (* D M) 2.0) d) -0.125)))
   (if (<= l 7.2e+78)
     (/ (/ (fma t_1 t_0 (* (pow (/ (pow l 3.0) h) 0.5) d)) l) l)
     (/ (/ (* t_0 t_1) l) 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 = pow((l * h), 0.5);
	double t_1 = (pow((D * M), 2.0) / d) * -0.125;
	double tmp;
	if (l <= 7.2e+78) {
		tmp = (fma(t_1, t_0, (pow((pow(l, 3.0) / h), 0.5) * d)) / l) / l;
	} else {
		tmp = ((t_0 * t_1) / l) / 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(l * h) ^ 0.5
	t_1 = Float64(Float64((Float64(D * M) ^ 2.0) / d) * -0.125)
	tmp = 0.0
	if (l <= 7.2e+78)
		tmp = Float64(Float64(fma(t_1, t_0, Float64((Float64((l ^ 3.0) / h) ^ 0.5) * d)) / l) / l);
	else
		tmp = Float64(Float64(Float64(t_0 * t_1) / l) / 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[Power[N[(l * h), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]}, If[LessEqual[l, 7.2e+78], N[(N[(N[(t$95$1 * t$95$0 + N[(N[Power[N[(N[Power[l, 3.0], $MachinePrecision] / h), $MachinePrecision], 0.5], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision], N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < 7.20000000000000039e78

   1. Initial program 69.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 53.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{\color{blue}{{\ell}^{2}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{\color{blue}{2}}} \]

   8. Applied rewrites 20.5%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\ell \cdot h\right)}^{0.5} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), -0.125, {\left(\frac{{\ell}^{3}}{h}\right)}^{0.5} \cdot d\right)}{\color{blue}{\ell \cdot \ell}} \]

   10. Applied rewrites 25.7%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125, {\left(\ell \cdot h\right)}^{0.5}, {\left(\frac{{\ell}^{3}}{h}\right)}^{0.5} \cdot d\right)}{\ell}}{\ell} \]

   if 7.20000000000000039e78 < l

   1. Initial program 56.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 46.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{\color{blue}{{\ell}^{2}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{\color{blue}{2}}} \]

   8. Applied rewrites 2.5%

      \[\leadsto \frac{\mathsf{fma}\left({\left(\ell \cdot h\right)}^{0.5} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), -0.125, {\left(\frac{{\ell}^{3}}{h}\right)}^{0.5} \cdot d\right)}{\color{blue}{\ell \cdot \ell}} \]

   9. Taylor expanded in d around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right)}{\ell \cdot \ell} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       3. *-commutative N/A

          \[\leadsto \frac{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       5. unpow-prod-down N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       6. lift-pow.f64 N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       8. lift-/.f64 N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

       9. pow1/2 N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(h \cdot \ell\right)}^{\frac{1}{2}}}{\ell \cdot \ell} \]

       10. *-commutative N/A

           \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell \cdot \ell} \]

       11. metadata-eval N/A

           \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\left(\frac{1}{2}\right)}}{\ell \cdot \ell} \]

       12. lift-pow.f64 N/A

           \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\left(\frac{1}{2}\right)}}{\ell \cdot \ell} \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\left(\frac{1}{2}\right)}}{\ell \cdot \ell} \]

       14. metadata-eval 7.9

           \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right) \cdot {\left(\ell \cdot h\right)}^{0.5}}{\ell \cdot \ell} \]

   11. Applied rewrites 7.9%

       \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right) \cdot {\left(\ell \cdot h\right)}^{0.5}}{\ell \cdot \ell} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell \cdot \ell} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell \cdot \color{blue}{\ell}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell}}{\ell} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell}}{\ell} \]

   13. Applied rewrites 20.6%

       \[\leadsto \frac{\frac{{\left(\ell \cdot h\right)}^{0.5} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right)}{\ell}}{\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 19.4% accurate, N/A× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \frac{\frac{{\left(\ell \cdot h\right)}^{0.5} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right)}{\ell}}{\ell} \end{array} \]

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
 (/ (/ (* (pow (* l h) 0.5) (* (/ (pow (* D M) 2.0) d) -0.125)) l) l))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	return ((pow((l * h), 0.5) * ((pow((D * M), 2.0) / d) * -0.125)) / l) / 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 = ((((l * h) ** 0.5d0) * ((((d_1 * m) ** 2.0d0) / d) * (-0.125d0))) / l) / 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 ((Math.pow((l * h), 0.5) * ((Math.pow((D * M), 2.0) / d) * -0.125)) / l) / l;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	return ((math.pow((l * h), 0.5) * ((math.pow((D * M), 2.0) / d) * -0.125)) / l) / l

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	return Float64(Float64(Float64((Float64(l * h) ^ 0.5) * Float64(Float64((Float64(D * M) ^ 2.0) / d) * -0.125)) / l) / l)
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = ((((l * h) ^ 0.5) * ((((D * M) ^ 2.0) / d) * -0.125)) / l) / 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[(N[(N[(N[Power[N[(l * h), $MachinePrecision], 0.5], $MachinePrecision] * N[(N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]

Derivation

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. Add Preprocessing

3. Taylor expanded in h around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

5. Applied rewrites 51.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]

6. Taylor expanded in l around 0

   \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{\color{blue}{{\ell}^{2}}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{\color{blue}{2}}} \]

8. Applied rewrites 17.1%

   \[\leadsto \frac{\mathsf{fma}\left({\left(\ell \cdot h\right)}^{0.5} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), -0.125, {\left(\frac{{\ell}^{3}}{h}\right)}^{0.5} \cdot d\right)}{\color{blue}{\ell \cdot \ell}} \]

9. Taylor expanded in d around 0

   \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right)}{\ell \cdot \ell} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \frac{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

    3. *-commutative N/A

       \[\leadsto \frac{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

    5. unpow-prod-down N/A

       \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

    6. lift-pow.f64 N/A

       \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

    7. lift-*.f64 N/A

       \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

    8. lift-/.f64 N/A

       \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

    9. pow1/2 N/A

       \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(h \cdot \ell\right)}^{\frac{1}{2}}}{\ell \cdot \ell} \]

    10. *-commutative N/A

        \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell \cdot \ell} \]

    11. metadata-eval N/A

        \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\left(\frac{1}{2}\right)}}{\ell \cdot \ell} \]

    12. lift-pow.f64 N/A

        \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\left(\frac{1}{2}\right)}}{\ell \cdot \ell} \]

    13. lift-*.f64 N/A

        \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\left(\frac{1}{2}\right)}}{\ell \cdot \ell} \]

    14. metadata-eval 16.6

        \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right) \cdot {\left(\ell \cdot h\right)}^{0.5}}{\ell \cdot \ell} \]

11. Applied rewrites 16.6%

    \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right) \cdot {\left(\ell \cdot h\right)}^{0.5}}{\ell \cdot \ell} \]
12. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell \cdot \ell} \]

    2. lift-/.f64 N/A

       \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell \cdot \color{blue}{\ell}} \]

    3. associate-/r* N/A

       \[\leadsto \frac{\frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell}}{\ell} \]

    4. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell}}{\ell} \]

13. Applied rewrites 19.6%

    \[\leadsto \frac{\frac{{\left(\ell \cdot h\right)}^{0.5} \cdot \left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right)}{\ell}}{\ell} \]
14. Add Preprocessing

Alternative 18: 17.0% accurate, N/A× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right) \cdot {\left(\ell \cdot h\right)}^{0.5}}{\ell \cdot \ell} \end{array} \]

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
 (/ (* (* (/ (pow (* D M) 2.0) d) -0.125) (pow (* l h) 0.5)) (* l l)))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	return (((pow((D * M), 2.0) / d) * -0.125) * pow((l * h), 0.5)) / (l * 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_1 * m) ** 2.0d0) / d) * (-0.125d0)) * ((l * h) ** 0.5d0)) / (l * 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 (((Math.pow((D * M), 2.0) / d) * -0.125) * Math.pow((l * h), 0.5)) / (l * l);
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	return (((math.pow((D * M), 2.0) / d) * -0.125) * math.pow((l * h), 0.5)) / (l * l)

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	return Float64(Float64(Float64(Float64((Float64(D * M) ^ 2.0) / d) * -0.125) * (Float64(l * h) ^ 0.5)) / Float64(l * 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 * M) ^ 2.0) / d) * -0.125) * ((l * h) ^ 0.5)) / (l * 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[(N[(N[(N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision] * N[Power[N[(l * h), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]

Derivation

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. Add Preprocessing

3. Taylor expanded in h around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

5. Applied rewrites 51.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, {\left({\left(\frac{h}{\ell}\right)}^{3}\right)}^{0.5}, {\left(\frac{h}{\ell}\right)}^{0.5} \cdot d\right)}{h}} \]

6. Taylor expanded in l around 0

   \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{\color{blue}{{\ell}^{2}}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{\color{blue}{2}}} \]

8. Applied rewrites 17.1%

   \[\leadsto \frac{\mathsf{fma}\left({\left(\ell \cdot h\right)}^{0.5} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right), -0.125, {\left(\frac{{\ell}^{3}}{h}\right)}^{0.5} \cdot d\right)}{\color{blue}{\ell \cdot \ell}} \]

9. Taylor expanded in d around 0

   \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right)}{\ell \cdot \ell} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \frac{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

    3. *-commutative N/A

       \[\leadsto \frac{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

    5. unpow-prod-down N/A

       \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

    6. lift-pow.f64 N/A

       \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

    7. lift-*.f64 N/A

       \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

    8. lift-/.f64 N/A

       \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot \sqrt{h \cdot \ell}}{\ell \cdot \ell} \]

    9. pow1/2 N/A

       \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(h \cdot \ell\right)}^{\frac{1}{2}}}{\ell \cdot \ell} \]

    10. *-commutative N/A

        \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\frac{1}{2}}}{\ell \cdot \ell} \]

    11. metadata-eval N/A

        \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\left(\frac{1}{2}\right)}}{\ell \cdot \ell} \]

    12. lift-pow.f64 N/A

        \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\left(\frac{1}{2}\right)}}{\ell \cdot \ell} \]

    13. lift-*.f64 N/A

        \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot \frac{-1}{8}\right) \cdot {\left(\ell \cdot h\right)}^{\left(\frac{1}{2}\right)}}{\ell \cdot \ell} \]

    14. metadata-eval 16.6

        \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right) \cdot {\left(\ell \cdot h\right)}^{0.5}}{\ell \cdot \ell} \]

11. Applied rewrites 16.6%

    \[\leadsto \frac{\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right) \cdot {\left(\ell \cdot h\right)}^{0.5}}{\ell \cdot \ell} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025066 
(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)))))