Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.8% → 74.9%

Time: 16.1s

Alternatives: 21

Speedup: 1.2×

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: 65.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 74.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - \frac{\frac{M\_m}{d} \cdot \left(\left(D\_m \cdot M\_m\right) \cdot \left(h \cdot 0.5\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D\_m}{d}}{2}\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-189}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{{\left(D\_m \cdot M\_m\right)}^{2}}{d} \cdot -0.125\right) \cdot \sqrt{\ell}, \sqrt{h}, \frac{{\ell}^{1.5}}{\sqrt{h}} \cdot d\right)}{\ell \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -5e-310)
   (*
    (* (pow (/ d h) (pow 2.0 -1.0)) (/ (sqrt (- d)) (sqrt (- l))))
    (-
     1.0
     (*
      (/ (* (/ M_m d) (* (* D_m M_m) (* h 0.5))) (* 2.0 l))
      (/ (/ D_m d) 2.0))))
   (if (<= d 7e-189)
     (/
      (fma
       (* (* (/ (pow (* D_m M_m) 2.0) d) -0.125) (sqrt l))
       (sqrt h)
       (* (/ (pow l 1.5) (sqrt h)) d))
      (* l l))
     (*
      (* (/ (sqrt d) (sqrt h)) (pow (/ d l) (pow 2.0 -1.0)))
      (-
       1.0
       (* (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -5e-310) {
		tmp = (pow((d / h), pow(2.0, -1.0)) * (sqrt(-d) / sqrt(-l))) * (1.0 - ((((M_m / d) * ((D_m * M_m) * (h * 0.5))) / (2.0 * l)) * ((D_m / d) / 2.0)));
	} else if (d <= 7e-189) {
		tmp = fma((((pow((D_m * M_m), 2.0) / d) * -0.125) * sqrt(l)), sqrt(h), ((pow(l, 1.5) / sqrt(h)) * d)) / (l * l);
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))) * Float64(1.0 - Float64(Float64(Float64(Float64(M_m / d) * Float64(Float64(D_m * M_m) * Float64(h * 0.5))) / Float64(2.0 * l)) * Float64(Float64(D_m / d) / 2.0))));
	elseif (d <= 7e-189)
		tmp = Float64(fma(Float64(Float64(Float64((Float64(D_m * M_m) ^ 2.0) / d) * -0.125) * sqrt(l)), sqrt(h), Float64(Float64((l ^ 1.5) / sqrt(h)) * d)) / Float64(l * l));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -5e-310], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(h * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7e-189], N[(N[(N[(N[(N[(N[Power[N[(D$95$m * M$95$m), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision] + N[(N[(N[Power[l, 1.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 69.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 65.7%

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

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 66.9

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

   6. Applied rewrites 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 70.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 70.9

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

   8. Applied rewrites 70.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 70.9

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. sqrt-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 78.6

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

   10. Applied rewrites 78.6%

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

   if -4.999999999999985e-310 < d < 7.0000000000000003e-189

   1. Initial program 29.3%

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

   3. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 25.3%

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

   7. Applied rewrites 68.6%

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

   if 7.0000000000000003e-189 < d

   1. Initial program 80.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      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 88.6

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

   4. Applied rewrites 88.6%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot 0.5\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-189}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right) \cdot \sqrt{\ell}, \sqrt{h}, \frac{{\ell}^{1.5}}{\sqrt{h}} \cdot d\right)}{\ell \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\_m\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{M\_m}{2} \cdot D\_m\right) \cdot t\_0, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot {t\_0}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-h\right) \cdot \left(\frac{\left(D\_m \cdot D\_m\right) \cdot 0.125}{\ell} \cdot \frac{M\_m \cdot M\_m}{d} - {h}^{-1}\right)\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ (/ D_m d) 2.0) M_m))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l)))))
        (t_2 (/ d (sqrt (* l h)))))
   (if (<= t_1 (- INFINITY))
     (* (fma (* (* (/ M_m 2.0) D_m) t_0) (* -0.5 (/ h l)) 1.0) t_2)
     (if (<= t_1 -2e-48)
       (* (fma (* -0.5 (pow t_0 2.0)) (/ h l) 1.0) (sqrt (* (/ d l) (/ d h))))
       (if (<= t_1 INFINITY)
         (* (sqrt (/ d l)) (sqrt (/ d h)))
         (*
          (*
           (- h)
           (- (* (/ (* (* D_m D_m) 0.125) l) (/ (* M_m M_m) d)) (pow h -1.0)))
          t_2))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M_m;
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = d / sqrt((l * h));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((((M_m / 2.0) * D_m) * t_0), (-0.5 * (h / l)), 1.0) * t_2;
	} else if (t_1 <= -2e-48) {
		tmp = fma((-0.5 * pow(t_0, 2.0)), (h / l), 1.0) * sqrt(((d / l) * (d / h)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = (-h * (((((D_m * D_m) * 0.125) / l) * ((M_m * M_m) / d)) - pow(h, -1.0))) * t_2;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(Float64(D_m / d) / 2.0) * M_m)
	t_1 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(Float64(M_m / 2.0) * D_m) * t_0), Float64(-0.5 * Float64(h / l)), 1.0) * t_2);
	elseif (t_1 <= -2e-48)
		tmp = Float64(fma(Float64(-0.5 * (t_0 ^ 2.0)), Float64(h / l), 1.0) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (t_1 <= Inf)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(Float64(-h) * Float64(Float64(Float64(Float64(Float64(D_m * D_m) * 0.125) / l) * Float64(Float64(M_m * M_m) / d)) - (h ^ -1.0))) * t_2);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(M$95$m / 2.0), $MachinePrecision] * D$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, -2e-48], N[(N[(N[(-0.5 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[((-h) * N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * 0.125), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 83.1%

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

   4. Applied rewrites 46.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 46.4

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

   6. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. associate-/l/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 76.3

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

   8. Applied rewrites 76.3%

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

   if -inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.9999999999999999e-48

   1. Initial program 99.0%

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

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 65.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 35.3

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

   6. Applied rewrites 35.3%

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

   8. Applied rewrites 65.4%

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

   if -1.9999999999999999e-48 < (*.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 89.0%

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

   4. Applied rewrites 83.3%

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

   5. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

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

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

      8. remove-double-neg N/A

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

      9. *-rgt-identity N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 87.7

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

   7. Applied rewrites 87.7%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 2.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 2.0

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

   6. Applied rewrites 22.1%

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

   7. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   9. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\left(\left(-h\right) \cdot \left(\frac{\left(D \cdot D\right) \cdot 0.125}{\ell} \cdot \frac{M \cdot M}{d} - \frac{1}{h}\right)\right)} \cdot \frac{d}{\sqrt{\ell \cdot h}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{M}{2} \cdot D\right) \cdot \left(\frac{\frac{D}{d}}{2} \cdot M\right), -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot {\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-h\right) \cdot \left(\frac{\left(D \cdot D\right) \cdot 0.125}{\ell} \cdot \frac{M \cdot M}{d} - {h}^{-1}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2} \cdot M\_m\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{M\_m}{2} \cdot D\_m\right) \cdot t\_0, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot {t\_0}^{2}, \frac{h}{\ell}, 1\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-h\right) \cdot \left(\frac{\left(D\_m \cdot D\_m\right) \cdot 0.125}{\ell} \cdot \frac{M\_m \cdot M\_m}{d} - {h}^{-1}\right)\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ (/ D_m d) 2.0) M_m))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l)))))
        (t_2 (/ d (sqrt (* l h)))))
   (if (<= t_1 (- INFINITY))
     (* (fma (* (* (/ M_m 2.0) D_m) t_0) (* -0.5 (/ h l)) 1.0) t_2)
     (if (<= t_1 0.0)
       (* (fma (* -0.5 (pow t_0 2.0)) (/ h l) 1.0) t_2)
       (if (<= t_1 INFINITY)
         (* (sqrt (/ d l)) (sqrt (/ d h)))
         (*
          (*
           (- h)
           (- (* (/ (* (* D_m D_m) 0.125) l) (/ (* M_m M_m) d)) (pow h -1.0)))
          t_2))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = ((D_m / d) / 2.0) * M_m;
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = d / sqrt((l * h));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((((M_m / 2.0) * D_m) * t_0), (-0.5 * (h / l)), 1.0) * t_2;
	} else if (t_1 <= 0.0) {
		tmp = fma((-0.5 * pow(t_0, 2.0)), (h / l), 1.0) * t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = (-h * (((((D_m * D_m) * 0.125) / l) * ((M_m * M_m) / d)) - pow(h, -1.0))) * t_2;
	}
	return tmp;
}

Julia

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

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(M$95$m / 2.0), $MachinePrecision] * D$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(-0.5 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[((-h) * N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * 0.125), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 83.1%

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

   4. Applied rewrites 46.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 46.4

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

   6. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. associate-/l/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 76.3

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

   8. Applied rewrites 76.3%

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

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

   1. Initial program 82.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 82.2

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

   4. Applied rewrites 82.2%

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

   6. Applied rewrites 47.2%

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

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

   1. Initial program 91.9%

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

   4. Applied rewrites 86.6%

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

   5. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

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

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

      8. remove-double-neg N/A

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

      9. *-rgt-identity N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 91.4

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

   7. Applied rewrites 91.4%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 2.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 2.0

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

   6. Applied rewrites 22.1%

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

   7. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   9. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\left(\left(-h\right) \cdot \left(\frac{\left(D \cdot D\right) \cdot 0.125}{\ell} \cdot \frac{M \cdot M}{d} - \frac{1}{h}\right)\right)} \cdot \frac{d}{\sqrt{\ell \cdot h}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.9%

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

Alternative 4: 66.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{M\_m}{2} \cdot D\_m\right) \cdot \left(t\_0 \cdot M\_m\right), -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D\_m}{2} \cdot \left(\frac{M\_m}{d} \cdot M\_m\right)\right)\right) \cdot t\_0\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-h\right) \cdot \left(\frac{\left(D\_m \cdot D\_m\right) \cdot 0.125}{\ell} \cdot \frac{M\_m \cdot M\_m}{d} - {h}^{-1}\right)\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (/ D_m d) 2.0))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l)))))
        (t_2 (/ d (sqrt (* l h)))))
   (if (<= t_1 (- INFINITY))
     (* (fma (* (* (/ M_m 2.0) D_m) (* t_0 M_m)) (* -0.5 (/ h l)) 1.0) t_2)
     (if (<= t_1 0.0)
       (*
        (* (/ (sqrt d) (sqrt h)) (/ (sqrt d) (sqrt l)))
        (- 1.0 (* (* (* (/ h l) 0.5) (* (/ D_m 2.0) (* (/ M_m d) M_m))) t_0)))
       (if (<= t_1 INFINITY)
         (* (sqrt (/ d l)) (sqrt (/ d h)))
         (*
          (*
           (- h)
           (- (* (/ (* (* D_m D_m) 0.125) l) (/ (* M_m M_m) d)) (pow h -1.0)))
          t_2))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m / d) / 2.0;
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = d / sqrt((l * h));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((((M_m / 2.0) * D_m) * (t_0 * M_m)), (-0.5 * (h / l)), 1.0) * t_2;
	} else if (t_1 <= 0.0) {
		tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0 - ((((h / l) * 0.5) * ((D_m / 2.0) * ((M_m / d) * M_m))) * t_0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = (-h * (((((D_m * D_m) * 0.125) / l) * ((M_m * M_m) / d)) - pow(h, -1.0))) * t_2;
	}
	return tmp;
}

Julia

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

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(M$95$m / 2.0), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(t$95$0 * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(h / l), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[((-h) * N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * 0.125), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 83.1%

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

   4. Applied rewrites 46.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 46.4

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

   6. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. associate-/l/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 76.3

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

   8. Applied rewrites 76.3%

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

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

   1. Initial program 82.2%

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

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 56.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 32.3

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

   6. Applied rewrites 32.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 32.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 38.7

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

   8. Applied rewrites 38.7%

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

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

   1. Initial program 91.9%

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

   4. Applied rewrites 86.6%

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

   5. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

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

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

      8. remove-double-neg N/A

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

      9. *-rgt-identity N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 91.4

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

   7. Applied rewrites 91.4%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 2.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 2.0

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

   6. Applied rewrites 22.1%

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

   7. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   9. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\left(\left(-h\right) \cdot \left(\frac{\left(D \cdot D\right) \cdot 0.125}{\ell} \cdot \frac{M \cdot M}{d} - \frac{1}{h}\right)\right)} \cdot \frac{d}{\sqrt{\ell \cdot h}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.0%

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

Alternative 5: 65.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{\frac{D\_m}{d}}{2}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{M\_m}{2} \cdot D\_m\right) \cdot \left(t\_1 \cdot M\_m\right), -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot t\_3\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t\_0\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D\_m}{2} \cdot \left(\frac{M\_m}{d} \cdot M\_m\right)\right)\right) \cdot t\_1\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-h\right) \cdot \left(\frac{\left(D\_m \cdot D\_m\right) \cdot 0.125}{\ell} \cdot \frac{M\_m \cdot M\_m}{d} - {h}^{-1}\right)\right) \cdot t\_3\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (/ (/ D_m d) 2.0))
        (t_2
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l)))))
        (t_3 (/ d (sqrt (* l h)))))
   (if (<= t_2 (- INFINITY))
     (* (fma (* (* (/ M_m 2.0) D_m) (* t_1 M_m)) (* -0.5 (/ h l)) 1.0) t_3)
     (if (<= t_2 -4e-42)
       (*
        (* (/ (sqrt d) (sqrt h)) t_0)
        (- 1.0 (* (* (* (/ h l) 0.5) (* (/ D_m 2.0) (* (/ M_m d) M_m))) t_1)))
       (if (<= t_2 INFINITY)
         (* t_0 (sqrt (/ d h)))
         (*
          (*
           (- h)
           (- (* (/ (* (* D_m D_m) 0.125) l) (/ (* M_m M_m) d)) (pow h -1.0)))
          t_3))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = (D_m / d) / 2.0;
	double t_2 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = d / sqrt((l * h));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma((((M_m / 2.0) * D_m) * (t_1 * M_m)), (-0.5 * (h / l)), 1.0) * t_3;
	} else if (t_2 <= -4e-42) {
		tmp = ((sqrt(d) / sqrt(h)) * t_0) * (1.0 - ((((h / l) * 0.5) * ((D_m / 2.0) * ((M_m / d) * M_m))) * t_1));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * sqrt((d / h));
	} else {
		tmp = (-h * (((((D_m * D_m) * 0.125) / l) * ((M_m * M_m) / d)) - pow(h, -1.0))) * t_3;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64(D_m / d) / 2.0)
	t_2 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_3 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(Float64(M_m / 2.0) * D_m) * Float64(t_1 * M_m)), Float64(-0.5 * Float64(h / l)), 1.0) * t_3);
	elseif (t_2 <= -4e-42)
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_0) * Float64(1.0 - Float64(Float64(Float64(Float64(h / l) * 0.5) * Float64(Float64(D_m / 2.0) * Float64(Float64(M_m / d) * M_m))) * t_1)));
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(Float64(-h) * Float64(Float64(Float64(Float64(Float64(D_m * D_m) * 0.125) / l) * Float64(Float64(M_m * M_m) / d)) - (h ^ -1.0))) * t_3);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(N[(N[(M$95$m / 2.0), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(t$95$1 * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$2, -4e-42], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(h / l), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[((-h) * N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * 0.125), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 83.1%

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

   4. Applied rewrites 46.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 46.4

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

   6. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. associate-/l/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 76.3

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

   8. Applied rewrites 76.3%

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

   if -inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.00000000000000015e-42

   1. Initial program 98.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 69.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 37.5

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

   6. Applied rewrites 37.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 37.5

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 37.5

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

   8. Applied rewrites 37.5%

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

   if -4.00000000000000015e-42 < (*.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 89.1%

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

   4. Applied rewrites 82.7%

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

   5. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

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

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

      8. remove-double-neg N/A

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

      9. *-rgt-identity N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 87.1

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

   7. Applied rewrites 87.1%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 2.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 2.0

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

   6. Applied rewrites 22.1%

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

   7. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   9. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\left(\left(-h\right) \cdot \left(\frac{\left(D \cdot D\right) \cdot 0.125}{\ell} \cdot \frac{M \cdot M}{d} - \frac{1}{h}\right)\right)} \cdot \frac{d}{\sqrt{\ell \cdot h}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{M}{2} \cdot D\right) \cdot \left(\frac{\frac{D}{d}}{2} \cdot M\right), -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-h\right) \cdot \left(\frac{\left(D \cdot D\right) \cdot 0.125}{\ell} \cdot \frac{M \cdot M}{d} - {h}^{-1}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 82.8%

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

   4. Applied rewrites 37.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 37.3

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

   6. Applied rewrites 59.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/r* N/A

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

      9. lower-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. associate-/l/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 61.5

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

   8. Applied rewrites 61.5%

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

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

   1. Initial program 91.9%

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

   4. Applied rewrites 86.6%

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

   5. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

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

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

      8. remove-double-neg N/A

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

      9. *-rgt-identity N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 91.4

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

   7. Applied rewrites 91.4%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 2.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 2.0

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

   6. Applied rewrites 22.1%

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

   7. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   9. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\left(\left(-h\right) \cdot \left(\frac{\left(D \cdot D\right) \cdot 0.125}{\ell} \cdot \frac{M \cdot M}{d} - \frac{1}{h}\right)\right)} \cdot \frac{d}{\sqrt{\ell \cdot h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.8%

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

Alternative 7: 65.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(0.25 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot M\_m\right) \cdot M\_m}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-h\right) \cdot \left(\frac{\left(D\_m \cdot D\_m\right) \cdot 0.125}{\ell} \cdot \frac{M\_m \cdot M\_m}{d} - {h}^{-1}\right)\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l)))))
        (t_1 (/ d (sqrt (* l h)))))
   (if (<= t_0 0.0)
     (*
      (fma (/ (* (* (* 0.25 (* D_m D_m)) M_m) M_m) d) (* -0.5 (/ h l)) 1.0)
      t_1)
     (if (<= t_0 INFINITY)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (*
        (*
         (- h)
         (- (* (/ (* (* D_m D_m) 0.125) l) (/ (* M_m M_m) d)) (pow h -1.0)))
        t_1)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = d / sqrt((l * h));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fma(((((0.25 * (D_m * D_m)) * M_m) * M_m) / d), (-0.5 * (h / l)), 1.0) * t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = (-h * (((((D_m * D_m) * 0.125) / l) * ((M_m * M_m) / d)) - pow(h, -1.0))) * t_1;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(0.25 * Float64(D_m * D_m)) * M_m) * M_m) / d), Float64(-0.5 * Float64(h / l)), 1.0) * t_1);
	elseif (t_0 <= Inf)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(Float64(-h) * Float64(Float64(Float64(Float64(Float64(D_m * D_m) * 0.125) / l) * Float64(Float64(M_m * M_m) / d)) - (h ^ -1.0))) * t_1);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(N[(N[(0.25 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[((-h) * N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * 0.125), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 82.8%

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

   4. Applied rewrites 37.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 37.3

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

   6. Applied rewrites 59.2%

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

   7. Taylor expanded in M around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 59.0

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(0.25 \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot M\right) \cdot M}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

   9. Applied rewrites 59.0%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\left(0.25 \cdot \left(D \cdot D\right)\right) \cdot M\right) \cdot M}}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

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

   1. Initial program 91.9%

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

   4. Applied rewrites 86.6%

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

   5. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

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

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

      8. remove-double-neg N/A

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

      9. *-rgt-identity N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 91.4

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

   7. Applied rewrites 91.4%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 2.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 2.0

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

   6. Applied rewrites 22.1%

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

   7. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   9. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\left(\left(-h\right) \cdot \left(\frac{\left(D \cdot D\right) \cdot 0.125}{\ell} \cdot \frac{M \cdot M}{d} - \frac{1}{h}\right)\right)} \cdot \frac{d}{\sqrt{\ell \cdot h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(0.25 \cdot \left(D \cdot D\right)\right) \cdot M\right) \cdot M}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-h\right) \cdot \left(\frac{\left(D \cdot D\right) \cdot 0.125}{\ell} \cdot \frac{M \cdot M}{d} - {h}^{-1}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(0.25 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot M\_m\right) \cdot M\_m}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(D\_m \cdot D\_m\right) \cdot -0.125}{\ell} \cdot \left(\frac{M\_m \cdot M\_m}{d} \cdot h\right)\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (*
            (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0))
            (/ h l)))))
        (t_1 (/ d (sqrt (* l h)))))
   (if (<= t_0 0.0)
     (*
      (fma (/ (* (* (* 0.25 (* D_m D_m)) M_m) M_m) d) (* -0.5 (/ h l)) 1.0)
      t_1)
     (if (<= t_0 INFINITY)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (* (* (/ (* (* D_m D_m) -0.125) l) (* (/ (* M_m M_m) d) h)) t_1)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = d / sqrt((l * h));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fma(((((0.25 * (D_m * D_m)) * M_m) * M_m) / d), (-0.5 * (h / l)), 1.0) * t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = ((((D_m * D_m) * -0.125) / l) * (((M_m * M_m) / d) * h)) * t_1;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(0.25 * Float64(D_m * D_m)) * M_m) * M_m) / d), Float64(-0.5 * Float64(h / l)), 1.0) * t_1);
	elseif (t_0 <= Inf)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(D_m * D_m) * -0.125) / l) * Float64(Float64(Float64(M_m * M_m) / d) * h)) * t_1);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(N[(N[(0.25 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 82.8%

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

   4. Applied rewrites 37.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 37.3

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

   6. Applied rewrites 59.2%

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

   7. Taylor expanded in M around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 59.0

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(0.25 \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot M\right) \cdot M}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

   9. Applied rewrites 59.0%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\left(0.25 \cdot \left(D \cdot D\right)\right) \cdot M\right) \cdot M}}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

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

   1. Initial program 91.9%

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

   4. Applied rewrites 86.6%

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

   5. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

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

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

      8. remove-double-neg N/A

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

      9. *-rgt-identity N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 91.4

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

   7. Applied rewrites 91.4%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 2.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 2.0

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

   6. Applied rewrites 22.1%

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

   7. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 26.8

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

   9. Applied rewrites 26.8%

      \[\leadsto \color{blue}{\left(\frac{\left(D \cdot D\right) \cdot -0.125}{\ell} \cdot \left(\frac{M \cdot M}{d} \cdot h\right)\right)} \cdot \frac{d}{\sqrt{\ell \cdot h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(0.25 \cdot \left(D \cdot D\right)\right) \cdot M\right) \cdot M}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(D \cdot D\right) \cdot -0.125}{\ell} \cdot \left(\frac{M \cdot M}{d} \cdot h\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 88.1

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

   4. Applied rewrites 88.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. lift-sqrt.f64 88.1

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

   6. Applied rewrites 88.1%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 2.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 2.0

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

   6. Applied rewrites 22.1%

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

   7. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   9. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\left(\left(-h\right) \cdot \left(\frac{\left(D \cdot D\right) \cdot 0.125}{\ell} \cdot \frac{M \cdot M}{d} - \frac{1}{h}\right)\right)} \cdot \frac{d}{\sqrt{\ell \cdot h}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

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

Alternative 10: 70.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.5 \cdot {\left(\frac{\frac{D\_m}{d}}{2} \cdot M\_m\right)}^{2}, \frac{h}{\ell}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-h\right) \cdot \left(\frac{\left(D\_m \cdot D\_m\right) \cdot 0.125}{\ell} \cdot \frac{M\_m \cdot M\_m}{d} - {h}^{-1}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 88.1

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

   4. Applied rewrites 88.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 87.1%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 2.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 2.0

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

   6. Applied rewrites 22.1%

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

   7. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   9. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\left(\left(-h\right) \cdot \left(\frac{\left(D \cdot D\right) \cdot 0.125}{\ell} \cdot \frac{M \cdot M}{d} - \frac{1}{h}\right)\right)} \cdot \frac{d}{\sqrt{\ell \cdot h}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

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

Alternative 11: 43.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if ((math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= 0.0:
		tmp = math.sqrt(math.pow((l * h), -1.0)) * d
	else:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 0.0)
		tmp = Float64(sqrt((Float64(l * h) ^ -1.0)) * d);
	else
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (((((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= 0.0)
		tmp = sqrt(((l * h) ^ -1.0)) * d;
	else
		tmp = sqrt((d / l)) * sqrt((d / h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 82.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 16.7

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 16.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 64.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   5. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \sqrt{\frac{d}{h}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \cdot \sqrt{\frac{d}{h}} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{d}{\ell}}\right)\right)} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{d}{h}} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{d}{\ell}}\right)\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \cdot \sqrt{\frac{d}{h}} \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{d}{\ell}}\right)\right) \cdot \color{blue}{-1}\right) \cdot \sqrt{\frac{d}{h}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{d}{\ell}}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \sqrt{\frac{d}{h}} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\frac{d}{\ell}}\right)\right) \cdot 1\right)\right)} \cdot \sqrt{\frac{d}{h}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\frac{d}{\ell}}\right)\right)\right)\right) \cdot 1\right)} \cdot \sqrt{\frac{d}{h}} \]

      8. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot 1\right) \cdot \sqrt{\frac{d}{h}} \]

      9. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

      11. lower-/.f64 69.4

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

   7. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2}\\ \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - \frac{\frac{M\_m}{d} \cdot \left(\left(D\_m \cdot M\_m\right) \cdot \left(h \cdot 0.5\right)\right)}{2 \cdot \ell} \cdot t\_0\right)\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-188}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{{\left(D\_m \cdot M\_m\right)}^{2}}{d} \cdot -0.125\right) \cdot \sqrt{\ell}, \sqrt{h}, \frac{{\ell}^{1.5}}{\sqrt{h}} \cdot d\right)}{\ell \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M\_m}{d} \cdot \left(D\_m \cdot \frac{M\_m}{2}\right)\right)\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (/ D_m d) 2.0)))
   (if (<= d -5e-310)
     (*
      (* (pow (/ d h) (pow 2.0 -1.0)) (/ (sqrt (- d)) (sqrt (- l))))
      (- 1.0 (* (/ (* (/ M_m d) (* (* D_m M_m) (* h 0.5))) (* 2.0 l)) t_0)))
     (if (<= d 1.35e-188)
       (/
        (fma
         (* (* (/ (pow (* D_m M_m) 2.0) d) -0.125) (sqrt l))
         (sqrt h)
         (* (/ (pow l 1.5) (sqrt h)) d))
        (* l l))
       (*
        (* (/ (sqrt d) (sqrt h)) (pow (/ d l) (pow 2.0 -1.0)))
        (-
         1.0
         (* (* (* (/ h l) 0.5) (* (/ M_m d) (* D_m (/ M_m 2.0)))) t_0)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m / d) / 2.0;
	double tmp;
	if (d <= -5e-310) {
		tmp = (pow((d / h), pow(2.0, -1.0)) * (sqrt(-d) / sqrt(-l))) * (1.0 - ((((M_m / d) * ((D_m * M_m) * (h * 0.5))) / (2.0 * l)) * t_0));
	} else if (d <= 1.35e-188) {
		tmp = fma((((pow((D_m * M_m), 2.0) / d) * -0.125) * sqrt(l)), sqrt(h), ((pow(l, 1.5) / sqrt(h)) * d)) / (l * l);
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((((h / l) * 0.5) * ((M_m / d) * (D_m * (M_m / 2.0)))) * t_0));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(D_m / d) / 2.0)
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))) * Float64(1.0 - Float64(Float64(Float64(Float64(M_m / d) * Float64(Float64(D_m * M_m) * Float64(h * 0.5))) / Float64(2.0 * l)) * t_0)));
	elseif (d <= 1.35e-188)
		tmp = Float64(fma(Float64(Float64(Float64((Float64(D_m * M_m) ^ 2.0) / d) * -0.125) * sqrt(l)), sqrt(h), Float64(Float64((l ^ 1.5) / sqrt(h)) * d)) / Float64(l * l));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64(Float64(Float64(h / l) * 0.5) * Float64(Float64(M_m / d) * Float64(D_m * Float64(M_m / 2.0)))) * t_0)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[d, -5e-310], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(h * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.35e-188], N[(N[(N[(N[(N[(N[Power[N[(D$95$m * M$95$m), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision] + N[(N[(N[Power[l, 1.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(h / l), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(D$95$m * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 69.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot M\right) \cdot \frac{D}{2 \cdot d}\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot M\right)\right) \cdot \frac{D}{2 \cdot d}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot M\right)\right) \cdot \frac{D}{2 \cdot d}}\right) \]

   4. Applied rewrites 65.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right)} \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right)} \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\color{blue}{\frac{D}{2}} \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D \cdot \left(\frac{M}{d} \cdot M\right)}{2}} \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D \cdot \left(\frac{M}{d} \cdot M\right)}{2} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right)}\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D \cdot \left(\frac{M}{d} \cdot M\right)}{2} \cdot \left(\color{blue}{\frac{h}{\ell}} \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D \cdot \left(\frac{M}{d} \cdot M\right)}{2} \cdot \color{blue}{\frac{h \cdot \frac{1}{2}}{\ell}}\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      10. frac-times N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(D \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{2 \cdot \ell}} \cdot \frac{\frac{D}{d}}{2}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(D \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{2 \cdot \ell}} \cdot \frac{\frac{D}{d}}{2}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(D \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right)} \cdot \left(h \cdot \frac{1}{2}\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right)} \cdot \left(h \cdot \frac{1}{2}\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      17. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      18. lower-*.f64 66.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(0.5 \cdot h\right)}{\color{blue}{2 \cdot \ell}} \cdot \frac{\frac{D}{d}}{2}\right) \]

   6. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(0.5 \cdot h\right)}{2 \cdot \ell}} \cdot \frac{\frac{D}{d}}{2}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\color{blue}{\left(\frac{M}{d} \cdot M\right)} \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M}{d} \cdot \left(M \cdot D\right)\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M}{d} \cdot \left(\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M}{d} \cdot \left(\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \left(\frac{1}{2} \cdot h\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. lower-*.f64 70.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \left(0.5 \cdot h\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \color{blue}{\left(h \cdot \frac{1}{2}\right)}\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      12. lower-*.f64 70.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \color{blue}{\left(h \cdot 0.5\right)}\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

   8. Applied rewrites 70.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot 0.5\right)\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. metadata-eval 70.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot 0.5\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lift-neg.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{\color{blue}{-d}}{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{\sqrt{-d}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      12. lower-neg.f64 78.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{\color{blue}{-\ell}}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot 0.5\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

   10. Applied rewrites 78.6%

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot 0.5\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

   if -4.999999999999985e-310 < d < 1.35e-188

   1. Initial program 29.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{2}}} \]

   5. Applied rewrites 25.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot D}{d}, \sqrt{\ell \cdot h}, \sqrt{\frac{{\ell}^{3}}{h}} \cdot d\right)}{\ell \cdot \ell}} \]
   6. Step-by-step derivation

   7. Applied rewrites 68.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right) \cdot \sqrt{\ell}, \sqrt{h}, \frac{{\ell}^{1.5}}{\sqrt{h}} \cdot d\right)}{\color{blue}{\ell} \cdot \ell} \]

   if 1.35e-188 < d

   1. Initial program 80.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot M\right) \cdot \frac{D}{2 \cdot d}\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot M\right)\right) \cdot \frac{D}{2 \cdot d}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot M\right)\right) \cdot \frac{D}{2 \cdot d}}\right) \]

   4. Applied rewrites 74.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      11. lower-sqrt.f64 82.4

          \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

   6. Applied rewrites 82.4%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)}\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{\frac{D}{2}} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{D \cdot \left(\frac{M}{d} \cdot M\right)}{2}}\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot M\right) \cdot D}}{2}\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot M\right) \cdot D}}{2}\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot M\right) \cdot D}}{2}\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot M\right)} \cdot D}{2}\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{\color{blue}{\frac{M}{d} \cdot \left(M \cdot D\right)}}{2}\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. associate-/l* N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M \cdot D}{2}\right)}\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M \cdot D}{2}\right)}\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d} \cdot \frac{\color{blue}{D \cdot M}}{2}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      12. associate-/l* N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d} \cdot \color{blue}{\left(D \cdot \frac{M}{2}\right)}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d} \cdot \color{blue}{\left(D \cdot \frac{M}{2}\right)}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      14. lower-/.f64 86.9

          \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d} \cdot \left(D \cdot \color{blue}{\frac{M}{2}}\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

   8. Applied rewrites 86.9%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot \left(D \cdot \frac{M}{2}\right)\right)}\right) \cdot \frac{\frac{D}{d}}{2}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot 0.5\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right)\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-188}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{{\left(D \cdot M\right)}^{2}}{d} \cdot -0.125\right) \cdot \sqrt{\ell}, \sqrt{h}, \frac{{\ell}^{1.5}}{\sqrt{h}} \cdot d\right)}{\ell \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d} \cdot \left(D \cdot \frac{M}{2}\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2}\\ \mathbf{if}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - \frac{\frac{M\_m}{d} \cdot \left(\left(D\_m \cdot M\_m\right) \cdot \left(h \cdot 0.5\right)\right)}{2 \cdot \ell} \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D\_m}{2} \cdot \left(\frac{M\_m}{d} \cdot M\_m\right)\right)\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (/ D_m d) 2.0)))
   (if (<= h -4e-310)
     (*
      (* (pow (/ d h) (pow 2.0 -1.0)) (/ (sqrt (- d)) (sqrt (- l))))
      (- 1.0 (* (/ (* (/ M_m d) (* (* D_m M_m) (* h 0.5))) (* 2.0 l)) t_0)))
     (*
      (* (/ (sqrt d) (sqrt h)) (/ (sqrt d) (sqrt l)))
      (- 1.0 (* (* (* (/ h l) 0.5) (* (/ D_m 2.0) (* (/ M_m d) M_m))) t_0))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m / d) / 2.0;
	double tmp;
	if (h <= -4e-310) {
		tmp = (pow((d / h), pow(2.0, -1.0)) * (sqrt(-d) / sqrt(-l))) * (1.0 - ((((M_m / d) * ((D_m * M_m) * (h * 0.5))) / (2.0 * l)) * t_0));
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0 - ((((h / l) * 0.5) * ((D_m / 2.0) * ((M_m / d) * M_m))) * t_0));
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_m / d) / 2.0d0
    if (h <= (-4d-310)) then
        tmp = (((d / h) ** (2.0d0 ** (-1.0d0))) * (sqrt(-d) / sqrt(-l))) * (1.0d0 - ((((m_m / d) * ((d_m * m_m) * (h * 0.5d0))) / (2.0d0 * l)) * t_0))
    else
        tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0d0 - ((((h / l) * 0.5d0) * ((d_m / 2.0d0) * ((m_m / d) * m_m))) * t_0))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m / d) / 2.0;
	double tmp;
	if (h <= -4e-310) {
		tmp = (Math.pow((d / h), Math.pow(2.0, -1.0)) * (Math.sqrt(-d) / Math.sqrt(-l))) * (1.0 - ((((M_m / d) * ((D_m * M_m) * (h * 0.5))) / (2.0 * l)) * t_0));
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * (Math.sqrt(d) / Math.sqrt(l))) * (1.0 - ((((h / l) * 0.5) * ((D_m / 2.0) * ((M_m / d) * M_m))) * t_0));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (D_m / d) / 2.0
	tmp = 0
	if h <= -4e-310:
		tmp = (math.pow((d / h), math.pow(2.0, -1.0)) * (math.sqrt(-d) / math.sqrt(-l))) * (1.0 - ((((M_m / d) * ((D_m * M_m) * (h * 0.5))) / (2.0 * l)) * t_0))
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * (math.sqrt(d) / math.sqrt(l))) * (1.0 - ((((h / l) * 0.5) * ((D_m / 2.0) * ((M_m / d) * M_m))) * t_0))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(D_m / d) / 2.0)
	tmp = 0.0
	if (h <= -4e-310)
		tmp = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))) * Float64(1.0 - Float64(Float64(Float64(Float64(M_m / d) * Float64(Float64(D_m * M_m) * Float64(h * 0.5))) / Float64(2.0 * l)) * t_0)));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - Float64(Float64(Float64(Float64(h / l) * 0.5) * Float64(Float64(D_m / 2.0) * Float64(Float64(M_m / d) * M_m))) * t_0)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (D_m / d) / 2.0;
	tmp = 0.0;
	if (h <= -4e-310)
		tmp = (((d / h) ^ (2.0 ^ -1.0)) * (sqrt(-d) / sqrt(-l))) * (1.0 - ((((M_m / d) * ((D_m * M_m) * (h * 0.5))) / (2.0 * l)) * t_0));
	else
		tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0 - ((((h / l) * 0.5) * ((D_m / 2.0) * ((M_m / d) * M_m))) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[h, -4e-310], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(h * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(h / l), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -3.999999999999988e-310

   1. Initial program 69.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot M\right) \cdot \frac{D}{2 \cdot d}\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot M\right)\right) \cdot \frac{D}{2 \cdot d}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot M\right)\right) \cdot \frac{D}{2 \cdot d}}\right) \]

   4. Applied rewrites 65.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right)} \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right)} \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\color{blue}{\frac{D}{2}} \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D \cdot \left(\frac{M}{d} \cdot M\right)}{2}} \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D \cdot \left(\frac{M}{d} \cdot M\right)}{2} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right)}\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D \cdot \left(\frac{M}{d} \cdot M\right)}{2} \cdot \left(\color{blue}{\frac{h}{\ell}} \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D \cdot \left(\frac{M}{d} \cdot M\right)}{2} \cdot \color{blue}{\frac{h \cdot \frac{1}{2}}{\ell}}\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      10. frac-times N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(D \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{2 \cdot \ell}} \cdot \frac{\frac{D}{d}}{2}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(D \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{2 \cdot \ell}} \cdot \frac{\frac{D}{d}}{2}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(D \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right)} \cdot \left(h \cdot \frac{1}{2}\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right)} \cdot \left(h \cdot \frac{1}{2}\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      17. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      18. lower-*.f64 66.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(0.5 \cdot h\right)}{\color{blue}{2 \cdot \ell}} \cdot \frac{\frac{D}{d}}{2}\right) \]

   6. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(0.5 \cdot h\right)}{2 \cdot \ell}} \cdot \frac{\frac{D}{d}}{2}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\color{blue}{\left(\frac{M}{d} \cdot M\right)} \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M}{d} \cdot \left(M \cdot D\right)\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M}{d} \cdot \left(\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M}{d} \cdot \left(\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \left(\frac{1}{2} \cdot h\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. lower-*.f64 70.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \left(0.5 \cdot h\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \color{blue}{\left(h \cdot \frac{1}{2}\right)}\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      12. lower-*.f64 70.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \color{blue}{\left(h \cdot 0.5\right)}\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

   8. Applied rewrites 70.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot 0.5\right)\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. metadata-eval 70.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot 0.5\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lift-neg.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{\color{blue}{-d}}{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{\sqrt{-d}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      12. lower-neg.f64 78.6

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{\color{blue}{-\ell}}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot 0.5\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

   10. Applied rewrites 78.6%

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot 0.5\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

   if -3.999999999999988e-310 < h

   1. Initial program 71.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot M\right) \cdot \frac{D}{2 \cdot d}\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot M\right)\right) \cdot \frac{D}{2 \cdot d}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot M\right)\right) \cdot \frac{D}{2 \cdot d}}\right) \]

   4. Applied rewrites 66.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      11. lower-sqrt.f64 76.0

          \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

   6. Applied rewrites 76.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. metadata-eval 76.0

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. lower-sqrt.f64 79.0

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

   8. Applied rewrites 79.0%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot 0.5\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2}\\ \mathbf{if}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \frac{\frac{M\_m}{d} \cdot \left(\left(D\_m \cdot M\_m\right) \cdot \left(h \cdot 0.5\right)\right)}{2 \cdot \ell} \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D\_m}{2} \cdot \left(\frac{M\_m}{d} \cdot M\_m\right)\right)\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (/ D_m d) 2.0)))
   (if (<= h -4e-310)
     (*
      (* (/ (sqrt (- d)) (sqrt (- h))) (pow (/ d l) (pow 2.0 -1.0)))
      (- 1.0 (* (/ (* (/ M_m d) (* (* D_m M_m) (* h 0.5))) (* 2.0 l)) t_0)))
     (*
      (* (/ (sqrt d) (sqrt h)) (/ (sqrt d) (sqrt l)))
      (- 1.0 (* (* (* (/ h l) 0.5) (* (/ D_m 2.0) (* (/ M_m d) M_m))) t_0))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m / d) / 2.0;
	double tmp;
	if (h <= -4e-310) {
		tmp = ((sqrt(-d) / sqrt(-h)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((((M_m / d) * ((D_m * M_m) * (h * 0.5))) / (2.0 * l)) * t_0));
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0 - ((((h / l) * 0.5) * ((D_m / 2.0) * ((M_m / d) * M_m))) * t_0));
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_m / d) / 2.0d0
    if (h <= (-4d-310)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - ((((m_m / d) * ((d_m * m_m) * (h * 0.5d0))) / (2.0d0 * l)) * t_0))
    else
        tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0d0 - ((((h / l) * 0.5d0) * ((d_m / 2.0d0) * ((m_m / d) * m_m))) * t_0))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m / d) / 2.0;
	double tmp;
	if (h <= -4e-310) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((((M_m / d) * ((D_m * M_m) * (h * 0.5))) / (2.0 * l)) * t_0));
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * (Math.sqrt(d) / Math.sqrt(l))) * (1.0 - ((((h / l) * 0.5) * ((D_m / 2.0) * ((M_m / d) * M_m))) * t_0));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (D_m / d) / 2.0
	tmp = 0
	if h <= -4e-310:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((((M_m / d) * ((D_m * M_m) * (h * 0.5))) / (2.0 * l)) * t_0))
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * (math.sqrt(d) / math.sqrt(l))) * (1.0 - ((((h / l) * 0.5) * ((D_m / 2.0) * ((M_m / d) * M_m))) * t_0))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(D_m / d) / 2.0)
	tmp = 0.0
	if (h <= -4e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64(Float64(Float64(M_m / d) * Float64(Float64(D_m * M_m) * Float64(h * 0.5))) / Float64(2.0 * l)) * t_0)));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - Float64(Float64(Float64(Float64(h / l) * 0.5) * Float64(Float64(D_m / 2.0) * Float64(Float64(M_m / d) * M_m))) * t_0)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (D_m / d) / 2.0;
	tmp = 0.0;
	if (h <= -4e-310)
		tmp = ((sqrt(-d) / sqrt(-h)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - ((((M_m / d) * ((D_m * M_m) * (h * 0.5))) / (2.0 * l)) * t_0));
	else
		tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0 - ((((h / l) * 0.5) * ((D_m / 2.0) * ((M_m / d) * M_m))) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[h, -4e-310], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(h * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(h / l), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -3.999999999999988e-310

   1. Initial program 69.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot M\right) \cdot \frac{D}{2 \cdot d}\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot M\right)\right) \cdot \frac{D}{2 \cdot d}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot M\right)\right) \cdot \frac{D}{2 \cdot d}}\right) \]

   4. Applied rewrites 65.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right)} \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right)} \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\color{blue}{\frac{D}{2}} \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D \cdot \left(\frac{M}{d} \cdot M\right)}{2}} \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D \cdot \left(\frac{M}{d} \cdot M\right)}{2} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right)}\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D \cdot \left(\frac{M}{d} \cdot M\right)}{2} \cdot \left(\color{blue}{\frac{h}{\ell}} \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D \cdot \left(\frac{M}{d} \cdot M\right)}{2} \cdot \color{blue}{\frac{h \cdot \frac{1}{2}}{\ell}}\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      10. frac-times N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(D \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{2 \cdot \ell}} \cdot \frac{\frac{D}{d}}{2}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(D \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{2 \cdot \ell}} \cdot \frac{\frac{D}{d}}{2}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(D \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right)} \cdot \left(h \cdot \frac{1}{2}\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right)} \cdot \left(h \cdot \frac{1}{2}\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      17. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      18. lower-*.f64 66.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(0.5 \cdot h\right)}{\color{blue}{2 \cdot \ell}} \cdot \frac{\frac{D}{d}}{2}\right) \]

   6. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(0.5 \cdot h\right)}{2 \cdot \ell}} \cdot \frac{\frac{D}{d}}{2}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\color{blue}{\left(\frac{M}{d} \cdot M\right)} \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\frac{M}{d} \cdot \left(M \cdot D\right)\right)} \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M}{d} \cdot \left(\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M}{d} \cdot \left(\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \left(\frac{1}{2} \cdot h\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. lower-*.f64 70.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \left(0.5 \cdot h\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \color{blue}{\left(h \cdot \frac{1}{2}\right)}\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      12. lower-*.f64 70.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \color{blue}{\left(h \cdot 0.5\right)}\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

   8. Applied rewrites 70.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot 0.5\right)\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. metadata-eval 70.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot 0.5\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lift-neg.f64 N/A

         \[\leadsto \left(\sqrt{\frac{\color{blue}{-d}}{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{-d}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot \frac{1}{2}\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      12. lower-neg.f64 75.6

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot 0.5\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

   10. Applied rewrites 75.6%

       \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot 0.5\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

   if -3.999999999999988e-310 < h

   1. Initial program 71.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot M\right) \cdot \frac{D}{2 \cdot d}\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot M\right)\right) \cdot \frac{D}{2 \cdot d}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot M\right)\right) \cdot \frac{D}{2 \cdot d}}\right) \]

   4. Applied rewrites 66.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      11. lower-sqrt.f64 76.0

          \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

   6. Applied rewrites 76.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. metadata-eval 76.0

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. lower-sqrt.f64 79.0

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

   8. Applied rewrites 79.0%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \frac{\frac{M}{d} \cdot \left(\left(D \cdot M\right) \cdot \left(h \cdot 0.5\right)\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 75.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{D\_m}{d}}{2}\\ t_1 := \frac{M\_m}{d} \cdot M\_m\\ \mathbf{if}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \frac{\left(t\_1 \cdot D\_m\right) \cdot \left(0.5 \cdot h\right)}{2 \cdot \ell} \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D\_m}{2} \cdot t\_1\right)\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (/ D_m d) 2.0)) (t_1 (* (/ M_m d) M_m)))
   (if (<= h -4e-310)
     (*
      (* (/ (sqrt (- d)) (sqrt (- h))) (pow (/ d l) (pow 2.0 -1.0)))
      (- 1.0 (* (/ (* (* t_1 D_m) (* 0.5 h)) (* 2.0 l)) t_0)))
     (*
      (* (/ (sqrt d) (sqrt h)) (/ (sqrt d) (sqrt l)))
      (- 1.0 (* (* (* (/ h l) 0.5) (* (/ D_m 2.0) t_1)) t_0))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m / d) / 2.0;
	double t_1 = (M_m / d) * M_m;
	double tmp;
	if (h <= -4e-310) {
		tmp = ((sqrt(-d) / sqrt(-h)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((((t_1 * D_m) * (0.5 * h)) / (2.0 * l)) * t_0));
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0 - ((((h / l) * 0.5) * ((D_m / 2.0) * t_1)) * t_0));
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (d_m / d) / 2.0d0
    t_1 = (m_m / d) * m_m
    if (h <= (-4d-310)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - ((((t_1 * d_m) * (0.5d0 * h)) / (2.0d0 * l)) * t_0))
    else
        tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0d0 - ((((h / l) * 0.5d0) * ((d_m / 2.0d0) * t_1)) * t_0))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m / d) / 2.0;
	double t_1 = (M_m / d) * M_m;
	double tmp;
	if (h <= -4e-310) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((((t_1 * D_m) * (0.5 * h)) / (2.0 * l)) * t_0));
	} else {
		tmp = ((Math.sqrt(d) / Math.sqrt(h)) * (Math.sqrt(d) / Math.sqrt(l))) * (1.0 - ((((h / l) * 0.5) * ((D_m / 2.0) * t_1)) * t_0));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (D_m / d) / 2.0
	t_1 = (M_m / d) * M_m
	tmp = 0
	if h <= -4e-310:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((((t_1 * D_m) * (0.5 * h)) / (2.0 * l)) * t_0))
	else:
		tmp = ((math.sqrt(d) / math.sqrt(h)) * (math.sqrt(d) / math.sqrt(l))) * (1.0 - ((((h / l) * 0.5) * ((D_m / 2.0) * t_1)) * t_0))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(D_m / d) / 2.0)
	t_1 = Float64(Float64(M_m / d) * M_m)
	tmp = 0.0
	if (h <= -4e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64(Float64(Float64(t_1 * D_m) * Float64(0.5 * h)) / Float64(2.0 * l)) * t_0)));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - Float64(Float64(Float64(Float64(h / l) * 0.5) * Float64(Float64(D_m / 2.0) * t_1)) * t_0)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (D_m / d) / 2.0;
	t_1 = (M_m / d) * M_m;
	tmp = 0.0;
	if (h <= -4e-310)
		tmp = ((sqrt(-d) / sqrt(-h)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - ((((t_1 * D_m) * (0.5 * h)) / (2.0 * l)) * t_0));
	else
		tmp = ((sqrt(d) / sqrt(h)) * (sqrt(d) / sqrt(l))) * (1.0 - ((((h / l) * 0.5) * ((D_m / 2.0) * t_1)) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[h, -4e-310], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$1 * D$95$m), $MachinePrecision] * N[(0.5 * h), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(h / l), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(D$95$m / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if h < -3.999999999999988e-310

   1. Initial program 69.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot M\right) \cdot \frac{D}{2 \cdot d}\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot M\right)\right) \cdot \frac{D}{2 \cdot d}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot M\right)\right) \cdot \frac{D}{2 \cdot d}}\right) \]

   4. Applied rewrites 65.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right)} \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right)} \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)} \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\color{blue}{\frac{D}{2}} \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\color{blue}{\frac{D \cdot \left(\frac{M}{d} \cdot M\right)}{2}} \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D \cdot \left(\frac{M}{d} \cdot M\right)}{2} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right)}\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D \cdot \left(\frac{M}{d} \cdot M\right)}{2} \cdot \left(\color{blue}{\frac{h}{\ell}} \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. associate-*l/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{D \cdot \left(\frac{M}{d} \cdot M\right)}{2} \cdot \color{blue}{\frac{h \cdot \frac{1}{2}}{\ell}}\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      10. frac-times N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(D \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{2 \cdot \ell}} \cdot \frac{\frac{D}{d}}{2}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(D \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}{2 \cdot \ell}} \cdot \frac{\frac{D}{d}}{2}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(D \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(h \cdot \frac{1}{2}\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      13. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right)} \cdot \left(h \cdot \frac{1}{2}\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right)} \cdot \left(h \cdot \frac{1}{2}\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot h\right)}}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      17. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      18. lower-*.f64 66.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(0.5 \cdot h\right)}{\color{blue}{2 \cdot \ell}} \cdot \frac{\frac{D}{d}}{2}\right) \]

   6. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(0.5 \cdot h\right)}{2 \cdot \ell}} \cdot \frac{\frac{D}{d}}{2}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(0.5 \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lift-neg.f64 N/A

         \[\leadsto \left(\sqrt{\frac{\color{blue}{-d}}{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{-d}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(\frac{1}{2} \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

      12. lower-neg.f64 72.5

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(0.5 \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

   8. Applied rewrites 72.5%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(0.5 \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right) \]

   if -3.999999999999988e-310 < h

   1. Initial program 71.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot M\right) \cdot \frac{D}{2 \cdot d}\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot M\right)\right) \cdot \frac{D}{2 \cdot d}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot M\right)\right) \cdot \frac{D}{2 \cdot d}}\right) \]

   4. Applied rewrites 66.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      11. lower-sqrt.f64 76.0

          \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

   6. Applied rewrites 76.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      2. metadata-eval 76.0

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

      9. lower-sqrt.f64 79.0

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]

   8. Applied rewrites 79.0%

      \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(0.5 \cdot h\right)}{2 \cdot \ell} \cdot \frac{\frac{D}{d}}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{D}{2} \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \frac{\frac{D}{d}}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 45.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4.2 \cdot 10^{-170}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d 4.2e-170)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (/ d (* (sqrt l) (sqrt h)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 4.2e-170) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 4.2d-170) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 4.2e-170) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= 4.2e-170:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= 4.2e-170)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= 4.2e-170)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, 4.2e-170], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.2000000000000001e-170

   1. Initial program 63.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(d \cdot -1\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-1 \cdot d\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      10. lower-neg.f64 N/A

          \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      13. *-commutative N/A

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      14. lower-*.f64 39.3

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 39.3%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 4.2000000000000001e-170 < d

   1. Initial program 81.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 48.9

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 48.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.8%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

   9. Applied rewrites 48.8%

      \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 62.7%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.2 \cdot 10^{-170}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 41.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{if}\;h \leq -6 \cdot 10^{-288}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (pow (* l h) -1.0))))
   (if (<= h -6e-288) (* (- d) t_0) (* t_0 d))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(pow((l * h), -1.0));
	double tmp;
	if (h <= -6e-288) {
		tmp = -d * t_0;
	} else {
		tmp = t_0 * d;
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((l * h) ** (-1.0d0)))
    if (h <= (-6d-288)) then
        tmp = -d * t_0
    else
        tmp = t_0 * d
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt(Math.pow((l * h), -1.0));
	double tmp;
	if (h <= -6e-288) {
		tmp = -d * t_0;
	} else {
		tmp = t_0 * d;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt(math.pow((l * h), -1.0))
	tmp = 0
	if h <= -6e-288:
		tmp = -d * t_0
	else:
		tmp = t_0 * d
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt((Float64(l * h) ^ -1.0))
	tmp = 0.0
	if (h <= -6e-288)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(t_0 * d);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt(((l * h) ^ -1.0));
	tmp = 0.0;
	if (h <= -6e-288)
		tmp = -d * t_0;
	else
		tmp = t_0 * d;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -6e-288], N[((-d) * t$95$0), $MachinePrecision], N[(t$95$0 * d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -5.99999999999999998e-288

   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 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. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(d \cdot -1\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-1 \cdot d\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      10. lower-neg.f64 N/A

          \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      13. *-commutative N/A

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      14. lower-*.f64 44.3

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -5.99999999999999998e-288 < h

   1. Initial program 72.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 42.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 42.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -6 \cdot 10^{-288}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 25.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (* (sqrt (pow (* l h) -1.0)) d))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return sqrt(pow((l * h), -1.0)) * d;
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = sqrt(((l * h) ** (-1.0d0))) * d
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return Math.sqrt(Math.pow((l * h), -1.0)) * d;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return math.sqrt(math.pow((l * h), -1.0)) * d

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(sqrt((Float64(l * h) ^ -1.0)) * d)
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = sqrt(((l * h) ^ -1.0)) * d;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 70.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   6. lower-*.f64 27.0

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

5. Applied rewrites 27.0%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

6. Final simplification 27.0%

   \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]
7. Add Preprocessing

Alternative 19: 55.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 5 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(D\_m \cdot D\_m\right) \cdot -0.125}{\ell} \cdot \left(\frac{M\_m \cdot M\_m}{d} \cdot h\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= (* M_m D_m) 5e+71)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (*
    (* (/ (* (* D_m D_m) -0.125) l) (* (/ (* M_m M_m) d) h))
    (/ d (sqrt (* l h))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if ((M_m * D_m) <= 5e+71) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = ((((D_m * D_m) * -0.125) / l) * (((M_m * M_m) / d) * h)) * (d / sqrt((l * h)));
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if ((m_m * d_m) <= 5d+71) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = ((((d_m * d_m) * (-0.125d0)) / l) * (((m_m * m_m) / d) * h)) * (d / sqrt((l * h)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if ((M_m * D_m) <= 5e+71) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = ((((D_m * D_m) * -0.125) / l) * (((M_m * M_m) / d) * h)) * (d / Math.sqrt((l * h)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if (M_m * D_m) <= 5e+71:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = ((((D_m * D_m) * -0.125) / l) * (((M_m * M_m) / d) * h)) * (d / math.sqrt((l * h)))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 5e+71)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(D_m * D_m) * -0.125) / l) * Float64(Float64(Float64(M_m * M_m) / d) * h)) * Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if ((M_m * D_m) <= 5e+71)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = ((((D_m * D_m) * -0.125) / l) * (((M_m * M_m) / d) * h)) * (d / sqrt((l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e+71], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 4.99999999999999972e71

   1. Initial program 73.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   5. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \sqrt{\frac{d}{h}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \cdot \sqrt{\frac{d}{h}} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{d}{\ell}}\right)\right)} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{d}{h}} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{d}{\ell}}\right)\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \cdot \sqrt{\frac{d}{h}} \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{d}{\ell}}\right)\right) \cdot \color{blue}{-1}\right) \cdot \sqrt{\frac{d}{h}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{d}{\ell}}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \sqrt{\frac{d}{h}} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\frac{d}{\ell}}\right)\right) \cdot 1\right)\right)} \cdot \sqrt{\frac{d}{h}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\frac{d}{\ell}}\right)\right)\right)\right) \cdot 1\right)} \cdot \sqrt{\frac{d}{h}} \]

      8. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot 1\right) \cdot \sqrt{\frac{d}{h}} \]

      9. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

      11. lower-/.f64 54.2

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

   7. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

   if 4.99999999999999972e71 < (*.f64 M D)

   1. Initial program 58.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 24.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{-1}{2}, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{-1}{2}, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right)} \cdot \sqrt{\frac{d}{h}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{-1}{2}, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

      4. pow1/2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{-1}{2}, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{-1}{2}, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot {\color{blue}{\left(\frac{d}{h}\right)}}^{\frac{1}{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{-1}{2}, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{-1}{2}, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot {\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{-1}{2}, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{-1}{2}, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \]

      10. associate-*l* N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{-1}{2}, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{-1}{2}, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{-1}{2}, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right)} \]

      13. lower-*.f64 24.9

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right)} \]

   6. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]

   7. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \ell}\right)} \cdot \frac{d}{\sqrt{\ell \cdot h}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{d \cdot \ell}} \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \left({M}^{2} \cdot h\right)}}{d \cdot \ell} \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{\ell \cdot d}} \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{-1}{8} \cdot {D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{d}\right)} \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{-1}{8} \cdot {D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{d}\right)} \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{-1}{8} \cdot {D}^{2}}{\ell}} \cdot \frac{{M}^{2} \cdot h}{d}\right) \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

      7. *-commutative N/A

         \[\leadsto \left(\frac{\color{blue}{{D}^{2} \cdot \frac{-1}{8}}}{\ell} \cdot \frac{{M}^{2} \cdot h}{d}\right) \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{{D}^{2} \cdot \frac{-1}{8}}}{\ell} \cdot \frac{{M}^{2} \cdot h}{d}\right) \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(D \cdot D\right)} \cdot \frac{-1}{8}}{\ell} \cdot \frac{{M}^{2} \cdot h}{d}\right) \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\left(D \cdot D\right)} \cdot \frac{-1}{8}}{\ell} \cdot \frac{{M}^{2} \cdot h}{d}\right) \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

      11. associate-*l/ N/A

          \[\leadsto \left(\frac{\left(D \cdot D\right) \cdot \frac{-1}{8}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot h\right)}\right) \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{\left(D \cdot D\right) \cdot \frac{-1}{8}}{\ell} \cdot \color{blue}{\left(\frac{{M}^{2}}{d} \cdot h\right)}\right) \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

      13. lower-/.f64 N/A

          \[\leadsto \left(\frac{\left(D \cdot D\right) \cdot \frac{-1}{8}}{\ell} \cdot \left(\color{blue}{\frac{{M}^{2}}{d}} \cdot h\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

      14. unpow2 N/A

          \[\leadsto \left(\frac{\left(D \cdot D\right) \cdot \frac{-1}{8}}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot h\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

      15. lower-*.f64 47.8

          \[\leadsto \left(\frac{\left(D \cdot D\right) \cdot -0.125}{\ell} \cdot \left(\frac{\color{blue}{M \cdot M}}{d} \cdot h\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}} \]

   9. Applied rewrites 47.8%

      \[\leadsto \color{blue}{\left(\frac{\left(D \cdot D\right) \cdot -0.125}{\ell} \cdot \left(\frac{M \cdot M}{d} \cdot h\right)\right)} \cdot \frac{d}{\sqrt{\ell \cdot h}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 44.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 10^{+78}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(M\_m \cdot M\_m\right) \cdot \left(\frac{D\_m \cdot D\_m}{d} \cdot -0.125\right)\right) \cdot \sqrt{\ell \cdot h}}{\ell \cdot \ell}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= (* M_m D_m) 1e+78)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (/
    (* (* (* M_m M_m) (* (/ (* D_m D_m) d) -0.125)) (sqrt (* l h)))
    (* l l))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if ((M_m * D_m) <= 1e+78) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = (((M_m * M_m) * (((D_m * D_m) / d) * -0.125)) * sqrt((l * h))) / (l * l);
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if ((m_m * d_m) <= 1d+78) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = (((m_m * m_m) * (((d_m * d_m) / d) * (-0.125d0))) * sqrt((l * h))) / (l * l)
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if ((M_m * D_m) <= 1e+78) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = (((M_m * M_m) * (((D_m * D_m) / d) * -0.125)) * Math.sqrt((l * h))) / (l * l);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if (M_m * D_m) <= 1e+78:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = (((M_m * M_m) * (((D_m * D_m) / d) * -0.125)) * math.sqrt((l * h))) / (l * l)
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 1e+78)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(Float64(Float64(M_m * M_m) * Float64(Float64(Float64(D_m * D_m) / d) * -0.125)) * sqrt(Float64(l * h))) / Float64(l * l));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if ((M_m * D_m) <= 1e+78)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = (((M_m * M_m) * (((D_m * D_m) / d) * -0.125)) * sqrt((l * h))) / (l * l);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e+78], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1.00000000000000001e78

   1. Initial program 73.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   5. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot \sqrt{\frac{d}{h}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \cdot \sqrt{\frac{d}{h}} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{d}{\ell}}\right)\right)} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{d}{h}} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{d}{\ell}}\right)\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \cdot \sqrt{\frac{d}{h}} \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{d}{\ell}}\right)\right) \cdot \color{blue}{-1}\right) \cdot \sqrt{\frac{d}{h}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{d}{\ell}}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \sqrt{\frac{d}{h}} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\frac{d}{\ell}}\right)\right) \cdot 1\right)\right)} \cdot \sqrt{\frac{d}{h}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\frac{d}{\ell}}\right)\right)\right)\right) \cdot 1\right)} \cdot \sqrt{\frac{d}{h}} \]

      8. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot 1\right) \cdot \sqrt{\frac{d}{h}} \]

      9. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

      11. lower-/.f64 54.2

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

   7. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}} \]

   if 1.00000000000000001e78 < (*.f64 M D)

   1. Initial program 58.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{2}}} \]

   5. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot \frac{D \cdot D}{d}, \sqrt{\ell \cdot h}, \sqrt{\frac{{\ell}^{3}}{h}} \cdot d\right)}{\ell \cdot \ell}} \]

   6. Taylor expanded in d around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right)}{\color{blue}{\ell} \cdot \ell} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.2%

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot \left(\frac{D \cdot D}{d} \cdot -0.125\right)\right) \cdot \sqrt{\ell \cdot h}}{\color{blue}{\ell} \cdot \ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 25.3% accurate, 15.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (/ d (sqrt (* l h))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d / sqrt((l * h));
}

Fortran

D_m =     private
M_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d / sqrt((l * h))
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d / Math.sqrt((l * h));
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d / math.sqrt((l * h))

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d / sqrt((l * h));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   6. lower-*.f64 27.0

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

5. Applied rewrites 27.0%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
6. Step-by-step derivation

7. Applied rewrites 26.6%

   \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
8. Step-by-step derivation

9. Applied rewrites 26.6%

   \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024353 
(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)))))