Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.8% → 80.8%

Time: 15.8s

Alternatives: 21

Speedup: 3.2×

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: 66.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;d \leq -2.4 \cdot 10^{-201}:\\ \;\;\;\;\left(\frac{t\_0}{\sqrt{-h}} \cdot \frac{t\_0}{\sqrt{-\ell}}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{-200}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right), \frac{M\_m}{d} \cdot M\_m, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \frac{D\_m}{2}\right) \cdot \frac{M\_m}{d}\right), \frac{D\_m}{-2} \cdot \frac{M\_m}{d}, 1\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= d -2.4e-201)
     (*
      (* (/ t_0 (sqrt (- h))) (/ t_0 (sqrt (- l))))
      (-
       1.0
       (* (* (pow 2.0 -1.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))
     (if (<= d 1.18e-200)
       (/
        (fma
         (* (pow (/ h l) 1.5) (* (* D_m D_m) -0.125))
         (* (/ M_m d) M_m)
         (* (sqrt (/ h l)) d))
        h)
       (*
        (/ (/ d (sqrt l)) (sqrt h))
        (fma
         (* (/ h l) (* (* 0.5 (/ D_m 2.0)) (/ M_m d)))
         (* (/ D_m -2.0) (/ M_m d))
         1.0))))))

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 = sqrt(-d);
	double tmp;
	if (d <= -2.4e-201) {
		tmp = ((t_0 / sqrt(-h)) * (t_0 / sqrt(-l))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	} else if (d <= 1.18e-200) {
		tmp = fma((pow((h / l), 1.5) * ((D_m * D_m) * -0.125)), ((M_m / d) * M_m), (sqrt((h / l)) * d)) / h;
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * fma(((h / l) * ((0.5 * (D_m / 2.0)) * (M_m / d))), ((D_m / -2.0) * (M_m / d)), 1.0);
	}
	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 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -2.4e-201)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(t_0 / sqrt(Float64(-l)))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	elseif (d <= 1.18e-200)
		tmp = Float64(fma(Float64((Float64(h / l) ^ 1.5) * Float64(Float64(D_m * D_m) * -0.125)), Float64(Float64(M_m / d) * M_m), Float64(sqrt(Float64(h / l)) * d)) / h);
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * fma(Float64(Float64(h / l) * Float64(Float64(0.5 * Float64(D_m / 2.0)) * Float64(M_m / d))), Float64(Float64(D_m / -2.0) * Float64(M_m / d)), 1.0));
	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[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -2.4e-201], N[(N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[(-l)], $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], If[LessEqual[d, 1.18e-200], N[(N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * N[(N[(0.5 * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / -2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.40000000000000009e-201

   1. Initial program 73.3%

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

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

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 78.5

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

   4. Applied rewrites 78.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. sqrt-div N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-neg.f64 89.8

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

   6. Applied rewrites 89.8%

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

   if -2.40000000000000009e-201 < d < 1.17999999999999996e-200

   1. Initial program 37.6%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 17.3

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

   5. Applied rewrites 17.3%

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

   7. Applied rewrites 17.4%

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

   8. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 33.4%

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

   12. Applied rewrites 55.2%

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

   if 1.17999999999999996e-200 < d

   1. Initial program 70.5%

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

      1. lift-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 70.5

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

   4. Applied rewrites 70.5%

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

   6. Applied rewrites 71.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 71.5

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 71.5

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

   8. Applied rewrites 71.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. frac-times N/A

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

      10. rem-square-sqrt N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 82.3

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

   10. Applied rewrites 82.3%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.4 \cdot 10^{-201}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{-200}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(D \cdot D\right) \cdot -0.125\right), \frac{M}{d} \cdot M, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right), \frac{D}{-2} \cdot \frac{M}{d}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\ell \cdot h}\\ t_2 := \frac{d}{t\_1}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(\left(0.25 \cdot D\_m\right) \cdot D\_m\right) \cdot M\_m\right) \cdot M\_m}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-242}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d \cdot \left(\frac{M\_m}{d} \cdot \left(D\_m \cdot 0.25\right)\right), \left(\frac{D\_m}{d} \cdot \frac{M\_m}{-2}\right) \cdot \frac{h}{\ell}, d\right)}{t\_1}\\ \mathbf{elif}\;t\_0 \leq 10^{+161}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(M\_m \cdot \left(D\_m \cdot 0.25\right)\right) \cdot h}{d \cdot \ell} \cdot \frac{D\_m}{d}, \frac{M\_m}{-2}, 1\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d 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 (sqrt (* l h)))
        (t_2 (/ d t_1)))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (/ (* (* (* (* 0.25 D_m) D_m) M_m) M_m) d) (* -0.5 (/ h l)) 1.0)
      t_2)
     (if (<= t_0 2e-242)
       (/
        (fma
         (* d (* (/ M_m d) (* D_m 0.25)))
         (* (* (/ D_m d) (/ M_m -2.0)) (/ h l))
         d)
        t_1)
       (if (<= t_0 1e+161)
         (* (sqrt (/ d l)) (sqrt (/ d h)))
         (if (<= t_0 INFINITY)
           (/ (* (sqrt (/ h l)) d) h)
           (*
            (fma
             (* (/ (* (* M_m (* D_m 0.25)) h) (* d l)) (/ D_m d))
             (/ M_m -2.0)
             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 = (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 = sqrt((l * h));
	double t_2 = d / t_1;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((((((0.25 * D_m) * D_m) * M_m) * M_m) / d), (-0.5 * (h / l)), 1.0) * t_2;
	} else if (t_0 <= 2e-242) {
		tmp = fma((d * ((M_m / d) * (D_m * 0.25))), (((D_m / d) * (M_m / -2.0)) * (h / l)), d) / t_1;
	} else if (t_0 <= 1e+161) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = fma(((((M_m * (D_m * 0.25)) * h) / (d * l)) * (D_m / d)), (M_m / -2.0), 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 / 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 = sqrt(Float64(l * h))
	t_2 = Float64(d / t_1)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(0.25 * D_m) * D_m) * M_m) * M_m) / d), Float64(-0.5 * Float64(h / l)), 1.0) * t_2);
	elseif (t_0 <= 2e-242)
		tmp = Float64(fma(Float64(d * Float64(Float64(M_m / d) * Float64(D_m * 0.25))), Float64(Float64(Float64(D_m / d) * Float64(M_m / -2.0)) * Float64(h / l)), d) / t_1);
	elseif (t_0 <= 1e+161)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(M_m * Float64(D_m * 0.25)) * h) / Float64(d * l)) * Float64(D_m / d)), Float64(M_m / -2.0), 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[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[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(d / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(N[(0.25 * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 2e-242], N[(N[(N[(d * N[(N[(M$95$m / d), $MachinePrecision] * N[(D$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m / -2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] + d), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 1e+161], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[(N[(N[(M$95$m * N[(D$95$m * 0.25), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / -2.0), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 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 83.4%

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

   4. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   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-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      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 32.8

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

   6. Applied rewrites 80.6%

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

   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. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 75.3

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

   9. Applied rewrites 75.3%

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

   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)))) < 2e-242

   1. Initial program 68.9%

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

      1. lift-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 68.9

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

   4. Applied rewrites 68.9%

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

   6. Applied rewrites 68.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 68.3

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 68.3

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

   8. Applied rewrites 68.3%

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

   10. Applied rewrites 47.4%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(d \cdot \left(\frac{M}{d} \cdot \left(D \cdot 0.25\right)\right), \left(\frac{D}{d} \cdot \frac{M}{-2}\right) \cdot \frac{h}{\ell}, d\right)}{\sqrt{\ell \cdot h}}} \]

   if 2e-242 < (*.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)))) < 1e161

   1. Initial program 99.2%

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

   4. Applied rewrites 92.4%

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

   5. 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. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 98.5

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

   7. Applied rewrites 98.5%

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

   if 1e161 < (*.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 53.5%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 37.0

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 37.1%

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

   8. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 48.9%

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

   11. Taylor expanded in d around inf

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

   13. Applied rewrites 79.3%

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

   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
   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 0.0

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 0.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 0.3

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 0.3

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

   8. Applied rewrites 0.3%

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

   10. Applied rewrites 26.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(M \cdot \left(D \cdot 0.25\right)\right) \cdot h}{d \cdot \ell} \cdot \frac{D}{d}, \frac{M}{-2}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(\left(0.25 \cdot D\right) \cdot D\right) \cdot M\right) \cdot M}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-242}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d \cdot \left(\frac{M}{d} \cdot \left(D \cdot 0.25\right)\right), \left(\frac{D}{d} \cdot \frac{M}{-2}\right) \cdot \frac{h}{\ell}, d\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+161}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(M \cdot \left(D \cdot 0.25\right)\right) \cdot h}{d \cdot \ell} \cdot \frac{D}{d}, \frac{M}{-2}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \frac{D\_m}{-2} \cdot \frac{M\_m}{d}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-51}:\\ \;\;\;\;\left(t\_0 \cdot t\_1\right) \cdot \mathsf{fma}\left(\frac{\frac{h}{\ell} \cdot \left(M\_m \cdot \left(D\_m \cdot 0.25\right)\right)}{d}, t\_3, 1\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-242} \lor \neg \left(t\_2 \leq 5 \cdot 10^{+260}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \frac{D\_m}{2}\right) \cdot \frac{M\_m}{d}\right), t\_3, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \end{array} \end{array} \]

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 h)))
        (t_1 (sqrt (/ d l)))
        (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_m -2.0) (/ M_m d))))
   (if (<= t_2 -4e-51)
     (* (* t_0 t_1) (fma (/ (* (/ h l) (* M_m (* D_m 0.25))) d) t_3 1.0))
     (if (or (<= t_2 2e-242) (not (<= t_2 5e+260)))
       (*
        (fabs (/ d (sqrt (* l h))))
        (fma (* (/ h l) (* (* 0.5 (/ D_m 2.0)) (/ M_m d))) t_3 1.0))
       (* 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 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	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_m / -2.0) * (M_m / d);
	double tmp;
	if (t_2 <= -4e-51) {
		tmp = (t_0 * t_1) * fma((((h / l) * (M_m * (D_m * 0.25))) / d), t_3, 1.0);
	} else if ((t_2 <= 2e-242) || !(t_2 <= 5e+260)) {
		tmp = fabs((d / sqrt((l * h)))) * fma(((h / l) * ((0.5 * (D_m / 2.0)) * (M_m / d))), t_3, 1.0);
	} else {
		tmp = 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 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	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(Float64(D_m / -2.0) * Float64(M_m / d))
	tmp = 0.0
	if (t_2 <= -4e-51)
		tmp = Float64(Float64(t_0 * t_1) * fma(Float64(Float64(Float64(h / l) * Float64(M_m * Float64(D_m * 0.25))) / d), t_3, 1.0));
	elseif ((t_2 <= 2e-242) || !(t_2 <= 5e+260))
		tmp = Float64(abs(Float64(d / sqrt(Float64(l * h)))) * fma(Float64(Float64(h / l) * Float64(Float64(0.5 * Float64(D_m / 2.0)) * Float64(M_m / d))), t_3, 1.0));
	else
		tmp = Float64(t_1 * 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[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $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[(N[(D$95$m / -2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-51], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(N[(N[(N[(h / l), $MachinePrecision] * N[(M$95$m * N[(D$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$2, 2e-242], N[Not[LessEqual[t$95$2, 5e+260]], $MachinePrecision]], N[(N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * N[(N[(0.5 * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4e-51

   1. Initial program 86.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 86.1

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

   4. Applied rewrites 86.1%

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

   6. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 90.3

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 90.3

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

   8. Applied rewrites 90.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

      10. frac-times N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 89.3

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l/ N/A

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

      18. lift-/.f64 N/A

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

      19. associate-*r/ N/A

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

   10. Applied rewrites 88.2%

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

   if -4e-51 < (*.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)))) < 2e-242 or 4.9999999999999996e260 < (*.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 21.6%

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

      1. lift-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 21.6

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

   4. Applied rewrites 21.6%

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

   6. Applied rewrites 22.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 22.5

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 22.5

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

   8. Applied rewrites 22.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. frac-times N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. rem-sqrt-square-rev N/A

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

      17. lower-fabs.f64 62.4

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

   10. Applied rewrites 62.4%

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

   if 2e-242 < (*.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)))) < 4.9999999999999996e260

   1. Initial program 99.2%

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

   4. Applied rewrites 91.8%

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

   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. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 98.5

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

   7. Applied rewrites 98.5%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -4 \cdot 10^{-51}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{\frac{h}{\ell} \cdot \left(M \cdot \left(D \cdot 0.25\right)\right)}{d}, \frac{D}{-2} \cdot \frac{M}{d}, 1\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-242} \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+260}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right), \frac{D}{-2} \cdot \frac{M}{d}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-242}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{M\_m}{2} \cdot D\_m\right) \cdot \left(\frac{\frac{D\_m}{d}}{2} \cdot M\_m\right), -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+161}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(M\_m \cdot \left(D\_m \cdot 0.25\right)\right) \cdot h}{d \cdot \ell} \cdot \frac{D\_m}{d}, \frac{M\_m}{-2}, 1\right) \cdot t\_1\\ \end{array} \end{array} \]

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 2e-242)
     (*
      (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 1e+161)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (if (<= t_0 INFINITY)
         (/ (* (sqrt (/ h l)) d) h)
         (*
          (fma
           (* (/ (* (* M_m (* D_m 0.25)) h) (* d l)) (/ D_m d))
           (/ M_m -2.0)
           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 <= 2e-242) {
		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 <= 1e+161) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = fma(((((M_m * (D_m * 0.25)) * h) / (d * l)) * (D_m / d)), (M_m / -2.0), 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 <= 2e-242)
		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 <= 1e+161)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(M_m * Float64(D_m * 0.25)) * h) / Float64(d * l)) * Float64(D_m / d)), Float64(M_m / -2.0), 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, 2e-242], 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, 1e+161], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[(N[(N[(M$95$m * N[(D$95$m * 0.25), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / -2.0), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $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)))) < 2e-242

   1. Initial program 78.9%

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

   4. Applied rewrites 27.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   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-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      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 27.2

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

   6. Applied rewrites 64.6%

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

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 64.6

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

   8. Applied rewrites 64.6%

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

   if 2e-242 < (*.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)))) < 1e161

   1. Initial program 99.2%

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

   4. Applied rewrites 92.4%

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

   5. 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. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 98.5

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

   7. Applied rewrites 98.5%

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

   if 1e161 < (*.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 53.5%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 37.0

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 37.1%

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

   8. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 48.9%

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

   11. Taylor expanded in d around inf

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

   13. Applied rewrites 79.3%

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

   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
   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 0.0

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 0.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 0.3

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 0.3

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

   8. Applied rewrites 0.3%

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

   10. Applied rewrites 26.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(M \cdot \left(D \cdot 0.25\right)\right) \cdot h}{d \cdot \ell} \cdot \frac{D}{d}, \frac{M}{-2}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-242}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{M}{2} \cdot D\right) \cdot \left(\frac{\frac{D}{d}}{2} \cdot M\right), -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+161}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(M \cdot \left(D \cdot 0.25\right)\right) \cdot h}{d \cdot \ell} \cdot \frac{D}{d}, \frac{M}{-2}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-242}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(\left(0.25 \cdot D\_m\right) \cdot D\_m\right) \cdot M\_m\right) \cdot M\_m}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+161}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(M\_m \cdot \left(D\_m \cdot 0.25\right)\right) \cdot h}{d \cdot \ell} \cdot \frac{D\_m}{d}, \frac{M\_m}{-2}, 1\right) \cdot t\_1\\ \end{array} \end{array} \]

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 2e-242)
     (*
      (fma (/ (* (* (* (* 0.25 D_m) D_m) M_m) M_m) d) (* -0.5 (/ h l)) 1.0)
      t_1)
     (if (<= t_0 1e+161)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (if (<= t_0 INFINITY)
         (/ (* (sqrt (/ h l)) d) h)
         (*
          (fma
           (* (/ (* (* M_m (* D_m 0.25)) h) (* d l)) (/ D_m d))
           (/ M_m -2.0)
           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 <= 2e-242) {
		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 <= 1e+161) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = fma(((((M_m * (D_m * 0.25)) * h) / (d * l)) * (D_m / d)), (M_m / -2.0), 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 <= 2e-242)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(0.25 * D_m) * D_m) * M_m) * M_m) / d), Float64(-0.5 * Float64(h / l)), 1.0) * t_1);
	elseif (t_0 <= 1e+161)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(M_m * Float64(D_m * 0.25)) * h) / Float64(d * l)) * Float64(D_m / d)), Float64(M_m / -2.0), 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, 2e-242], N[(N[(N[(N[(N[(N[(N[(0.25 * D$95$m), $MachinePrecision] * D$95$m), $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, 1e+161], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[(N[(N[(M$95$m * N[(D$95$m * 0.25), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / -2.0), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $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)))) < 2e-242

   1. Initial program 78.9%

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

   4. Applied rewrites 27.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   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-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      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 27.2

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

   6. Applied rewrites 64.6%

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

   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. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 60.7

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

   9. Applied rewrites 60.7%

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

   if 2e-242 < (*.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)))) < 1e161

   1. Initial program 99.2%

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

   4. Applied rewrites 92.4%

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

   5. 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. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 98.5

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

   7. Applied rewrites 98.5%

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

   if 1e161 < (*.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 53.5%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 37.0

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 37.1%

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

   8. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 48.9%

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

   11. Taylor expanded in d around inf

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

   13. Applied rewrites 79.3%

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

   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
   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 0.0

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 0.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 0.3

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 0.3

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

   8. Applied rewrites 0.3%

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

   10. Applied rewrites 26.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(M \cdot \left(D \cdot 0.25\right)\right) \cdot h}{d \cdot \ell} \cdot \frac{D}{d}, \frac{M}{-2}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-242}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(\left(0.25 \cdot D\right) \cdot D\right) \cdot M\right) \cdot M}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+161}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(M \cdot \left(D \cdot 0.25\right)\right) \cdot h}{d \cdot \ell} \cdot \frac{D}{d}, \frac{M}{-2}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \mathsf{fma}\left(\frac{\left(\left(\left(0.25 \cdot D\_m\right) \cdot D\_m\right) \cdot M\_m\right) \cdot M\_m}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-242}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+161}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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
         (*
          (fma (/ (* (* (* (* 0.25 D_m) D_m) M_m) M_m) d) (* -0.5 (/ h l)) 1.0)
          (/ d (sqrt (* l h))))))
   (if (<= t_0 2e-242)
     t_1
     (if (<= t_0 1e+161)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (if (<= t_0 INFINITY) (/ (* (sqrt (/ h l)) 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 = fma((((((0.25 * D_m) * D_m) * M_m) * M_m) / d), (-0.5 * (h / l)), 1.0) * (d / sqrt((l * h)));
	double tmp;
	if (t_0 <= 2e-242) {
		tmp = t_1;
	} else if (t_0 <= 1e+161) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = 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(fma(Float64(Float64(Float64(Float64(Float64(0.25 * D_m) * D_m) * M_m) * M_m) / d), Float64(-0.5 * Float64(h / l)), 1.0) * Float64(d / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_0 <= 2e-242)
		tmp = t_1;
	elseif (t_0 <= 1e+161)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = 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[(N[(N[(N[(N[(N[(N[(0.25 * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-242], t$95$1, If[LessEqual[t$95$0, 1e+161], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], t$95$1]]]]]

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)))) < 2e-242 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 55.5%

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

   4. Applied rewrites 19.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   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-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      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 19.4

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

   6. Applied rewrites 53.0%

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

   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. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 49.0

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

   9. Applied rewrites 49.0%

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

   if 2e-242 < (*.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)))) < 1e161

   1. Initial program 99.2%

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

   4. Applied rewrites 92.4%

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

   5. 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. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 98.5

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

   7. Applied rewrites 98.5%

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

   if 1e161 < (*.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 53.5%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 37.0

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 37.1%

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

   8. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 48.9%

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

   11. Taylor expanded in d around inf

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

   13. Applied rewrites 79.3%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-242}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(\left(0.25 \cdot D\right) \cdot D\right) \cdot M\right) \cdot M}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+161}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(\left(0.25 \cdot D\right) \cdot D\right) \cdot M\right) \cdot M}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-242}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(0.25 \cdot D\_m\right) \cdot D\_m\right) \cdot \frac{M\_m \cdot M\_m}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+161}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 2e-242)
     (*
      (fma (* (* (* 0.25 D_m) D_m) (/ (* M_m M_m) d)) (* -0.5 (/ h l)) 1.0)
      t_1)
     (if (<= t_0 1e+161)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (if (<= t_0 INFINITY) (/ (* (sqrt (/ h l)) 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 <= 2e-242) {
		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 <= 1e+161) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = 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 <= 2e-242)
		tmp = Float64(fma(Float64(Float64(Float64(0.25 * D_m) * D_m) * Float64(Float64(M_m * M_m) / d)), Float64(-0.5 * Float64(h / l)), 1.0) * t_1);
	elseif (t_0 <= 1e+161)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = 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, 2e-242], N[(N[(N[(N[(N[(0.25 * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 1e+161], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], t$95$1]]]]]

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)))) < 2e-242

   1. Initial program 78.9%

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

   4. Applied rewrites 27.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, \frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   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-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      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 27.2

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

   6. Applied rewrites 64.6%

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

   7. Taylor expanded in d around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 56.5

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

   9. Applied rewrites 56.5%

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

   if 2e-242 < (*.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)))) < 1e161

   1. Initial program 99.2%

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

   4. Applied rewrites 92.4%

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

   5. 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. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 98.5

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

   7. Applied rewrites 98.5%

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

   if 1e161 < (*.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 53.5%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 37.0

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 37.1%

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

   8. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 48.9%

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

   11. Taylor expanded in d around inf

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

   13. Applied rewrites 79.3%

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

   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

   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 21.5

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

   5. Applied rewrites 21.5%

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

   7. Applied rewrites 21.5%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-242}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(0.25 \cdot D\right) \cdot D\right) \cdot \frac{M \cdot M}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 10^{+161}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ t_1 := \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \frac{D\_m}{2}\right) \cdot \frac{M\_m}{d}\right), \frac{D\_m}{-2} \cdot \frac{M\_m}{d}, 1\right)\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\left|t\_0\right| \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \left(\frac{{\left(\frac{M\_m}{2} \cdot D\_m\right)}^{2}}{d} \cdot h\right) \cdot \frac{-0.5}{\ell}, t\_0\right)\\ \end{array} \end{array} \]

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 (sqrt (* l h))))
        (t_1
         (fma
          (* (/ h l) (* (* 0.5 (/ D_m 2.0)) (/ M_m d)))
          (* (/ D_m -2.0) (/ M_m d))
          1.0))
        (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))))))
   (if (<= t_2 5e+260)
     (* (* (sqrt (/ d h)) (sqrt (/ d l))) t_1)
     (if (<= t_2 INFINITY)
       (* (fabs t_0) t_1)
       (fma
        t_0
        (* (* (/ (pow (* (/ M_m 2.0) D_m) 2.0) d) h) (/ -0.5 l))
        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 / sqrt((l * h));
	double t_1 = fma(((h / l) * ((0.5 * (D_m / 2.0)) * (M_m / d))), ((D_m / -2.0) * (M_m / d)), 1.0);
	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 tmp;
	if (t_2 <= 5e+260) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fabs(t_0) * t_1;
	} else {
		tmp = fma(t_0, (((pow(((M_m / 2.0) * D_m), 2.0) / d) * h) * (-0.5 / l)), 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(d / sqrt(Float64(l * h)))
	t_1 = fma(Float64(Float64(h / l) * Float64(Float64(0.5 * Float64(D_m / 2.0)) * Float64(M_m / d))), Float64(Float64(D_m / -2.0) * Float64(M_m / d)), 1.0)
	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))))
	tmp = 0.0
	if (t_2 <= 5e+260)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * t_1);
	elseif (t_2 <= Inf)
		tmp = Float64(abs(t_0) * t_1);
	else
		tmp = fma(t_0, Float64(Float64(Float64((Float64(Float64(M_m / 2.0) * D_m) ^ 2.0) / d) * h) * Float64(-0.5 / l)), 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[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(h / l), $MachinePrecision] * N[(N[(0.5 * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / -2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] + 1.0), $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]}, If[LessEqual[t$95$2, 5e+260], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Abs[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$0 * N[(N[(N[(N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.9999999999999996e260

   1. Initial program 87.0%

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

      1. lift-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 87.0

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

   4. Applied rewrites 87.0%

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

   6. Applied rewrites 89.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 89.0

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 89.0

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

   8. Applied rewrites 89.0%

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

   if 4.9999999999999996e260 < (*.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 41.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 41.1

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

   4. Applied rewrites 41.1%

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

   6. Applied rewrites 41.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 41.1

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 41.1

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

   8. Applied rewrites 41.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. frac-times N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. rem-sqrt-square-rev N/A

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

      17. lower-fabs.f64 99.7

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

   10. Applied rewrites 99.7%

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

   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 0.7%

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

   6. Applied rewrites 47.6%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right), \frac{D}{-2} \cdot \frac{M}{d}, 1\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right), \frac{D}{-2} \cdot \frac{M}{d}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{\sqrt{\ell \cdot h}}, \left(\frac{{\left(\frac{M}{2} \cdot D\right)}^{2}}{d} \cdot h\right) \cdot \frac{-0.5}{\ell}, \frac{d}{\sqrt{\ell \cdot h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-242} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+260}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \frac{D\_m}{2}\right) \cdot \frac{M\_m}{d}\right), \frac{D\_m}{-2} \cdot \frac{M\_m}{d}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \end{array} \]

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
 (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))))))
   (if (or (<= t_0 2e-242) (not (<= t_0 5e+260)))
     (*
      (fabs (/ d (sqrt (* l h))))
      (fma
       (* (/ h l) (* (* 0.5 (/ D_m 2.0)) (/ M_m d)))
       (* (/ D_m -2.0) (/ M_m d))
       1.0))
     (* (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 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 tmp;
	if ((t_0 <= 2e-242) || !(t_0 <= 5e+260)) {
		tmp = fabs((d / sqrt((l * h)))) * fma(((h / l) * ((0.5 * (D_m / 2.0)) * (M_m / d))), ((D_m / -2.0) * (M_m / d)), 1.0);
	} else {
		tmp = sqrt((d / l)) * 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)
	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))))
	tmp = 0.0
	if ((t_0 <= 2e-242) || !(t_0 <= 5e+260))
		tmp = Float64(abs(Float64(d / sqrt(Float64(l * h)))) * fma(Float64(Float64(h / l) * Float64(Float64(0.5 * Float64(D_m / 2.0)) * Float64(M_m / d))), Float64(Float64(D_m / -2.0) * Float64(M_m / d)), 1.0));
	else
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[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]}, If[Or[LessEqual[t$95$0, 2e-242], N[Not[LessEqual[t$95$0, 5e+260]], $MachinePrecision]], N[(N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * N[(N[(0.5 * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / -2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $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)))) < 2e-242 or 4.9999999999999996e260 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 53.0%

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

      1. lift-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 53.0

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

   4. Applied rewrites 53.0%

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

   6. Applied rewrites 55.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 55.5

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 55.5

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

   8. Applied rewrites 55.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. frac-times N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. rem-sqrt-square-rev N/A

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

      17. lower-fabs.f64 72.4

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

   10. Applied rewrites 72.4%

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

   if 2e-242 < (*.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)))) < 4.9999999999999996e260

   1. Initial program 99.2%

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

   4. Applied rewrites 91.8%

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

   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. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 98.5

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

   7. Applied rewrites 98.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-242} \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+260}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right), \frac{D}{-2} \cdot \frac{M}{d}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \frac{D\_m}{2}\right) \cdot \frac{M\_m}{d}\right), \frac{D\_m}{-2} \cdot \frac{M\_m}{d}, 1\right)\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot t\_0\\ \end{array} \end{array} \]

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
         (fma
          (* (/ h l) (* (* 0.5 (/ D_m 2.0)) (/ M_m d)))
          (* (/ D_m -2.0) (/ M_m d))
          1.0)))
   (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))))
        5e+260)
     (* (* (sqrt (/ d h)) (sqrt (/ d l))) t_0)
     (* (fabs (/ d (sqrt (* l h)))) 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 = fma(((h / l) * ((0.5 * (D_m / 2.0)) * (M_m / d))), ((D_m / -2.0) * (M_m / d)), 1.0);
	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)))) <= 5e+260) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * t_0;
	} else {
		tmp = fabs((d / sqrt((l * h)))) * 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 = fma(Float64(Float64(h / l) * Float64(Float64(0.5 * Float64(D_m / 2.0)) * Float64(M_m / d))), Float64(Float64(D_m / -2.0) * Float64(M_m / d)), 1.0)
	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)))) <= 5e+260)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * t_0);
	else
		tmp = Float64(abs(Float64(d / sqrt(Float64(l * h)))) * 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[(N[(h / l), $MachinePrecision] * N[(N[(0.5 * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / -2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] + 1.0), $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] * 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], 5e+260], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $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)))) < 4.9999999999999996e260

   1. Initial program 87.0%

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

      1. lift-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 87.0

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

   4. Applied rewrites 87.0%

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

   6. Applied rewrites 89.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 89.0

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 89.0

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

   8. Applied rewrites 89.0%

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

   if 4.9999999999999996e260 < (*.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 17.4%

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

      1. lift-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 17.4

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

   4. Applied rewrites 17.4%

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

   6. Applied rewrites 17.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 17.5

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 17.5

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

   8. Applied rewrites 17.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. frac-times N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. rem-sqrt-square-rev N/A

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

      17. lower-fabs.f64 58.0

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

   10. Applied rewrites 58.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right), \frac{D}{-2} \cdot \frac{M}{d}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \frac{D}{2}\right) \cdot \frac{M}{d}\right), \frac{D}{-2} \cdot \frac{M}{d}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \frac{D\_m}{2}\right) \cdot \frac{M\_m}{d}\right), \frac{D\_m}{-2} \cdot \frac{M\_m}{d}, 1\right)\\ \mathbf{if}\;d \leq -1.2 \cdot 10^{-246}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_0\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{-200}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right), \frac{M\_m}{d} \cdot M\_m, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot t\_0\\ \end{array} \end{array} \]

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
         (fma
          (* (/ h l) (* (* 0.5 (/ D_m 2.0)) (/ M_m d)))
          (* (/ D_m -2.0) (/ M_m d))
          1.0)))
   (if (<= d -1.2e-246)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l))) t_0)
     (if (<= d 1.18e-200)
       (/
        (fma
         (* (pow (/ h l) 1.5) (* (* D_m D_m) -0.125))
         (* (/ M_m d) M_m)
         (* (sqrt (/ h l)) d))
        h)
       (* (/ (/ d (sqrt l)) (sqrt h)) 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 = fma(((h / l) * ((0.5 * (D_m / 2.0)) * (M_m / d))), ((D_m / -2.0) * (M_m / d)), 1.0);
	double tmp;
	if (d <= -1.2e-246) {
		tmp = ((sqrt(-d) / sqrt(-h)) * sqrt((d / l))) * t_0;
	} else if (d <= 1.18e-200) {
		tmp = fma((pow((h / l), 1.5) * ((D_m * D_m) * -0.125)), ((M_m / d) * M_m), (sqrt((h / l)) * d)) / h;
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * 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 = fma(Float64(Float64(h / l) * Float64(Float64(0.5 * Float64(D_m / 2.0)) * Float64(M_m / d))), Float64(Float64(D_m / -2.0) * Float64(M_m / d)), 1.0)
	tmp = 0.0
	if (d <= -1.2e-246)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * t_0);
	elseif (d <= 1.18e-200)
		tmp = Float64(fma(Float64((Float64(h / l) ^ 1.5) * Float64(Float64(D_m * D_m) * -0.125)), Float64(Float64(M_m / d) * M_m), Float64(sqrt(Float64(h / l)) * d)) / h);
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * 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[(N[(h / l), $MachinePrecision] * N[(N[(0.5 * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / -2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -1.2e-246], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, 1.18e-200], N[(N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.1999999999999999e-246

   1. Initial program 69.6%

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

      1. lift-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 69.6

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

   4. Applied rewrites 69.6%

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

   6. Applied rewrites 71.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 71.9

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 85.0

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

   8. Applied rewrites 85.0%

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

   if -1.1999999999999999e-246 < d < 1.17999999999999996e-200

   1. Initial program 34.1%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      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 15.5

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

   5. Applied rewrites 15.5%

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 30.9%

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

   12. Applied rewrites 47.9%

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

   if 1.17999999999999996e-200 < d

   1. Initial program 70.5%

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

      1. lift-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 70.5

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

   4. Applied rewrites 70.5%

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

   6. Applied rewrites 71.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 71.5

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 71.5

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

   8. Applied rewrites 71.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. frac-times N/A

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

      10. rem-square-sqrt N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 82.3

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

   10. Applied rewrites 82.3%

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

Alternative 12: 80.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \frac{D\_m}{2}\right) \cdot \frac{M\_m}{d}\right), \frac{D\_m}{-2} \cdot \frac{M\_m}{d}, 1\right)\\ \mathbf{if}\;d \leq -1.2 \cdot 10^{-246}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_0\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{-200}:\\ \;\;\;\;\frac{\mathsf{fma}\left(D\_m, \left(-0.125 \cdot D\_m\right) \cdot \left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\frac{M\_m}{d} \cdot M\_m\right)\right), \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot t\_0\\ \end{array} \end{array} \]

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
         (fma
          (* (/ h l) (* (* 0.5 (/ D_m 2.0)) (/ M_m d)))
          (* (/ D_m -2.0) (/ M_m d))
          1.0)))
   (if (<= d -1.2e-246)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l))) t_0)
     (if (<= d 1.18e-200)
       (/
        (fma
         D_m
         (* (* -0.125 D_m) (* (pow (/ h l) 1.5) (* (/ M_m d) M_m)))
         (* (sqrt (/ h l)) d))
        h)
       (* (/ (/ d (sqrt l)) (sqrt h)) 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 = fma(((h / l) * ((0.5 * (D_m / 2.0)) * (M_m / d))), ((D_m / -2.0) * (M_m / d)), 1.0);
	double tmp;
	if (d <= -1.2e-246) {
		tmp = ((sqrt(-d) / sqrt(-h)) * sqrt((d / l))) * t_0;
	} else if (d <= 1.18e-200) {
		tmp = fma(D_m, ((-0.125 * D_m) * (pow((h / l), 1.5) * ((M_m / d) * M_m))), (sqrt((h / l)) * d)) / h;
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * 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 = fma(Float64(Float64(h / l) * Float64(Float64(0.5 * Float64(D_m / 2.0)) * Float64(M_m / d))), Float64(Float64(D_m / -2.0) * Float64(M_m / d)), 1.0)
	tmp = 0.0
	if (d <= -1.2e-246)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * t_0);
	elseif (d <= 1.18e-200)
		tmp = Float64(fma(D_m, Float64(Float64(-0.125 * D_m) * Float64((Float64(h / l) ^ 1.5) * Float64(Float64(M_m / d) * M_m))), Float64(sqrt(Float64(h / l)) * d)) / h);
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * 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[(N[(h / l), $MachinePrecision] * N[(N[(0.5 * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / -2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -1.2e-246], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, 1.18e-200], N[(N[(D$95$m * N[(N[(-0.125 * D$95$m), $MachinePrecision] * N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.1999999999999999e-246

   1. Initial program 69.6%

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

      1. lift-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 69.6

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

   4. Applied rewrites 69.6%

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

   6. Applied rewrites 71.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 71.9

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 85.0

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

   8. Applied rewrites 85.0%

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

   if -1.1999999999999999e-246 < d < 1.17999999999999996e-200

   1. Initial program 34.1%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      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 15.5

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

   5. Applied rewrites 15.5%

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 30.9%

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

   12. Applied rewrites 54.7%

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

   if 1.17999999999999996e-200 < d

   1. Initial program 70.5%

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

      1. lift-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 70.5

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

   4. Applied rewrites 70.5%

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

   6. Applied rewrites 71.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 71.5

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 71.5

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

   8. Applied rewrites 71.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. frac-times N/A

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

      10. rem-square-sqrt N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 82.3

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

   10. Applied rewrites 82.3%

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

Alternative 13: 80.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \frac{D\_m}{2}\right) \cdot \frac{M\_m}{d}\right), \frac{D\_m}{-2} \cdot \frac{M\_m}{d}, 1\right)\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot t\_0\\ \end{array} \end{array} \]

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
         (fma
          (* (/ h l) (* (* 0.5 (/ D_m 2.0)) (/ M_m d)))
          (* (/ D_m -2.0) (/ M_m d))
          1.0)))
   (if (<= h -5e-310)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l))) t_0)
     (* (/ (/ d (sqrt l)) (sqrt h)) 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 = fma(((h / l) * ((0.5 * (D_m / 2.0)) * (M_m / d))), ((D_m / -2.0) * (M_m / d)), 1.0);
	double tmp;
	if (h <= -5e-310) {
		tmp = ((sqrt(-d) / sqrt(-h)) * sqrt((d / l))) * t_0;
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * 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 = fma(Float64(Float64(h / l) * Float64(Float64(0.5 * Float64(D_m / 2.0)) * Float64(M_m / d))), Float64(Float64(D_m / -2.0) * Float64(M_m / d)), 1.0)
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * t_0);
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * 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[(N[(h / l), $MachinePrecision] * N[(N[(0.5 * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / -2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. 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 66.7

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

   4. Applied rewrites 66.7%

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

   6. Applied rewrites 69.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 69.3

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 81.9

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

   8. Applied rewrites 81.9%

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

   if -4.999999999999985e-310 < h

   1. Initial program 64.8%

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

      1. lift-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 64.8

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

   4. Applied rewrites 64.8%

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

   6. Applied rewrites 64.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 64.8

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 64.8

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

   8. Applied rewrites 64.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. frac-times N/A

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

      10. rem-square-sqrt N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 77.6

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

   10. Applied rewrites 77.6%

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

Alternative 14: 77.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \frac{D\_m}{2}\right) \cdot \frac{M\_m}{d}\right), \frac{D\_m}{-2} \cdot \frac{M\_m}{d}, 1\right)\\ \mathbf{if}\;d \leq 7.5 \cdot 10^{-211}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right| \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot t\_0\\ \end{array} \end{array} \]

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
         (fma
          (* (/ h l) (* (* 0.5 (/ D_m 2.0)) (/ M_m d)))
          (* (/ D_m -2.0) (/ M_m d))
          1.0)))
   (if (<= d 7.5e-211)
     (* (fabs (/ d (sqrt (* l h)))) t_0)
     (* (/ (/ d (sqrt l)) (sqrt h)) 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 = fma(((h / l) * ((0.5 * (D_m / 2.0)) * (M_m / d))), ((D_m / -2.0) * (M_m / d)), 1.0);
	double tmp;
	if (d <= 7.5e-211) {
		tmp = fabs((d / sqrt((l * h)))) * t_0;
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * 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 = fma(Float64(Float64(h / l) * Float64(Float64(0.5 * Float64(D_m / 2.0)) * Float64(M_m / d))), Float64(Float64(D_m / -2.0) * Float64(M_m / d)), 1.0)
	tmp = 0.0
	if (d <= 7.5e-211)
		tmp = Float64(abs(Float64(d / sqrt(Float64(l * h)))) * t_0);
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * 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[(N[(h / l), $MachinePrecision] * N[(N[(0.5 * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / -2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, 7.5e-211], N[(N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 7.5000000000000003e-211

   1. Initial program 62.8%

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

      1. lift-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 62.8

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

   4. Applied rewrites 62.8%

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

   6. Applied rewrites 64.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 64.6

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 64.6

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

   8. Applied rewrites 64.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. frac-times N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. rem-sqrt-square-rev N/A

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

      17. lower-fabs.f64 76.3

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

   10. Applied rewrites 76.3%

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

   if 7.5000000000000003e-211 < d

   1. Initial program 70.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 70.1

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

   4. Applied rewrites 70.1%

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

   6. Applied rewrites 71.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 71.1

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 71.1

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

   8. Applied rewrites 71.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. frac-times N/A

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

      10. rem-square-sqrt N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 81.7

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

   10. Applied rewrites 81.7%

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

Alternative 15: 45.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-212}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{elif}\;\ell \leq 10^{-307}:\\ \;\;\;\;t\_0 \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (pow (* l h) -1.0))))
   (if (<= l -6.5e-212)
     (* (- d) t_0)
     (if (<= l 1e-307) (* t_0 d) (/ d (* (sqrt l) (sqrt h)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(pow((l * h), -1.0));
	double tmp;
	if (l <= -6.5e-212) {
		tmp = -d * t_0;
	} else if (l <= 1e-307) {
		tmp = t_0 * d;
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((l * h) ** (-1.0d0)))
    if (l <= (-6.5d-212)) then
        tmp = -d * t_0
    else if (l <= 1d-307) then
        tmp = t_0 * d
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt(Math.pow((l * h), -1.0));
	double tmp;
	if (l <= -6.5e-212) {
		tmp = -d * t_0;
	} else if (l <= 1e-307) {
		tmp = t_0 * d;
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt(math.pow((l * h), -1.0))
	tmp = 0
	if l <= -6.5e-212:
		tmp = -d * t_0
	elif l <= 1e-307:
		tmp = t_0 * d
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt((Float64(l * h) ^ -1.0))
	tmp = 0.0
	if (l <= -6.5e-212)
		tmp = Float64(Float64(-d) * t_0);
	elseif (l <= 1e-307)
		tmp = Float64(t_0 * d);
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt(((l * h) ^ -1.0));
	tmp = 0.0;
	if (l <= -6.5e-212)
		tmp = -d * t_0;
	elseif (l <= 1e-307)
		tmp = t_0 * d;
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -6.5e-212], N[((-d) * t$95$0), $MachinePrecision], If[LessEqual[l, 1e-307], N[(t$95$0 * d), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.5e-212

   1. Initial program 62.8%

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

   3. Taylor expanded in l around -inf

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

      1. *-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 45.9

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

   5. Applied rewrites 45.9%

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

   if -6.5e-212 < l < 9.99999999999999909e-308

   1. Initial program 80.7%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 40.2

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

   5. Applied rewrites 40.2%

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

   if 9.99999999999999909e-308 < l

   1. Initial program 65.3%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 45.1

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

   5. Applied rewrites 45.1%

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

   7. Applied rewrites 45.2%

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

   9. Applied rewrites 51.5%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-212}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq 10^{-307}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{-202}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq 5.7 \cdot 10^{-259}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (<= l -2.5e-202)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (if (<= l 5.7e-259)
     (/ (* (- d) (sqrt (/ h l))) h)
     (/ 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 (l <= -2.5e-202) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (l <= 5.7e-259) {
		tmp = (-d * sqrt((h / l))) / h;
	} 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 (l <= (-2.5d-202)) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else if (l <= 5.7d-259) then
        tmp = (-d * sqrt((h / l))) / h
    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 (l <= -2.5e-202) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else if (l <= 5.7e-259) {
		tmp = (-d * Math.sqrt((h / l))) / h;
	} 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 l <= -2.5e-202:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	elif l <= 5.7e-259:
		tmp = (-d * math.sqrt((h / l))) / h
	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 (l <= -2.5e-202)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (l <= 5.7e-259)
		tmp = Float64(Float64(Float64(-d) * sqrt(Float64(h / l))) / h);
	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 (l <= -2.5e-202)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	elseif (l <= 5.7e-259)
		tmp = (-d * sqrt((h / l))) / h;
	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[l, -2.5e-202], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.7e-259], N[(N[((-d) * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.49999999999999986e-202

   1. Initial program 63.0%

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

   3. Taylor expanded in l around -inf

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

      1. *-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 45.9

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

   5. Applied rewrites 45.9%

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

   if -2.49999999999999986e-202 < l < 5.69999999999999947e-259

   1. Initial program 83.5%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   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 32.1

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

   5. Applied rewrites 32.1%

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

   7. Applied rewrites 29.5%

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

   8. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 41.7%

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

   11. Taylor expanded in l around -inf

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

   13. Applied rewrites 37.9%

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

   if 5.69999999999999947e-259 < l

   1. Initial program 62.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 47.4

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

   5. Applied rewrites 47.4%

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

   7. Applied rewrites 47.5%

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

   9. Applied rewrites 53.4%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{-202}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\ell \leq 5.7 \cdot 10^{-259}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 41.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-212}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot d\\ \end{array} \end{array} \]

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 (<= l -6.5e-212) (* (- 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 (l <= -6.5e-212) {
		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 (l <= (-6.5d-212)) 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 (l <= -6.5e-212) {
		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 l <= -6.5e-212:
		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 (l <= -6.5e-212)
		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 (l <= -6.5e-212)
		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[l, -6.5e-212], N[((-d) * t$95$0), $MachinePrecision], N[(t$95$0 * d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -6.5e-212

   1. Initial program 62.8%

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

   3. Taylor expanded in l around -inf

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

      1. *-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 45.9

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

   5. Applied rewrites 45.9%

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

   if -6.5e-212 < l

   1. Initial program 68.1%

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

   3. 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 44.2

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

   5. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{-212}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 73.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\left(M\_m \cdot \left(D\_m \cdot 0.25\right)\right) \cdot h}{d \cdot \ell}\\ t_1 := \frac{d}{\sqrt{\ell \cdot h}}\\ t_2 := \frac{D\_m}{-2} \cdot \frac{M\_m}{d}\\ \mathbf{if}\;d \leq 7.8 \cdot 10^{-165}:\\ \;\;\;\;\left|t\_1\right| \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \frac{D\_m}{2}\right) \cdot \frac{M\_m}{d}\right), t\_2, 1\right)\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{+100}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(t\_0, t\_2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \frac{D\_m}{d}, \frac{M\_m}{-2}, 1\right) \cdot t\_1\\ \end{array} \end{array} \]

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 (/ (* (* M_m (* D_m 0.25)) h) (* d l)))
        (t_1 (/ d (sqrt (* l h))))
        (t_2 (* (/ D_m -2.0) (/ M_m d))))
   (if (<= d 7.8e-165)
     (* (fabs t_1) (fma (* (/ h l) (* (* 0.5 (/ D_m 2.0)) (/ M_m d))) t_2 1.0))
     (if (<= d 5.8e+100)
       (* (* (sqrt (/ d h)) (sqrt (/ d l))) (fma t_0 t_2 1.0))
       (* (fma (* t_0 (/ D_m d)) (/ M_m -2.0) 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 = ((M_m * (D_m * 0.25)) * h) / (d * l);
	double t_1 = d / sqrt((l * h));
	double t_2 = (D_m / -2.0) * (M_m / d);
	double tmp;
	if (d <= 7.8e-165) {
		tmp = fabs(t_1) * fma(((h / l) * ((0.5 * (D_m / 2.0)) * (M_m / d))), t_2, 1.0);
	} else if (d <= 5.8e+100) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * fma(t_0, t_2, 1.0);
	} else {
		tmp = fma((t_0 * (D_m / d)), (M_m / -2.0), 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(M_m * Float64(D_m * 0.25)) * h) / Float64(d * l))
	t_1 = Float64(d / sqrt(Float64(l * h)))
	t_2 = Float64(Float64(D_m / -2.0) * Float64(M_m / d))
	tmp = 0.0
	if (d <= 7.8e-165)
		tmp = Float64(abs(t_1) * fma(Float64(Float64(h / l) * Float64(Float64(0.5 * Float64(D_m / 2.0)) * Float64(M_m / d))), t_2, 1.0));
	elseif (d <= 5.8e+100)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * fma(t_0, t_2, 1.0));
	else
		tmp = Float64(fma(Float64(t_0 * Float64(D_m / d)), Float64(M_m / -2.0), 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[(M$95$m * N[(D$95$m * 0.25), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(D$95$m / -2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 7.8e-165], N[(N[Abs[t$95$1], $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * N[(N[(0.5 * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.8e+100], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / -2.0), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < 7.7999999999999997e-165

   1. Initial program 60.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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 60.8

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

   4. Applied rewrites 60.8%

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

   6. Applied rewrites 62.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 62.4

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 62.4

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

   8. Applied rewrites 62.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. frac-times N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. rem-sqrt-square-rev N/A

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

      17. lower-fabs.f64 75.0

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

   10. Applied rewrites 75.0%

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

   if 7.7999999999999997e-165 < d < 5.8000000000000001e100

   1. Initial program 82.6%

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

      1. lift-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 82.6

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

   4. Applied rewrites 82.6%

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

   6. Applied rewrites 84.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 84.3

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 84.3

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

   8. Applied rewrites 84.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*r/ N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-*.f64 87.9

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

   10. Applied rewrites 87.9%

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

   if 5.8000000000000001e100 < d

   1. Initial program 64.3%

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

      1. lift-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 64.3

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

   4. Applied rewrites 64.3%

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

   6. Applied rewrites 64.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 64.5

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 64.5

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

   8. Applied rewrites 64.5%

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

   10. Applied rewrites 85.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(M \cdot \left(D \cdot 0.25\right)\right) \cdot h}{d \cdot \ell} \cdot \frac{D}{d}, \frac{M}{-2}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 73.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ t_1 := \frac{\left(M\_m \cdot \left(D\_m \cdot 0.25\right)\right) \cdot h}{d \cdot \ell}\\ \mathbf{if}\;d \leq 2.7 \cdot 10^{-123}:\\ \;\;\;\;\left|t\_0\right| \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\left(0.5 \cdot \frac{D\_m}{2}\right) \cdot \frac{M\_m}{d}\right), \frac{D\_m}{-2} \cdot \frac{M\_m}{d}, 1\right)\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{+100}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{D\_m}{d}, \frac{M\_m}{-2} \cdot t\_1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \frac{D\_m}{d}, \frac{M\_m}{-2}, 1\right) \cdot t\_0\\ \end{array} \end{array} \]

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 (sqrt (* l h))))
        (t_1 (/ (* (* M_m (* D_m 0.25)) h) (* d l))))
   (if (<= d 2.7e-123)
     (*
      (fabs t_0)
      (fma
       (* (/ h l) (* (* 0.5 (/ D_m 2.0)) (/ M_m d)))
       (* (/ D_m -2.0) (/ M_m d))
       1.0))
     (if (<= d 5.8e+100)
       (*
        (* (sqrt (/ d h)) (sqrt (/ d l)))
        (fma (/ D_m d) (* (/ M_m -2.0) t_1) 1.0))
       (* (fma (* t_1 (/ D_m d)) (/ M_m -2.0) 1.0) 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 / sqrt((l * h));
	double t_1 = ((M_m * (D_m * 0.25)) * h) / (d * l);
	double tmp;
	if (d <= 2.7e-123) {
		tmp = fabs(t_0) * fma(((h / l) * ((0.5 * (D_m / 2.0)) * (M_m / d))), ((D_m / -2.0) * (M_m / d)), 1.0);
	} else if (d <= 5.8e+100) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * fma((D_m / d), ((M_m / -2.0) * t_1), 1.0);
	} else {
		tmp = fma((t_1 * (D_m / d)), (M_m / -2.0), 1.0) * 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(d / sqrt(Float64(l * h)))
	t_1 = Float64(Float64(Float64(M_m * Float64(D_m * 0.25)) * h) / Float64(d * l))
	tmp = 0.0
	if (d <= 2.7e-123)
		tmp = Float64(abs(t_0) * fma(Float64(Float64(h / l) * Float64(Float64(0.5 * Float64(D_m / 2.0)) * Float64(M_m / d))), Float64(Float64(D_m / -2.0) * Float64(M_m / d)), 1.0));
	elseif (d <= 5.8e+100)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * fma(Float64(D_m / d), Float64(Float64(M_m / -2.0) * t_1), 1.0));
	else
		tmp = Float64(fma(Float64(t_1 * Float64(D_m / d)), Float64(M_m / -2.0), 1.0) * 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[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(M$95$m * N[(D$95$m * 0.25), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 2.7e-123], N[(N[Abs[t$95$0], $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * N[(N[(0.5 * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / -2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.8e+100], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(M$95$m / -2.0), $MachinePrecision] * t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / -2.0), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < 2.7000000000000001e-123

   1. Initial program 61.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 61.1

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

   4. Applied rewrites 61.1%

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

   6. Applied rewrites 62.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 62.6

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 62.6

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

   8. Applied rewrites 62.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. frac-times N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. rem-sqrt-square-rev N/A

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

      17. lower-fabs.f64 74.7

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

   10. Applied rewrites 74.7%

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

   if 2.7000000000000001e-123 < d < 5.8000000000000001e100

   1. Initial program 84.5%

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

      1. lift-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 84.5

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

   4. Applied rewrites 84.5%

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

   6. Applied rewrites 86.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 86.6

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 86.6

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

   8. Applied rewrites 86.6%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. frac-times N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

   10. Applied rewrites 88.6%

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

   if 5.8000000000000001e100 < d

   1. Initial program 64.3%

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

      1. lift-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 64.3

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

   4. Applied rewrites 64.3%

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

   6. Applied rewrites 64.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 64.5

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 64.5

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

   8. Applied rewrites 64.5%

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

   10. Applied rewrites 85.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(M \cdot \left(D \cdot 0.25\right)\right) \cdot h}{d \cdot \ell} \cdot \frac{D}{d}, \frac{M}{-2}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 26.0% 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 65.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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 26.3

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

5. Applied rewrites 26.3%

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

6. Final simplification 26.3%

   \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]
7. Add Preprocessing

Alternative 21: 25.9% 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 65.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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 26.3

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

5. Applied rewrites 26.3%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
6. Step-by-step derivation

7. Applied rewrites 26.0%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024361 
(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)))))