Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.3% → 74.3%

Time: 8.1s

Alternatives: 16

Speedup: 2.1×

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: 66.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 74.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 - \frac{\left({\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\\ \mathbf{if}\;d \leq -3.4 \cdot 10^{-91}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_0\\ \mathbf{elif}\;d \leq -2.05 \cdot 10^{-282}:\\ \;\;\;\;\left(\left(-1 \cdot d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - \frac{\left({\left(\frac{D\_m \cdot M}{2 \cdot d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-166}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D\_m \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (* (* (pow (* (/ M 2.0) (/ D_m d)) 2.0) 0.5) h) l))))
   (if (<= d -3.4e-91)
     (* (* (sqrt (/ d l)) (sqrt (/ d h))) t_0)
     (if (<= d -2.05e-282)
       (*
        (* (* -1.0 d) (pow (* l h) -0.5))
        (- 1.0 (/ (* (* (pow (/ (* D_m M) (* 2.0 d)) 2.0) 0.5) h) l)))
       (if (<= d 1.05e-166)
         (/
          (fma
           (* -0.125 (/ (pow (* D_m M) 2.0) d))
           (sqrt (pow (/ h l) 3.0))
           (* (sqrt (/ h l)) d))
          h)
         (* (* (/ (sqrt d) (sqrt h)) (pow (/ d l) (/ 1.0 2.0))) t_0))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - (((pow(((M / 2.0) * (D_m / d)), 2.0) * 0.5) * h) / l);
	double tmp;
	if (d <= -3.4e-91) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * t_0;
	} else if (d <= -2.05e-282) {
		tmp = ((-1.0 * d) * pow((l * h), -0.5)) * (1.0 - (((pow(((D_m * M) / (2.0 * d)), 2.0) * 0.5) * h) / l));
	} else if (d <= 1.05e-166) {
		tmp = fma((-0.125 * (pow((D_m * M), 2.0) / d)), sqrt(pow((h / l), 3.0)), (sqrt((h / l)) * d)) / h;
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), (1.0 / 2.0))) * t_0;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(1.0 - Float64(Float64(Float64((Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0) * 0.5) * h) / l))
	tmp = 0.0
	if (d <= -3.4e-91)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * t_0);
	elseif (d <= -2.05e-282)
		tmp = Float64(Float64(Float64(-1.0 * d) * (Float64(l * h) ^ -0.5)) * Float64(1.0 - Float64(Float64(Float64((Float64(Float64(D_m * M) / Float64(2.0 * d)) ^ 2.0) * 0.5) * h) / l)));
	elseif (d <= 1.05e-166)
		tmp = Float64(fma(Float64(-0.125 * Float64((Float64(D_m * M) ^ 2.0) / d)), sqrt((Float64(h / l) ^ 3.0)), Float64(sqrt(Float64(h / l)) * d)) / h);
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * t_0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.4e-91], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, -2.05e-282], N[(N[(N[(-1.0 * d), $MachinePrecision] * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.05e-166], N[(N[(N[(-0.125 * N[(N[Power[N[(D$95$m * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(h / l), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.40000000000000027e-91

   1. Initial program 76.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 78.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 78.8

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

   6. Applied rewrites 78.8%

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

   if -3.40000000000000027e-91 < d < -2.04999999999999989e-282

   1. Initial program 49.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 48.3%

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

      1. lift-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 49.7

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in h around -inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. pow1/2 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-pow2 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sqrt-div N/A

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

      13. *-commutative N/A

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

      14. sqrt-div N/A

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

      15. inv-pow N/A

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

      16. sqrt-pow1 N/A

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

      17. lower-pow.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. metadata-eval 65.6

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

   9. Applied rewrites 65.6%

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

   if -2.04999999999999989e-282 < d < 1.05e-166

   1. Initial program 39.2%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 44.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   if 1.05e-166 < d

   1. Initial program 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 75.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 84.9

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

   6. Applied rewrites 84.9%

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

Alternative 2: 60.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-62}:\\ \;\;\;\;t\_1 \cdot \left(t\_0 \cdot \left(\frac{\left(\left(M \cdot D\_m\right) \cdot \left(M \cdot D\_m\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (sqrt (/ d h))))
   (if (<=
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))
        -2e-62)
     (* t_1 (* t_0 (* (/ (* (* (* M D_m) (* M D_m)) h) (* (* d d) l)) -0.125)))
     (* t_1 t_0))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = sqrt((d / h));
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -2e-62) {
		tmp = t_1 * (t_0 * (((((M * D_m) * (M * D_m)) * h) / ((d * d) * l)) * -0.125));
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Fortran

D_m =     private
NOTE: d, h, l, 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, 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
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = sqrt((d / h))
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-2d-62)) then
        tmp = t_1 * (t_0 * (((((m * d_m) * (m * d_m)) * h) / ((d * d) * l)) * (-0.125d0)))
    else
        tmp = t_1 * t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = Math.sqrt((d / h));
	double tmp;
	if (((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_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -2e-62) {
		tmp = t_1 * (t_0 * (((((M * D_m) * (M * D_m)) * h) / ((d * d) * l)) * -0.125));
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((d / l))
	t_1 = math.sqrt((d / h))
	tmp = 0
	if ((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_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -2e-62:
		tmp = t_1 * (t_0 * (((((M * D_m) * (M * D_m)) * h) / ((d * d) * l)) * -0.125))
	else:
		tmp = t_1 * t_0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(d / h))
	tmp = 0.0
	if (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_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -2e-62)
		tmp = Float64(t_1 * Float64(t_0 * Float64(Float64(Float64(Float64(Float64(M * D_m) * Float64(M * D_m)) * h) / Float64(Float64(d * d) * l)) * -0.125)));
	else
		tmp = Float64(t_1 * t_0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((d / l));
	t_1 = sqrt((d / h));
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -2e-62)
		tmp = t_1 * (t_0 * (((((M * D_m) * (M * D_m)) * h) / ((d * d) * l)) * -0.125));
	else
		tmp = t_1 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-62], N[(t$95$1 * N[(t$95$0 * N[(N[(N[(N[(N[(M * D$95$m), $MachinePrecision] * N[(M * D$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * 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)))) < -2.0000000000000001e-62

   1. Initial program 85.2%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. *-commutative N/A

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

      8. lower-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 64.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\right)\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 64.2

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

   9. Applied rewrites 64.2%

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

   if -2.0000000000000001e-62 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 57.0%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. *-commutative N/A

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

      8. lower-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 3.0

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 3.1%

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

   8. Taylor expanded in d around inf

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 58.2

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

   10. Applied rewrites 58.2%

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

Alternative 3: 56.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{+41}:\\ \;\;\;\;t\_1 \cdot \left(t\_0 \cdot \left(\left(\left(D\_m \cdot D\_m\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (sqrt (/ d h))))
   (if (<=
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))
        -5e+41)
     (* t_1 (* t_0 (* (* (* D_m D_m) (* (* M M) (/ h (* (* d d) l)))) -0.125)))
     (* t_1 t_0))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = sqrt((d / h));
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e+41) {
		tmp = t_1 * (t_0 * (((D_m * D_m) * ((M * M) * (h / ((d * d) * l)))) * -0.125));
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Fortran

D_m =     private
NOTE: d, h, l, 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, 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
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = sqrt((d / h))
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-5d+41)) then
        tmp = t_1 * (t_0 * (((d_m * d_m) * ((m * m) * (h / ((d * d) * l)))) * (-0.125d0)))
    else
        tmp = t_1 * t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = Math.sqrt((d / h));
	double tmp;
	if (((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_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e+41) {
		tmp = t_1 * (t_0 * (((D_m * D_m) * ((M * M) * (h / ((d * d) * l)))) * -0.125));
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((d / l))
	t_1 = math.sqrt((d / h))
	tmp = 0
	if ((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_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e+41:
		tmp = t_1 * (t_0 * (((D_m * D_m) * ((M * M) * (h / ((d * d) * l)))) * -0.125))
	else:
		tmp = t_1 * t_0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(d / h))
	tmp = 0.0
	if (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_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e+41)
		tmp = Float64(t_1 * Float64(t_0 * Float64(Float64(Float64(D_m * D_m) * Float64(Float64(M * M) * Float64(h / Float64(Float64(d * d) * l)))) * -0.125)));
	else
		tmp = Float64(t_1 * t_0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((d / l));
	t_1 = sqrt((d / h));
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -5e+41)
		tmp = t_1 * (t_0 * (((D_m * D_m) * ((M * M) * (h / ((d * d) * l)))) * -0.125));
	else
		tmp = t_1 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+41], N[(t$95$1 * N[(t$95$0 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(h / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * 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)))) < -5.00000000000000022e41

   1. Initial program 84.7%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. *-commutative N/A

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

      8. lower-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

   7. Applied rewrites 64.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\right)\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 54.4

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

   9. Applied rewrites 54.4%

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

   if -5.00000000000000022e41 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 57.8%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. *-commutative N/A

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

      8. lower-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 3.9

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

   5. Applied rewrites 3.9%

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

   7. Applied rewrites 3.9%

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

   8. Taylor expanded in d around inf

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 57.2

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

   10. Applied rewrites 57.2%

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

Alternative 4: 44.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

D_m =     private
NOTE: d, h, l, 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, 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
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = sqrt((d / h))
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= 0.0d0) then
        tmp = t_1 * (t_0 * (-1.0d0))
    else
        tmp = t_1 * t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = Math.sqrt((d / h));
	double tmp;
	if (((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_m) / (2.0 * d)), 2.0)) * (h / l)))) <= 0.0) {
		tmp = t_1 * (t_0 * -1.0);
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((d / l))
	t_1 = math.sqrt((d / h))
	tmp = 0
	if ((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_m) / (2.0 * d)), 2.0)) * (h / l)))) <= 0.0:
		tmp = t_1 * (t_0 * -1.0)
	else:
		tmp = t_1 * t_0
	return tmp

Julia

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

MATLAB

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

Wolfram

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

   1. Initial program 78.6%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. *-commutative N/A

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

      8. lower-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 55.5%

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

   8. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 19.2

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

   10. Applied rewrites 19.2%

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

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

   1. Initial program 58.2%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. *-commutative N/A

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

      8. lower-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 1.9

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

   5. Applied rewrites 1.9%

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

   7. Applied rewrites 1.9%

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

   8. Taylor expanded in d around inf

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 61.1

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

   10. Applied rewrites 61.1%

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

Alternative 5: 71.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 - \frac{\left({\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\\ t_1 := {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;d \leq -3.4 \cdot 10^{-91}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_0\\ \mathbf{elif}\;d \leq -2.05 \cdot 10^{-282}:\\ \;\;\;\;\left(\left(-1 \cdot d\right) \cdot t\_1\right) \cdot \left(1 - \frac{\left({\left(\frac{D\_m \cdot M}{2 \cdot d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D\_m \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot d\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (* (* (pow (* (/ M 2.0) (/ D_m d)) 2.0) 0.5) h) l)))
        (t_1 (pow (* l h) -0.5)))
   (if (<= d -3.4e-91)
     (* (* (sqrt (/ d l)) (sqrt (/ d h))) t_0)
     (if (<= d -2.05e-282)
       (*
        (* (* -1.0 d) t_1)
        (- 1.0 (/ (* (* (pow (/ (* D_m M) (* 2.0 d)) 2.0) 0.5) h) l)))
       (if (<= d 1.15e-146)
         (/
          (fma
           (* -0.125 (/ (pow (* D_m M) 2.0) d))
           (sqrt (pow (/ h l) 3.0))
           (* (sqrt (/ h l)) d))
          h)
         (* (* t_1 d) t_0))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - (((pow(((M / 2.0) * (D_m / d)), 2.0) * 0.5) * h) / l);
	double t_1 = pow((l * h), -0.5);
	double tmp;
	if (d <= -3.4e-91) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * t_0;
	} else if (d <= -2.05e-282) {
		tmp = ((-1.0 * d) * t_1) * (1.0 - (((pow(((D_m * M) / (2.0 * d)), 2.0) * 0.5) * h) / l));
	} else if (d <= 1.15e-146) {
		tmp = fma((-0.125 * (pow((D_m * M), 2.0) / d)), sqrt(pow((h / l), 3.0)), (sqrt((h / l)) * d)) / h;
	} else {
		tmp = (t_1 * d) * t_0;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(1.0 - Float64(Float64(Float64((Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0) * 0.5) * h) / l))
	t_1 = Float64(l * h) ^ -0.5
	tmp = 0.0
	if (d <= -3.4e-91)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * t_0);
	elseif (d <= -2.05e-282)
		tmp = Float64(Float64(Float64(-1.0 * d) * t_1) * Float64(1.0 - Float64(Float64(Float64((Float64(Float64(D_m * M) / Float64(2.0 * d)) ^ 2.0) * 0.5) * h) / l)));
	elseif (d <= 1.15e-146)
		tmp = Float64(fma(Float64(-0.125 * Float64((Float64(D_m * M) ^ 2.0) / d)), sqrt((Float64(h / l) ^ 3.0)), Float64(sqrt(Float64(h / l)) * d)) / h);
	else
		tmp = Float64(Float64(t_1 * d) * t_0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[d, -3.4e-91], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, -2.05e-282], N[(N[(N[(-1.0 * d), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.15e-146], N[(N[(N[(-0.125 * N[(N[Power[N[(D$95$m * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(h / l), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(t$95$1 * d), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.40000000000000027e-91

   1. Initial program 76.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 78.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 78.8

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

   6. Applied rewrites 78.8%

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

   if -3.40000000000000027e-91 < d < -2.04999999999999989e-282

   1. Initial program 49.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 48.3%

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

      1. lift-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 49.7

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in h around -inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. pow1/2 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-pow2 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sqrt-div N/A

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

      13. *-commutative N/A

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

      14. sqrt-div N/A

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

      15. inv-pow N/A

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

      16. sqrt-pow1 N/A

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

      17. lower-pow.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. metadata-eval 65.6

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

   9. Applied rewrites 65.6%

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

   if -2.04999999999999989e-282 < d < 1.15e-146

   1. Initial program 40.7%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 44.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   if 1.15e-146 < d

   1. Initial program 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. inv-pow N/A

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

      4. *-commutative N/A

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

      5. sqrt-pow1 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. metadata-eval 79.0

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

   7. Applied rewrites 79.0%

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

Alternative 6: 73.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 - \frac{\left({\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\\ t_1 := {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;d \leq -3.4 \cdot 10^{-91}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_0\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(\left(-1 \cdot d\right) \cdot t\_1\right) \cdot \left(1 - \frac{\left({\left(\frac{D\_m \cdot M}{2 \cdot d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot d\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (* (* (pow (* (/ M 2.0) (/ D_m d)) 2.0) 0.5) h) l)))
        (t_1 (pow (* l h) -0.5)))
   (if (<= d -3.4e-91)
     (* (* (sqrt (/ d l)) (sqrt (/ d h))) t_0)
     (if (<= d -2e-311)
       (*
        (* (* -1.0 d) t_1)
        (- 1.0 (/ (* (* (pow (/ (* D_m M) (* 2.0 d)) 2.0) 0.5) h) l)))
       (* (* t_1 d) t_0)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - (((pow(((M / 2.0) * (D_m / d)), 2.0) * 0.5) * h) / l);
	double t_1 = pow((l * h), -0.5);
	double tmp;
	if (d <= -3.4e-91) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * t_0;
	} else if (d <= -2e-311) {
		tmp = ((-1.0 * d) * t_1) * (1.0 - (((pow(((D_m * M) / (2.0 * d)), 2.0) * 0.5) * h) / l));
	} else {
		tmp = (t_1 * d) * t_0;
	}
	return tmp;
}

Fortran

D_m =     private
NOTE: d, h, l, 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, 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
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - ((((((m / 2.0d0) * (d_m / d)) ** 2.0d0) * 0.5d0) * h) / l)
    t_1 = (l * h) ** (-0.5d0)
    if (d <= (-3.4d-91)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * t_0
    else if (d <= (-2d-311)) then
        tmp = (((-1.0d0) * d) * t_1) * (1.0d0 - ((((((d_m * m) / (2.0d0 * d)) ** 2.0d0) * 0.5d0) * h) / l))
    else
        tmp = (t_1 * d) * t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - (((Math.pow(((M / 2.0) * (D_m / d)), 2.0) * 0.5) * h) / l);
	double t_1 = Math.pow((l * h), -0.5);
	double tmp;
	if (d <= -3.4e-91) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * t_0;
	} else if (d <= -2e-311) {
		tmp = ((-1.0 * d) * t_1) * (1.0 - (((Math.pow(((D_m * M) / (2.0 * d)), 2.0) * 0.5) * h) / l));
	} else {
		tmp = (t_1 * d) * t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = 1.0 - (((math.pow(((M / 2.0) * (D_m / d)), 2.0) * 0.5) * h) / l)
	t_1 = math.pow((l * h), -0.5)
	tmp = 0
	if d <= -3.4e-91:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * t_0
	elif d <= -2e-311:
		tmp = ((-1.0 * d) * t_1) * (1.0 - (((math.pow(((D_m * M) / (2.0 * d)), 2.0) * 0.5) * h) / l))
	else:
		tmp = (t_1 * d) * t_0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(1.0 - Float64(Float64(Float64((Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0) * 0.5) * h) / l))
	t_1 = Float64(l * h) ^ -0.5
	tmp = 0.0
	if (d <= -3.4e-91)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * t_0);
	elseif (d <= -2e-311)
		tmp = Float64(Float64(Float64(-1.0 * d) * t_1) * Float64(1.0 - Float64(Float64(Float64((Float64(Float64(D_m * M) / Float64(2.0 * d)) ^ 2.0) * 0.5) * h) / l)));
	else
		tmp = Float64(Float64(t_1 * d) * t_0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = 1.0 - ((((((M / 2.0) * (D_m / d)) ^ 2.0) * 0.5) * h) / l);
	t_1 = (l * h) ^ -0.5;
	tmp = 0.0;
	if (d <= -3.4e-91)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * t_0;
	elseif (d <= -2e-311)
		tmp = ((-1.0 * d) * t_1) * (1.0 - ((((((D_m * M) / (2.0 * d)) ^ 2.0) * 0.5) * h) / l));
	else
		tmp = (t_1 * d) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[d, -3.4e-91], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, -2e-311], N[(N[(N[(-1.0 * d), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * d), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.40000000000000027e-91

   1. Initial program 76.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 78.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 78.8

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

   6. Applied rewrites 78.8%

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

   if -3.40000000000000027e-91 < d < -1.9999999999999e-311

   1. Initial program 47.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 45.7%

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

      1. lift-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 47.1

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

   6. Applied rewrites 47.1%

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

   7. Taylor expanded in h around -inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. pow1/2 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-pow2 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sqrt-div N/A

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

      13. *-commutative N/A

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

      14. sqrt-div N/A

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

      15. inv-pow N/A

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

      16. sqrt-pow1 N/A

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

      17. lower-pow.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. metadata-eval 63.1

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

   9. Applied rewrites 63.1%

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

   if -1.9999999999999e-311 < d

   1. Initial program 66.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 67.6%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. inv-pow N/A

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

      4. *-commutative N/A

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

      5. sqrt-pow1 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. metadata-eval 73.9

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

   7. Applied rewrites 73.9%

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

Alternative 7: 71.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left({\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{if}\;h \leq -2.2 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-1 \cdot d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - \frac{\left({\left(\frac{D\_m \cdot M}{2 \cdot d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{elif}\;h \leq 1.3 \cdot 10^{+79}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (sqrt (/ d l)) (sqrt (/ d h)))
          (- 1.0 (/ (* (* (pow (* (/ M 2.0) (/ D_m d)) 2.0) 0.5) h) l)))))
   (if (<= h -2.2e-17)
     t_0
     (if (<= h -5e-310)
       (*
        (* (* -1.0 d) (pow (* l h) -0.5))
        (- 1.0 (/ (* (* (pow (/ (* D_m M) (* 2.0 d)) 2.0) 0.5) h) l)))
       (if (<= h 1.3e+79)
         (*
          (* (sqrt (/ 1.0 (* l h))) d)
          (- 1.0 (* (* 0.5 (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))
         t_0)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((pow(((M / 2.0) * (D_m / d)), 2.0) * 0.5) * h) / l));
	double tmp;
	if (h <= -2.2e-17) {
		tmp = t_0;
	} else if (h <= -5e-310) {
		tmp = ((-1.0 * d) * pow((l * h), -0.5)) * (1.0 - (((pow(((D_m * M) / (2.0 * d)), 2.0) * 0.5) * h) / l));
	} else if (h <= 1.3e+79) {
		tmp = (sqrt((1.0 / (l * h))) * d) * (1.0 - ((0.5 * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

D_m =     private
NOTE: d, h, l, 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, 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
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((((((m / 2.0d0) * (d_m / d)) ** 2.0d0) * 0.5d0) * h) / l))
    if (h <= (-2.2d-17)) then
        tmp = t_0
    else if (h <= (-5d-310)) then
        tmp = (((-1.0d0) * d) * ((l * h) ** (-0.5d0))) * (1.0d0 - ((((((d_m * m) / (2.0d0 * d)) ** 2.0d0) * 0.5d0) * h) / l))
    else if (h <= 1.3d+79) then
        tmp = (sqrt((1.0d0 / (l * h))) * d) * (1.0d0 - ((0.5d0 * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (((Math.pow(((M / 2.0) * (D_m / d)), 2.0) * 0.5) * h) / l));
	double tmp;
	if (h <= -2.2e-17) {
		tmp = t_0;
	} else if (h <= -5e-310) {
		tmp = ((-1.0 * d) * Math.pow((l * h), -0.5)) * (1.0 - (((Math.pow(((D_m * M) / (2.0 * d)), 2.0) * 0.5) * h) / l));
	} else if (h <= 1.3e+79) {
		tmp = (Math.sqrt((1.0 / (l * h))) * d) * (1.0 - ((0.5 * Math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (((math.pow(((M / 2.0) * (D_m / d)), 2.0) * 0.5) * h) / l))
	tmp = 0
	if h <= -2.2e-17:
		tmp = t_0
	elif h <= -5e-310:
		tmp = ((-1.0 * d) * math.pow((l * h), -0.5)) * (1.0 - (((math.pow(((D_m * M) / (2.0 * d)), 2.0) * 0.5) * h) / l))
	elif h <= 1.3e+79:
		tmp = (math.sqrt((1.0 / (l * h))) * d) * (1.0 - ((0.5 * math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	else:
		tmp = t_0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64((Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0) * 0.5) * h) / l)))
	tmp = 0.0
	if (h <= -2.2e-17)
		tmp = t_0;
	elseif (h <= -5e-310)
		tmp = Float64(Float64(Float64(-1.0 * d) * (Float64(l * h) ^ -0.5)) * Float64(1.0 - Float64(Float64(Float64((Float64(Float64(D_m * M) / Float64(2.0 * d)) ^ 2.0) * 0.5) * h) / l)));
	elseif (h <= 1.3e+79)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d) * Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((((M / 2.0) * (D_m / d)) ^ 2.0) * 0.5) * h) / l));
	tmp = 0.0;
	if (h <= -2.2e-17)
		tmp = t_0;
	elseif (h <= -5e-310)
		tmp = ((-1.0 * d) * ((l * h) ^ -0.5)) * (1.0 - ((((((D_m * M) / (2.0 * d)) ^ 2.0) * 0.5) * h) / l));
	elseif (h <= 1.3e+79)
		tmp = (sqrt((1.0 / (l * h))) * d) * (1.0 - ((0.5 * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2.2e-17], t$95$0, If[LessEqual[h, -5e-310], N[(N[(N[(-1.0 * d), $MachinePrecision] * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.3e+79], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.2e-17 or 1.30000000000000007e79 < h

   1. Initial program 59.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 61.3

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

   6. Applied rewrites 61.3%

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

   if -2.2e-17 < h < -4.999999999999985e-310

   1. Initial program 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 70.9%

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

      1. lift-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 71.6

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

   6. Applied rewrites 71.6%

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

   7. Taylor expanded in h around -inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. pow1/2 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-pow2 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. sqrt-div N/A

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

      13. *-commutative N/A

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

      14. sqrt-div N/A

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

      15. inv-pow N/A

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

      16. sqrt-pow1 N/A

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

      17. lower-pow.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. metadata-eval 81.4

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

   9. Applied rewrites 81.4%

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

   if -4.999999999999985e-310 < h < 1.30000000000000007e79

   1. Initial program 73.2%

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

   3. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 80.4

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

   5. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 80.4

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

   7. Applied rewrites 80.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 80.4

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

   9. Applied rewrites 80.4%

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

Alternative 8: 71.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 - \left(0.5 \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_1 := \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left({\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{if}\;h \leq -2.2 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}\right) \cdot t\_0\\ \mathbf{elif}\;h \leq 1.3 \cdot 10^{+79}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (* 0.5 (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))
        (t_1
         (*
          (* (sqrt (/ d l)) (sqrt (/ d h)))
          (- 1.0 (/ (* (* (pow (* (/ M 2.0) (/ D_m d)) 2.0) 0.5) h) l)))))
   (if (<= h -2.2e-17)
     t_1
     (if (<= h -5e-310)
       (* (* (* -1.0 d) (/ 1.0 (sqrt (* l h)))) t_0)
       (if (<= h 1.3e+79) (* (* (sqrt (/ 1.0 (* l h))) d) t_0) t_1)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - ((0.5 * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l));
	double t_1 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((pow(((M / 2.0) * (D_m / d)), 2.0) * 0.5) * h) / l));
	double tmp;
	if (h <= -2.2e-17) {
		tmp = t_1;
	} else if (h <= -5e-310) {
		tmp = ((-1.0 * d) * (1.0 / sqrt((l * h)))) * t_0;
	} else if (h <= 1.3e+79) {
		tmp = (sqrt((1.0 / (l * h))) * d) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

D_m =     private
NOTE: d, h, l, 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, 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
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - ((0.5d0 * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l))
    t_1 = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((((((m / 2.0d0) * (d_m / d)) ** 2.0d0) * 0.5d0) * h) / l))
    if (h <= (-2.2d-17)) then
        tmp = t_1
    else if (h <= (-5d-310)) then
        tmp = (((-1.0d0) * d) * (1.0d0 / sqrt((l * h)))) * t_0
    else if (h <= 1.3d+79) then
        tmp = (sqrt((1.0d0 / (l * h))) * d) * t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - ((0.5 * Math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l));
	double t_1 = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (((Math.pow(((M / 2.0) * (D_m / d)), 2.0) * 0.5) * h) / l));
	double tmp;
	if (h <= -2.2e-17) {
		tmp = t_1;
	} else if (h <= -5e-310) {
		tmp = ((-1.0 * d) * (1.0 / Math.sqrt((l * h)))) * t_0;
	} else if (h <= 1.3e+79) {
		tmp = (Math.sqrt((1.0 / (l * h))) * d) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = 1.0 - ((0.5 * math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l))
	t_1 = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (((math.pow(((M / 2.0) * (D_m / d)), 2.0) * 0.5) * h) / l))
	tmp = 0
	if h <= -2.2e-17:
		tmp = t_1
	elif h <= -5e-310:
		tmp = ((-1.0 * d) * (1.0 / math.sqrt((l * h)))) * t_0
	elif h <= 1.3e+79:
		tmp = (math.sqrt((1.0 / (l * h))) * d) * t_0
	else:
		tmp = t_1
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))
	t_1 = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64((Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0) * 0.5) * h) / l)))
	tmp = 0.0
	if (h <= -2.2e-17)
		tmp = t_1;
	elseif (h <= -5e-310)
		tmp = Float64(Float64(Float64(-1.0 * d) * Float64(1.0 / sqrt(Float64(l * h)))) * t_0);
	elseif (h <= 1.3e+79)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d) * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = 1.0 - ((0.5 * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l));
	t_1 = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((((M / 2.0) * (D_m / d)) ^ 2.0) * 0.5) * h) / l));
	tmp = 0.0;
	if (h <= -2.2e-17)
		tmp = t_1;
	elseif (h <= -5e-310)
		tmp = ((-1.0 * d) * (1.0 / sqrt((l * h)))) * t_0;
	elseif (h <= 1.3e+79)
		tmp = (sqrt((1.0 / (l * h))) * d) * t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2.2e-17], t$95$1, If[LessEqual[h, -5e-310], N[(N[(N[(-1.0 * d), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[h, 1.3e+79], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] * t$95$0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.2e-17 or 1.30000000000000007e79 < h

   1. Initial program 59.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 61.3

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

   6. Applied rewrites 61.3%

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

   if -2.2e-17 < h < -4.999999999999985e-310

   1. Initial program 70.8%

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

   3. Taylor expanded in h around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 78.5

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

   5. Applied rewrites 78.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. *-commutative N/A

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

      5. inv-pow N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 79.1

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

   7. Applied rewrites 79.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 79.1

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

   9. Applied rewrites 79.1%

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

   if -4.999999999999985e-310 < h < 1.30000000000000007e79

   1. Initial program 73.2%

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

   3. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 80.4

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

   5. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 80.4

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

   7. Applied rewrites 80.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 80.4

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

   9. Applied rewrites 80.4%

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

Alternative 9: 70.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 - \left(0.5 \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ \mathbf{if}\;d \leq -2.6 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (* 0.5 (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l)))))
   (if (<= d -2.6e-91)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (- 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D_m d)) 2.0) 0.5)))))
     (if (<= d -2e-311)
       (* (* (* -1.0 d) (/ 1.0 (sqrt (* l h)))) t_0)
       (* (* (sqrt (/ 1.0 (* l h))) d) t_0)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - ((0.5 * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l));
	double tmp;
	if (d <= -2.6e-91) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * (pow(((M / 2.0) * (D_m / d)), 2.0) * 0.5))));
	} else if (d <= -2e-311) {
		tmp = ((-1.0 * d) * (1.0 / sqrt((l * h)))) * t_0;
	} else {
		tmp = (sqrt((1.0 / (l * h))) * d) * t_0;
	}
	return tmp;
}

Fortran

D_m =     private
NOTE: d, h, l, 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, 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
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((0.5d0 * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l))
    if (d <= (-2.6d-91)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - ((h / l) * ((((m / 2.0d0) * (d_m / d)) ** 2.0d0) * 0.5d0))))
    else if (d <= (-2d-311)) then
        tmp = (((-1.0d0) * d) * (1.0d0 / sqrt((l * h)))) * t_0
    else
        tmp = (sqrt((1.0d0 / (l * h))) * d) * t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - ((0.5 * Math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l));
	double tmp;
	if (d <= -2.6e-91) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - ((h / l) * (Math.pow(((M / 2.0) * (D_m / d)), 2.0) * 0.5))));
	} else if (d <= -2e-311) {
		tmp = ((-1.0 * d) * (1.0 / Math.sqrt((l * h)))) * t_0;
	} else {
		tmp = (Math.sqrt((1.0 / (l * h))) * d) * t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = 1.0 - ((0.5 * math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l))
	tmp = 0
	if d <= -2.6e-91:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - ((h / l) * (math.pow(((M / 2.0) * (D_m / d)), 2.0) * 0.5))))
	elif d <= -2e-311:
		tmp = ((-1.0 * d) * (1.0 / math.sqrt((l * h)))) * t_0
	else:
		tmp = (math.sqrt((1.0 / (l * h))) * d) * t_0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))
	tmp = 0.0
	if (d <= -2.6e-91)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0) * 0.5)))));
	elseif (d <= -2e-311)
		tmp = Float64(Float64(Float64(-1.0 * d) * Float64(1.0 / sqrt(Float64(l * h)))) * t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d) * t_0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = 1.0 - ((0.5 * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l));
	tmp = 0.0;
	if (d <= -2.6e-91)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((M / 2.0) * (D_m / d)) ^ 2.0) * 0.5))));
	elseif (d <= -2e-311)
		tmp = ((-1.0 * d) * (1.0 / sqrt((l * h)))) * t_0;
	else
		tmp = (sqrt((1.0 / (l * h))) * d) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.6e-91], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2e-311], N[(N[(N[(-1.0 * d), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.60000000000000014e-91

   1. Initial program 76.7%

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

   4. Applied rewrites 76.8%

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

   if -2.60000000000000014e-91 < d < -1.9999999999999e-311

   1. Initial program 47.0%

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

   3. Taylor expanded in h around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 60.0

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

   5. Applied rewrites 60.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. *-commutative N/A

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

      5. inv-pow N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 60.3

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

   7. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 60.3

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

   9. Applied rewrites 60.3%

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

   if -1.9999999999999e-311 < d

   1. Initial program 66.6%

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

   3. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 69.6

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

   7. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 69.6

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

   9. Applied rewrites 69.6%

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

Alternative 10: 70.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 - \left(0.5 \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ \mathbf{if}\;d \leq -2.6 \cdot 10^{-91}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_0\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (* 0.5 (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l)))))
   (if (<= d -2.6e-91)
     (* (* (sqrt (/ d l)) (sqrt (/ d h))) t_0)
     (if (<= d -2e-311)
       (* (* (* -1.0 d) (/ 1.0 (sqrt (* l h)))) t_0)
       (* (* (sqrt (/ 1.0 (* l h))) d) t_0)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - ((0.5 * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l));
	double tmp;
	if (d <= -2.6e-91) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * t_0;
	} else if (d <= -2e-311) {
		tmp = ((-1.0 * d) * (1.0 / sqrt((l * h)))) * t_0;
	} else {
		tmp = (sqrt((1.0 / (l * h))) * d) * t_0;
	}
	return tmp;
}

Fortran

D_m =     private
NOTE: d, h, l, 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, 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
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((0.5d0 * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l))
    if (d <= (-2.6d-91)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * t_0
    else if (d <= (-2d-311)) then
        tmp = (((-1.0d0) * d) * (1.0d0 / sqrt((l * h)))) * t_0
    else
        tmp = (sqrt((1.0d0 / (l * h))) * d) * t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - ((0.5 * Math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l));
	double tmp;
	if (d <= -2.6e-91) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * t_0;
	} else if (d <= -2e-311) {
		tmp = ((-1.0 * d) * (1.0 / Math.sqrt((l * h)))) * t_0;
	} else {
		tmp = (Math.sqrt((1.0 / (l * h))) * d) * t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = 1.0 - ((0.5 * math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l))
	tmp = 0
	if d <= -2.6e-91:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * t_0
	elif d <= -2e-311:
		tmp = ((-1.0 * d) * (1.0 / math.sqrt((l * h)))) * t_0
	else:
		tmp = (math.sqrt((1.0 / (l * h))) * d) * t_0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))
	tmp = 0.0
	if (d <= -2.6e-91)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * t_0);
	elseif (d <= -2e-311)
		tmp = Float64(Float64(Float64(-1.0 * d) * Float64(1.0 / sqrt(Float64(l * h)))) * t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d) * t_0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = 1.0 - ((0.5 * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l));
	tmp = 0.0;
	if (d <= -2.6e-91)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * t_0;
	elseif (d <= -2e-311)
		tmp = ((-1.0 * d) * (1.0 / sqrt((l * h)))) * t_0;
	else
		tmp = (sqrt((1.0 / (l * h))) * d) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.6e-91], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, -2e-311], N[(N[(N[(-1.0 * d), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.60000000000000014e-91

   1. Initial program 76.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 76.7

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

   4. Applied rewrites 76.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 76.7

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

   6. Applied rewrites 76.7%

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

   if -2.60000000000000014e-91 < d < -1.9999999999999e-311

   1. Initial program 47.0%

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

   3. Taylor expanded in h around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 60.0

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

   5. Applied rewrites 60.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. *-commutative N/A

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

      5. inv-pow N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 60.3

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

   7. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 60.3

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

   9. Applied rewrites 60.3%

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

   if -1.9999999999999e-311 < d

   1. Initial program 66.6%

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

   3. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 69.6

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

   7. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 69.6

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

   9. Applied rewrites 69.6%

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

Alternative 11: 69.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 - \left(0.5 \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ \mathbf{if}\;h \leq -2.8 \cdot 10^{+253}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - \frac{\left({\left(\frac{D\_m \cdot M}{2 \cdot d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}\right) \cdot t\_0\\ \mathbf{elif}\;h \leq 2.3 \cdot 10^{+196}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (* 0.5 (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l)))))
   (if (<= h -2.8e+253)
     (*
      (sqrt (* (/ d l) (/ d h)))
      (- 1.0 (/ (* (* (pow (/ (* D_m M) (* 2.0 d)) 2.0) 0.5) h) l)))
     (if (<= h -5e-310)
       (* (* (* -1.0 d) (/ 1.0 (sqrt (* l h)))) t_0)
       (if (<= h 2.3e+196)
         (* (* (sqrt (/ 1.0 (* l h))) d) t_0)
         (* (/ 1.0 (* (sqrt l) (sqrt h))) d))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - ((0.5 * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l));
	double tmp;
	if (h <= -2.8e+253) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (((pow(((D_m * M) / (2.0 * d)), 2.0) * 0.5) * h) / l));
	} else if (h <= -5e-310) {
		tmp = ((-1.0 * d) * (1.0 / sqrt((l * h)))) * t_0;
	} else if (h <= 2.3e+196) {
		tmp = (sqrt((1.0 / (l * h))) * d) * t_0;
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

D_m =     private
NOTE: d, h, l, 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, 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
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((0.5d0 * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l))
    if (h <= (-2.8d+253)) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - ((((((d_m * m) / (2.0d0 * d)) ** 2.0d0) * 0.5d0) * h) / l))
    else if (h <= (-5d-310)) then
        tmp = (((-1.0d0) * d) * (1.0d0 / sqrt((l * h)))) * t_0
    else if (h <= 2.3d+196) then
        tmp = (sqrt((1.0d0 / (l * h))) * d) * t_0
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - ((0.5 * Math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l));
	double tmp;
	if (h <= -2.8e+253) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - (((Math.pow(((D_m * M) / (2.0 * d)), 2.0) * 0.5) * h) / l));
	} else if (h <= -5e-310) {
		tmp = ((-1.0 * d) * (1.0 / Math.sqrt((l * h)))) * t_0;
	} else if (h <= 2.3e+196) {
		tmp = (Math.sqrt((1.0 / (l * h))) * d) * t_0;
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = 1.0 - ((0.5 * math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l))
	tmp = 0
	if h <= -2.8e+253:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - (((math.pow(((D_m * M) / (2.0 * d)), 2.0) * 0.5) * h) / l))
	elif h <= -5e-310:
		tmp = ((-1.0 * d) * (1.0 / math.sqrt((l * h)))) * t_0
	elif h <= 2.3e+196:
		tmp = (math.sqrt((1.0 / (l * h))) * d) * t_0
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))
	tmp = 0.0
	if (h <= -2.8e+253)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64((Float64(Float64(D_m * M) / Float64(2.0 * d)) ^ 2.0) * 0.5) * h) / l)));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(Float64(-1.0 * d) * Float64(1.0 / sqrt(Float64(l * h)))) * t_0);
	elseif (h <= 2.3e+196)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d) * t_0);
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = 1.0 - ((0.5 * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l));
	tmp = 0.0;
	if (h <= -2.8e+253)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - ((((((D_m * M) / (2.0 * d)) ^ 2.0) * 0.5) * h) / l));
	elseif (h <= -5e-310)
		tmp = ((-1.0 * d) * (1.0 / sqrt((l * h)))) * t_0;
	elseif (h <= 2.3e+196)
		tmp = (sqrt((1.0 / (l * h))) * d) * t_0;
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2.8e+253], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(N[Power[N[(N[(D$95$m * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(N[(-1.0 * d), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[h, 2.3e+196], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -2.8e253

   1. Initial program 42.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 46.2%

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

      1. lift-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 46.2

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

   6. Applied rewrites 46.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. pow1/2 N/A

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

      11. pow1/2 N/A

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

      12. *-commutative N/A

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

      13. sqrt-unprod N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 43.5

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

   8. Applied rewrites 43.5%

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

   if -2.8e253 < h < -4.999999999999985e-310

   1. Initial program 68.3%

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

   3. Taylor expanded in h around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 71.9

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

   5. Applied rewrites 71.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. inv-pow N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-*.f64 72.2

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 72.2%

      \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{1}{\color{blue}{\sqrt{\ell \cdot h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 72.2

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}\right) \cdot \left(1 - \left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   9. Applied rewrites 72.2%

      \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}\right) \cdot \left(1 - \left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   if -4.999999999999985e-310 < h < 2.2999999999999998e196

   1. Initial program 70.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. inv-pow N/A

         \[\leadsto \left(\sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 76.6

         \[\leadsto \left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 76.6

         \[\leadsto \left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 76.6%

      \[\leadsto \left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. unpow-1 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-*.f64 76.6

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   9. Applied rewrites 76.6%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   if 2.2999999999999998e196 < h

   1. Initial program 48.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 22.5

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   5. Applied rewrites 22.5%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      3. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      4. *-commutative N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. inv-pow N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      6. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      11. lift-*.f64 22.4

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   7. Applied rewrites 22.4%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lower-sqrt.f64 37.2

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   9. Applied rewrites 37.2%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 68.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 - \left(0.5 \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}\right) \cdot t\_0\\ \mathbf{elif}\;h \leq 2.3 \cdot 10^{+196}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (* 0.5 (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l)))))
   (if (<= h -5e-310)
     (* (* (* -1.0 d) (/ 1.0 (sqrt (* l h)))) t_0)
     (if (<= h 2.3e+196)
       (* (* (sqrt (/ 1.0 (* l h))) d) t_0)
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - ((0.5 * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l));
	double tmp;
	if (h <= -5e-310) {
		tmp = ((-1.0 * d) * (1.0 / sqrt((l * h)))) * t_0;
	} else if (h <= 2.3e+196) {
		tmp = (sqrt((1.0 / (l * h))) * d) * t_0;
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

D_m =     private
NOTE: d, h, l, 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, 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
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((0.5d0 * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l))
    if (h <= (-5d-310)) then
        tmp = (((-1.0d0) * d) * (1.0d0 / sqrt((l * h)))) * t_0
    else if (h <= 2.3d+196) then
        tmp = (sqrt((1.0d0 / (l * h))) * d) * t_0
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - ((0.5 * Math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l));
	double tmp;
	if (h <= -5e-310) {
		tmp = ((-1.0 * d) * (1.0 / Math.sqrt((l * h)))) * t_0;
	} else if (h <= 2.3e+196) {
		tmp = (Math.sqrt((1.0 / (l * h))) * d) * t_0;
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = 1.0 - ((0.5 * math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l))
	tmp = 0
	if h <= -5e-310:
		tmp = ((-1.0 * d) * (1.0 / math.sqrt((l * h)))) * t_0
	elif h <= 2.3e+196:
		tmp = (math.sqrt((1.0 / (l * h))) * d) * t_0
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(Float64(Float64(-1.0 * d) * Float64(1.0 / sqrt(Float64(l * h)))) * t_0);
	elseif (h <= 2.3e+196)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d) * t_0);
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = 1.0 - ((0.5 * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l));
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = ((-1.0 * d) * (1.0 / sqrt((l * h)))) * t_0;
	elseif (h <= 2.3e+196)
		tmp = (sqrt((1.0 / (l * h))) * d) * t_0;
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(N[(N[(-1.0 * d), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[h, 2.3e+196], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 66.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(\left(d \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(d \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{h \cdot \color{blue}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. inv-pow N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-*.f64 68.4

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 68.4%

      \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. inv-pow N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-*.f64 68.7

          \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 68.7%

      \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{1}{\color{blue}{\sqrt{\ell \cdot h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 68.7

         \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}\right) \cdot \left(1 - \left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   9. Applied rewrites 68.7%

      \[\leadsto \left(\left(-1 \cdot d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}\right) \cdot \left(1 - \left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   if -4.999999999999985e-310 < h < 2.2999999999999998e196

   1. Initial program 70.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. inv-pow N/A

         \[\leadsto \left(\sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 76.6

         \[\leadsto \left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{\frac{1}{2}} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. metadata-eval 76.6

         \[\leadsto \left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 76.6%

      \[\leadsto \left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\color{blue}{0.5} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. unpow-1 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-*.f64 76.6

         \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   9. Applied rewrites 76.6%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   if 2.2999999999999998e196 < h

   1. Initial program 48.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 22.5

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   5. Applied rewrites 22.5%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      3. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      4. *-commutative N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. inv-pow N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      6. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      11. lift-*.f64 22.4

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   7. Applied rewrites 22.4%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lower-sqrt.f64 37.2

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   9. Applied rewrites 37.2%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 47.3% accurate, 7.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2.2 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(-d\right) \cdot 1}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= h -2.2e-17)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= h -5e-310)
     (/ (* (- d) 1.0) (sqrt (* l h)))
     (* (/ 1.0 (* (sqrt l) (sqrt h))) d))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (h <= -2.2e-17) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (h <= -5e-310) {
		tmp = (-d * 1.0) / sqrt((l * h));
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

D_m =     private
NOTE: d, h, l, 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, 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
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (h <= (-2.2d-17)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (h <= (-5d-310)) then
        tmp = (-d * 1.0d0) / sqrt((l * h))
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (h <= -2.2e-17) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (h <= -5e-310) {
		tmp = (-d * 1.0) / Math.sqrt((l * h));
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if h <= -2.2e-17:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif h <= -5e-310:
		tmp = (-d * 1.0) / math.sqrt((l * h))
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (h <= -2.2e-17)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(Float64(-d) * 1.0) / sqrt(Float64(l * h)));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (h <= -2.2e-17)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (h <= -5e-310)
		tmp = (-d * 1.0) / sqrt((l * h));
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[h, -2.2e-17], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[((-d) * 1.0), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.2e-17

   1. Initial program 61.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2}} \cdot \ell} \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}{{d}^{\color{blue}{2}} \cdot \ell} \]

      5. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}{{d}^{\color{blue}{2}} \cdot \ell} \]

      6. pow-prod-down N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\frac{-1}{8} \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \]

      7. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\frac{-1}{8} \cdot \left({\left(M \cdot D\right)}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \]

      8. lower-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\frac{-1}{8} \cdot \left({\left(M \cdot D\right)}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \]

      9. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\frac{-1}{8} \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \]

      10. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\frac{-1}{8} \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\frac{-1}{8} \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{{d}^{2} \cdot \color{blue}{\ell}} \]

      12. unpow2 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\frac{-1}{8} \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \]

      13. lower-*.f64 22.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{-0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \]

   5. Applied rewrites 22.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\frac{-0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{\left(d \cdot d\right) \cdot \ell}} \]

   7. Applied rewrites 22.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\right)\right)} \]

   8. Taylor expanded in d around inf

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lift-/.f64 36.3

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 36.3%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if -2.2e-17 < h < -4.999999999999985e-310

   1. Initial program 70.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(d \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto \left(d \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{h \cdot \color{blue}{\ell}}} \]

      4. metadata-eval N/A

         \[\leadsto \left(d \cdot -1\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. inv-pow N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      10. *-commutative N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      11. lower-*.f64 52.8

          \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

   5. Applied rewrites 52.8%

      \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      8. inv-pow N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. sqrt-div N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      10. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      11. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\sqrt{\ell \cdot h}} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(d\right)\right) \cdot 1}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(d\right)\right) \cdot 1}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(d\right)\right) \cdot 1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      15. lower-neg.f64 N/A

          \[\leadsto \frac{\left(-d\right) \cdot 1}{\sqrt{\color{blue}{\ell} \cdot h}} \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left(-d\right) \cdot 1}{\sqrt{\ell \cdot h}} \]

      17. lift-*.f64 53.4

          \[\leadsto \frac{\left(-d\right) \cdot 1}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 53.4%

      \[\leadsto \frac{\left(-d\right) \cdot 1}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if -4.999999999999985e-310 < h

   1. Initial program 66.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 42.2

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   5. Applied rewrites 42.2%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      3. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      4. *-commutative N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. inv-pow N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      6. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      11. lift-*.f64 42.5

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   7. Applied rewrites 42.5%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lower-sqrt.f64 50.4

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   9. Applied rewrites 50.4%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 46.2% accurate, 8.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{-290}:\\ \;\;\;\;\frac{\left(-d\right) \cdot 1}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l 2.2e-290)
   (/ (* (- d) 1.0) (sqrt (* l h)))
   (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= 2.2e-290) {
		tmp = (-d * 1.0) / sqrt((l * h));
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

D_m =     private
NOTE: d, h, l, 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, 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
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 2.2d-290) then
        tmp = (-d * 1.0d0) / sqrt((l * h))
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= 2.2e-290) {
		tmp = (-d * 1.0) / Math.sqrt((l * h));
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if l <= 2.2e-290:
		tmp = (-d * 1.0) / math.sqrt((l * h))
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= 2.2e-290)
		tmp = Float64(Float64(Float64(-d) * 1.0) / sqrt(Float64(l * h)));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (l <= 2.2e-290)
		tmp = (-d * 1.0) / sqrt((l * h));
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, 2.2e-290], N[(N[((-d) * 1.0), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.2000000000000001e-290

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(d \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto \left(d \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{h \cdot \color{blue}{\ell}}} \]

      4. metadata-eval N/A

         \[\leadsto \left(d \cdot -1\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. inv-pow N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      10. *-commutative N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      11. lower-*.f64 41.7

          \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

   5. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      8. inv-pow N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. sqrt-div N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      10. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      11. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\sqrt{\ell \cdot h}} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(d\right)\right) \cdot 1}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(d\right)\right) \cdot 1}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(d\right)\right) \cdot 1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      15. lower-neg.f64 N/A

          \[\leadsto \frac{\left(-d\right) \cdot 1}{\sqrt{\color{blue}{\ell} \cdot h}} \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left(-d\right) \cdot 1}{\sqrt{\ell \cdot h}} \]

      17. lift-*.f64 41.9

          \[\leadsto \frac{\left(-d\right) \cdot 1}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 41.9%

      \[\leadsto \frac{\left(-d\right) \cdot 1}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 2.2000000000000001e-290 < l

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 42.8

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   5. Applied rewrites 42.8%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      3. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      4. *-commutative N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. inv-pow N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      6. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      11. lift-*.f64 43.0

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   7. Applied rewrites 43.0%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lower-sqrt.f64 50.8

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   9. Applied rewrites 50.8%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 42.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\ell \leq -9.2 \cdot 10^{-308}:\\ \;\;\;\;\frac{\left(-d\right) \cdot 1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{t\_0}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= l -9.2e-308) (/ (* (- d) 1.0) t_0) (/ (* 1.0 d) t_0))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (l <= -9.2e-308) {
		tmp = (-d * 1.0) / t_0;
	} else {
		tmp = (1.0 * d) / t_0;
	}
	return tmp;
}

Fortran

D_m =     private
NOTE: d, h, l, 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, 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
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (l <= (-9.2d-308)) then
        tmp = (-d * 1.0d0) / t_0
    else
        tmp = (1.0d0 * d) / t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (l <= -9.2e-308) {
		tmp = (-d * 1.0) / t_0;
	} else {
		tmp = (1.0 * d) / t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if l <= -9.2e-308:
		tmp = (-d * 1.0) / t_0
	else:
		tmp = (1.0 * d) / t_0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (l <= -9.2e-308)
		tmp = Float64(Float64(Float64(-d) * 1.0) / t_0);
	else
		tmp = Float64(Float64(1.0 * d) / t_0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (l <= -9.2e-308)
		tmp = (-d * 1.0) / t_0;
	else
		tmp = (1.0 * d) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -9.2e-308], N[(N[((-d) * 1.0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 * d), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -9.1999999999999996e-308

   1. Initial program 66.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      2. sqrt-pow2 N/A

         \[\leadsto \left(d \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. metadata-eval N/A

         \[\leadsto \left(d \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{h \cdot \color{blue}{\ell}}} \]

      4. metadata-eval N/A

         \[\leadsto \left(d \cdot -1\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. inv-pow N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      10. *-commutative N/A

          \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      11. lower-*.f64 42.2

          \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

   5. Applied rewrites 42.2%

      \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{{\left(h \cdot \ell\right)}^{-1}} \]

      8. inv-pow N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. sqrt-div N/A

         \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      10. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      11. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\sqrt{\ell \cdot h}} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(d\right)\right) \cdot 1}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(d\right)\right) \cdot 1}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(d\right)\right) \cdot 1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      15. lower-neg.f64 N/A

          \[\leadsto \frac{\left(-d\right) \cdot 1}{\sqrt{\color{blue}{\ell} \cdot h}} \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left(-d\right) \cdot 1}{\sqrt{\ell \cdot h}} \]

      17. lift-*.f64 42.5

          \[\leadsto \frac{\left(-d\right) \cdot 1}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 42.5%

      \[\leadsto \frac{\left(-d\right) \cdot 1}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if -9.1999999999999996e-308 < l

   1. Initial program 66.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 42.2

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   5. Applied rewrites 42.2%

      \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      3. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      4. *-commutative N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. inv-pow N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      6. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      11. lift-*.f64 42.5

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   7. Applied rewrites 42.5%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot \color{blue}{d} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      5. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{\ell \cdot h}} \]

      9. lift-*.f64 42.5

         \[\leadsto \frac{1 \cdot d}{\sqrt{\ell \cdot h}} \]

   9. Applied rewrites 42.5%

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 26.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m) :precision binary64 (* (sqrt (/ 1.0 (* l h))) d))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	return sqrt((1.0 / (l * h))) * d;
}

Fortran

D_m =     private
NOTE: d, h, l, 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, 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
    real(8), intent (in) :: d_m
    code = sqrt((1.0d0 / (l * h))) * d
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	return Math.sqrt((1.0 / (l * h))) * d;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	return math.sqrt((1.0 / (l * h))) * d

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	return Float64(sqrt(Float64(1.0 / Float64(l * h))) * d)
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp = code(d, h, l, M, D_m)
	tmp = sqrt((1.0 / (l * h))) * d;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 66.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   4. inv-pow N/A

      \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

   5. lower-pow.f64 N/A

      \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

   6. *-commutative N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   7. lower-*.f64 26.1

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

5. Applied rewrites 26.1%

   \[\leadsto \color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   2. lift-pow.f64 N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   3. unpow-1 N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   5. lift-*.f64 26.1

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

7. Applied rewrites 26.1%

   \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025091 
(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)))))