Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 90.6%

Time: 4.7s

Alternatives: 6

Speedup: 0.7×

Specification

Math

\[\begin{array}{l} \\ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (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(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (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(w0, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{d\_m + d\_m}\\ t_1 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2}\\ t_2 := \frac{D\_m}{d\_m + d\_m} \cdot M\_m\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-19}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t\_2 \cdot \left(t\_2 \cdot h\right)}{\ell}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+266}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\left(-\sqrt{-0.25 \cdot \frac{h}{\ell}} \cdot D\_m\right) \cdot M\_m}{d\_m}\right) \cdot w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ (* M_m D_m) (+ d_m d_m)))
        (t_1 (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0))
        (t_2 (* (/ D_m (+ d_m d_m)) M_m)))
   (if (<= t_1 4e-19)
     (* w0 (sqrt (- 1.0 (/ (* t_2 (* t_2 h)) l))))
     (if (<= t_1 2e+266)
       (* w0 (sqrt (- 1.0 (* (* (/ h l) t_0) t_0))))
       (* (- (/ (* (- (* (sqrt (* -0.25 (/ h l))) D_m)) M_m) d_m)) w0)))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * D_m) / (d_m + d_m);
	double t_1 = pow(((M_m * D_m) / (2.0 * d_m)), 2.0);
	double t_2 = (D_m / (d_m + d_m)) * M_m;
	double tmp;
	if (t_1 <= 4e-19) {
		tmp = w0 * sqrt((1.0 - ((t_2 * (t_2 * h)) / l)));
	} else if (t_1 <= 2e+266) {
		tmp = w0 * sqrt((1.0 - (((h / l) * t_0) * t_0)));
	} else {
		tmp = -((-(sqrt((-0.25 * (h / l))) * D_m) * M_m) / d_m) * w0;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
d_m =     private
NOTE: w0, M_m, D_m, h, l, 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(w0, m_m, d_m, h, l, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (m_m * d_m) / (d_m_1 + d_m_1)
    t_1 = ((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0
    t_2 = (d_m / (d_m_1 + d_m_1)) * m_m
    if (t_1 <= 4d-19) then
        tmp = w0 * sqrt((1.0d0 - ((t_2 * (t_2 * h)) / l)))
    else if (t_1 <= 2d+266) then
        tmp = w0 * sqrt((1.0d0 - (((h / l) * t_0) * t_0)))
    else
        tmp = -((-(sqrt(((-0.25d0) * (h / l))) * d_m) * m_m) / d_m_1) * w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * D_m) / (d_m + d_m);
	double t_1 = Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0);
	double t_2 = (D_m / (d_m + d_m)) * M_m;
	double tmp;
	if (t_1 <= 4e-19) {
		tmp = w0 * Math.sqrt((1.0 - ((t_2 * (t_2 * h)) / l)));
	} else if (t_1 <= 2e+266) {
		tmp = w0 * Math.sqrt((1.0 - (((h / l) * t_0) * t_0)));
	} else {
		tmp = -((-(Math.sqrt((-0.25 * (h / l))) * D_m) * M_m) / d_m) * w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = (M_m * D_m) / (d_m + d_m)
	t_1 = math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)
	t_2 = (D_m / (d_m + d_m)) * M_m
	tmp = 0
	if t_1 <= 4e-19:
		tmp = w0 * math.sqrt((1.0 - ((t_2 * (t_2 * h)) / l)))
	elif t_1 <= 2e+266:
		tmp = w0 * math.sqrt((1.0 - (((h / l) * t_0) * t_0)))
	else:
		tmp = -((-(math.sqrt((-0.25 * (h / l))) * D_m) * M_m) / d_m) * w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(M_m * D_m) / Float64(d_m + d_m))
	t_1 = Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0
	t_2 = Float64(Float64(D_m / Float64(d_m + d_m)) * M_m)
	tmp = 0.0
	if (t_1 <= 4e-19)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_2 * Float64(t_2 * h)) / l))));
	elseif (t_1 <= 2e+266)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h / l) * t_0) * t_0))));
	else
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(sqrt(Float64(-0.25 * Float64(h / l))) * D_m)) * M_m) / d_m)) * w0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = (M_m * D_m) / (d_m + d_m);
	t_1 = ((M_m * D_m) / (2.0 * d_m)) ^ 2.0;
	t_2 = (D_m / (d_m + d_m)) * M_m;
	tmp = 0.0;
	if (t_1 <= 4e-19)
		tmp = w0 * sqrt((1.0 - ((t_2 * (t_2 * h)) / l)));
	elseif (t_1 <= 2e+266)
		tmp = w0 * sqrt((1.0 - (((h / l) * t_0) * t_0)));
	else
		tmp = -((-(sqrt((-0.25 * (h / l))) * D_m) * M_m) / d_m) * w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(D$95$m / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-19], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(t$95$2 * N[(t$95$2 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+266], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[(N[((-N[(N[Sqrt[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * D$95$m), $MachinePrecision]) * M$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]) * w0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 3.9999999999999999e-19

   1. Initial program 81.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{h}{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}} \]

   3. Applied rewrites 71.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d + d\right) \cdot \left(d + d\right)} \cdot h}{\ell}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d + d\right) \cdot \left(d + d\right)}} \cdot h}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{\left(d + d\right) \cdot \left(d + d\right)} \cdot h}{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\color{blue}{\left(d + d\right) \cdot \left(d + d\right)}} \cdot h}{\ell}} \]

      4. times-frac N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\frac{D \cdot M}{d + d} \cdot \frac{D \cdot M}{d + d}\right)} \cdot h}{\ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{\color{blue}{D \cdot M}}{d + d} \cdot \frac{D \cdot M}{d + d}\right) \cdot h}{\ell}} \]

      6. associate-*l/ N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot \frac{D \cdot M}{d + d}\right) \cdot h}{\ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\color{blue}{\frac{D}{d + d}} \cdot M\right) \cdot \frac{D \cdot M}{d + d}\right) \cdot h}{\ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot \frac{D \cdot M}{d + d}\right) \cdot h}{\ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \frac{\color{blue}{D \cdot M}}{d + d}\right) \cdot h}{\ell}} \]

      10. associate-*l/ N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \cdot h}{\ell}} \]

      11. lift-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\color{blue}{\frac{D}{d + d}} \cdot M\right)\right) \cdot h}{\ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \cdot h}{\ell}} \]

      13. lower-*.f64 85.7

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right)} \cdot h}{\ell}} \]

      14. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}} \]

      15. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(M \cdot \frac{D}{d + d}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}} \]

      16. lower-*.f64 85.7

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(M \cdot \frac{D}{d + d}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}} \]

      17. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \cdot h}{\ell}} \]

      18. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \cdot h}{\ell}} \]

      19. lower-*.f64 85.7

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \cdot h}{\ell}} \]

   5. Applied rewrites 85.7%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right)} \cdot h}{\ell}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot h}}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right)} \cdot h}{\ell}} \]

      3. associate-*l* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(M \cdot \frac{D}{d + d}\right) \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot h\right)}}{\ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(M \cdot \frac{D}{d + d}\right) \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot h\right)}}{\ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(M \cdot \frac{D}{d + d}\right)} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot h\right)}{\ell}} \]

      6. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot h\right)}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot h\right)}{\ell}} \]

      8. lower-*.f64 87.4

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot h\right)}}{\ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\color{blue}{\left(M \cdot \frac{D}{d + d}\right)} \cdot h\right)}{\ell}} \]

      10. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot h\right)}{\ell}} \]

      11. lower-*.f64 87.4

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot h\right)}{\ell}} \]

   7. Applied rewrites 87.4%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot h\right)}}{\ell}} \]

   if 3.9999999999999999e-19 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 2.0000000000000001e266

   1. Initial program 81.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]

      2. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}} \]

      3. lift-pow.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}} \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}} \]

      7. lower-*.f64 83.4

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{2 \cdot d}} \]

      8. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \frac{M \cdot D}{2 \cdot d}} \]

      9. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}} \]

      10. associate-/l* N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}\right) \cdot \frac{M \cdot D}{2 \cdot d}} \]

      11. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}\right) \cdot \frac{M \cdot D}{2 \cdot d}} \]

      12. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}\right) \cdot \frac{M \cdot D}{2 \cdot d}} \]

      13. lower-/.f64 81.1

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}} \]

      14. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}} \]

      15. count-2-rev N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\frac{D}{\color{blue}{d + d}} \cdot M\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}} \]

      16. lower-+.f64 81.1

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\frac{D}{\color{blue}{d + d}} \cdot M\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}} \]

      17. lift-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}} \]

      18. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}} \]

      19. associate-/l* N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}} \]

      20. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}} \]

      21. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}} \]

   3. Applied rewrites 82.4%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \left(\frac{D}{d + d} \cdot M\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\color{blue}{\frac{D}{d + d}} \cdot M\right)\right) \cdot \left(\frac{D}{d + d} \cdot M\right)} \]

      3. associate-*l/ N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \color{blue}{\frac{D \cdot M}{d + d}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{\color{blue}{D \cdot M}}{d + d}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)} \]

      5. lower-/.f64 81.1

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \color{blue}{\frac{D \cdot M}{d + d}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{\color{blue}{D \cdot M}}{d + d}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)} \]

      7. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{\color{blue}{M \cdot D}}{d + d}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)} \]

      8. lower-*.f64 81.1

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{\color{blue}{M \cdot D}}{d + d}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)} \]

   5. Applied rewrites 81.1%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \color{blue}{\frac{M \cdot D}{d + d}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d + d}\right) \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d + d}\right) \cdot \left(\color{blue}{\frac{D}{d + d}} \cdot M\right)} \]

      3. associate-*l/ N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d + d}\right) \cdot \color{blue}{\frac{D \cdot M}{d + d}}} \]

      4. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d + d}\right) \cdot \frac{\color{blue}{D \cdot M}}{d + d}} \]

      5. lower-/.f64 83.4

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d + d}\right) \cdot \color{blue}{\frac{D \cdot M}{d + d}}} \]

      6. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d + d}\right) \cdot \frac{\color{blue}{D \cdot M}}{d + d}} \]

      7. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d + d}\right) \cdot \frac{\color{blue}{M \cdot D}}{d + d}} \]

      8. lower-*.f64 83.4

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d + d}\right) \cdot \frac{\color{blue}{M \cdot D}}{d + d}} \]

   7. Applied rewrites 83.4%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d + d}\right) \cdot \color{blue}{\frac{M \cdot D}{d + d}}} \]

   if 2.0000000000000001e266 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 81.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Taylor expanded in M around inf

      \[\leadsto w0 \cdot \color{blue}{\left(M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right) \]

      3. lower-neg.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      9. lower-pow.f64 17.0

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

   4. Applied rewrites 17.0%

      \[\leadsto w0 \cdot \color{blue}{\left(M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)} \]

   5. Taylor expanded in d around -inf

      \[\leadsto w0 \cdot \left(-1 \cdot \color{blue}{\frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{\color{blue}{d}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}\right) \]

      5. lower-neg.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      9. lower-pow.f64 1.3

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

   7. Applied rewrites 1.3%

      \[\leadsto w0 \cdot \left(-1 \cdot \color{blue}{\frac{M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}}\right) \]

   8. Taylor expanded in D around -inf

      \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-\frac{1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-\frac{1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]

      6. lower-/.f64 26.7

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]

   10. Applied rewrites 26.7%

       \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-\frac{1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d}\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-\frac{1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \cdot w0} \]

       3. lower-*.f64 26.7

          \[\leadsto \color{blue}{\left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \cdot w0} \]

   12. Applied rewrites 26.7%

       \[\leadsto \color{blue}{\left(-\frac{\left(-\sqrt{-0.25 \cdot \frac{h}{\ell}} \cdot D\right) \cdot M}{d}\right) \cdot w0} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 90.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{d\_m + d\_m}\\ \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \leq 2 \cdot 10^{+266}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t\_0 \cdot \left(t\_0 \cdot h\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\left(-\sqrt{-0.25 \cdot \frac{h}{\ell}} \cdot D\_m\right) \cdot M\_m}{d\_m}\right) \cdot w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ (* M_m D_m) (+ d_m d_m))))
   (if (<= (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) 2e+266)
     (* w0 (sqrt (- 1.0 (/ (* t_0 (* t_0 h)) l))))
     (* (- (/ (* (- (* (sqrt (* -0.25 (/ h l))) D_m)) M_m) d_m)) w0))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * D_m) / (d_m + d_m);
	double tmp;
	if (pow(((M_m * D_m) / (2.0 * d_m)), 2.0) <= 2e+266) {
		tmp = w0 * sqrt((1.0 - ((t_0 * (t_0 * h)) / l)));
	} else {
		tmp = -((-(sqrt((-0.25 * (h / l))) * D_m) * M_m) / d_m) * w0;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
d_m =     private
NOTE: w0, M_m, D_m, h, l, 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(w0, m_m, d_m, h, l, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m_m * d_m) / (d_m_1 + d_m_1)
    if ((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) <= 2d+266) then
        tmp = w0 * sqrt((1.0d0 - ((t_0 * (t_0 * h)) / l)))
    else
        tmp = -((-(sqrt(((-0.25d0) * (h / l))) * d_m) * m_m) / d_m_1) * w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * D_m) / (d_m + d_m);
	double tmp;
	if (Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) <= 2e+266) {
		tmp = w0 * Math.sqrt((1.0 - ((t_0 * (t_0 * h)) / l)));
	} else {
		tmp = -((-(Math.sqrt((-0.25 * (h / l))) * D_m) * M_m) / d_m) * w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = (M_m * D_m) / (d_m + d_m)
	tmp = 0
	if math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) <= 2e+266:
		tmp = w0 * math.sqrt((1.0 - ((t_0 * (t_0 * h)) / l)))
	else:
		tmp = -((-(math.sqrt((-0.25 * (h / l))) * D_m) * M_m) / d_m) * w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(M_m * D_m) / Float64(d_m + d_m))
	tmp = 0.0
	if ((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) <= 2e+266)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * h)) / l))));
	else
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(sqrt(Float64(-0.25 * Float64(h / l))) * D_m)) * M_m) / d_m)) * w0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = (M_m * D_m) / (d_m + d_m);
	tmp = 0.0;
	if ((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) <= 2e+266)
		tmp = w0 * sqrt((1.0 - ((t_0 * (t_0 * h)) / l)));
	else
		tmp = -((-(sqrt((-0.25 * (h / l))) * D_m) * M_m) / d_m) * w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 2e+266], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[(N[((-N[(N[Sqrt[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * D$95$m), $MachinePrecision]) * M$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]) * w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 2.0000000000000001e266

   1. Initial program 81.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{h}{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}} \]

   3. Applied rewrites 71.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d + d\right) \cdot \left(d + d\right)} \cdot h}{\ell}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d + d\right) \cdot \left(d + d\right)}} \cdot h}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{\left(d + d\right) \cdot \left(d + d\right)} \cdot h}{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\color{blue}{\left(d + d\right) \cdot \left(d + d\right)}} \cdot h}{\ell}} \]

      4. times-frac N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\frac{D \cdot M}{d + d} \cdot \frac{D \cdot M}{d + d}\right)} \cdot h}{\ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{\color{blue}{D \cdot M}}{d + d} \cdot \frac{D \cdot M}{d + d}\right) \cdot h}{\ell}} \]

      6. associate-*l/ N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot \frac{D \cdot M}{d + d}\right) \cdot h}{\ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\color{blue}{\frac{D}{d + d}} \cdot M\right) \cdot \frac{D \cdot M}{d + d}\right) \cdot h}{\ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot \frac{D \cdot M}{d + d}\right) \cdot h}{\ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \frac{\color{blue}{D \cdot M}}{d + d}\right) \cdot h}{\ell}} \]

      10. associate-*l/ N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \cdot h}{\ell}} \]

      11. lift-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\color{blue}{\frac{D}{d + d}} \cdot M\right)\right) \cdot h}{\ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \cdot h}{\ell}} \]

      13. lower-*.f64 85.7

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right)} \cdot h}{\ell}} \]

      14. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}} \]

      15. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(M \cdot \frac{D}{d + d}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}} \]

      16. lower-*.f64 85.7

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(M \cdot \frac{D}{d + d}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}} \]

      17. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \cdot h}{\ell}} \]

      18. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \cdot h}{\ell}} \]

      19. lower-*.f64 85.7

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \cdot h}{\ell}} \]

   5. Applied rewrites 85.7%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right)} \cdot h}{\ell}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot h}}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right)} \cdot h}{\ell}} \]

      3. associate-*l* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(M \cdot \frac{D}{d + d}\right) \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot h\right)}}{\ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(M \cdot \frac{D}{d + d}\right) \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot h\right)}}{\ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(M \cdot \frac{D}{d + d}\right)} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot h\right)}{\ell}} \]

      6. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot h\right)}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot h\right)}{\ell}} \]

      8. lower-*.f64 87.4

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot h\right)}}{\ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\color{blue}{\left(M \cdot \frac{D}{d + d}\right)} \cdot h\right)}{\ell}} \]

      10. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot h\right)}{\ell}} \]

      11. lower-*.f64 87.4

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot h\right)}{\ell}} \]

   7. Applied rewrites 87.4%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot h\right)}}{\ell}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot h\right)}{\ell}} \]

      2. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(M \cdot \frac{D}{d + d}\right)} \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot h\right)}{\ell}} \]

      3. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(M \cdot \color{blue}{\frac{D}{d + d}}\right) \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot h\right)}{\ell}} \]

      4. associate-/l* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{M \cdot D}{d + d}} \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot h\right)}{\ell}} \]

      5. lower-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{M \cdot D}{d + d}} \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot h\right)}{\ell}} \]

      6. lower-*.f64 86.0

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{M \cdot D}}{d + d} \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot h\right)}{\ell}} \]

   9. Applied rewrites 86.0%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{M \cdot D}{d + d}} \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot h\right)}{\ell}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{d + d} \cdot \left(\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot h\right)}{\ell}} \]

       2. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{d + d} \cdot \left(\color{blue}{\left(M \cdot \frac{D}{d + d}\right)} \cdot h\right)}{\ell}} \]

       3. lift-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{d + d} \cdot \left(\left(M \cdot \color{blue}{\frac{D}{d + d}}\right) \cdot h\right)}{\ell}} \]

       4. associate-/l* N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{d + d} \cdot \left(\color{blue}{\frac{M \cdot D}{d + d}} \cdot h\right)}{\ell}} \]

       5. lower-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{d + d} \cdot \left(\color{blue}{\frac{M \cdot D}{d + d}} \cdot h\right)}{\ell}} \]

       6. lower-*.f64 88.1

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{d + d} \cdot \left(\frac{\color{blue}{M \cdot D}}{d + d} \cdot h\right)}{\ell}} \]

   11. Applied rewrites 88.1%

       \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{d + d} \cdot \left(\color{blue}{\frac{M \cdot D}{d + d}} \cdot h\right)}{\ell}} \]

   if 2.0000000000000001e266 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 81.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Taylor expanded in M around inf

      \[\leadsto w0 \cdot \color{blue}{\left(M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right) \]

      3. lower-neg.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      9. lower-pow.f64 17.0

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

   4. Applied rewrites 17.0%

      \[\leadsto w0 \cdot \color{blue}{\left(M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)} \]

   5. Taylor expanded in d around -inf

      \[\leadsto w0 \cdot \left(-1 \cdot \color{blue}{\frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{\color{blue}{d}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}\right) \]

      5. lower-neg.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      9. lower-pow.f64 1.3

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

   7. Applied rewrites 1.3%

      \[\leadsto w0 \cdot \left(-1 \cdot \color{blue}{\frac{M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}}\right) \]

   8. Taylor expanded in D around -inf

      \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-\frac{1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-\frac{1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]

      6. lower-/.f64 26.7

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]

   10. Applied rewrites 26.7%

       \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-\frac{1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d}\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-\frac{1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \cdot w0} \]

       3. lower-*.f64 26.7

          \[\leadsto \color{blue}{\left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \cdot w0} \]

   12. Applied rewrites 26.7%

       \[\leadsto \color{blue}{\left(-\frac{\left(-\sqrt{-0.25 \cdot \frac{h}{\ell}} \cdot D\right) \cdot M}{d}\right) \cdot w0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{d\_m + d\_m}\\ t_1 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2}\\ t_2 := \frac{M\_m \cdot D\_m}{d\_m + d\_m}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{\frac{\ell - \left(\left(\left(M\_m \cdot t\_0\right) \cdot t\_0\right) \cdot M\_m\right) \cdot h}{\ell}} \cdot w0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+266}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot t\_2\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\left(-\sqrt{-0.25 \cdot \frac{h}{\ell}} \cdot D\_m\right) \cdot M\_m}{d\_m}\right) \cdot w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ D_m (+ d_m d_m)))
        (t_1 (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0))
        (t_2 (/ (* M_m D_m) (+ d_m d_m))))
   (if (<= t_1 4e-19)
     (* (sqrt (/ (- l (* (* (* (* M_m t_0) t_0) M_m) h)) l)) w0)
     (if (<= t_1 2e+266)
       (* w0 (sqrt (- 1.0 (* (* (/ h l) t_2) t_2))))
       (* (- (/ (* (- (* (sqrt (* -0.25 (/ h l))) D_m)) M_m) d_m)) w0)))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = D_m / (d_m + d_m);
	double t_1 = pow(((M_m * D_m) / (2.0 * d_m)), 2.0);
	double t_2 = (M_m * D_m) / (d_m + d_m);
	double tmp;
	if (t_1 <= 4e-19) {
		tmp = sqrt(((l - ((((M_m * t_0) * t_0) * M_m) * h)) / l)) * w0;
	} else if (t_1 <= 2e+266) {
		tmp = w0 * sqrt((1.0 - (((h / l) * t_2) * t_2)));
	} else {
		tmp = -((-(sqrt((-0.25 * (h / l))) * D_m) * M_m) / d_m) * w0;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
d_m =     private
NOTE: w0, M_m, D_m, h, l, 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(w0, m_m, d_m, h, l, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = d_m / (d_m_1 + d_m_1)
    t_1 = ((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0
    t_2 = (m_m * d_m) / (d_m_1 + d_m_1)
    if (t_1 <= 4d-19) then
        tmp = sqrt(((l - ((((m_m * t_0) * t_0) * m_m) * h)) / l)) * w0
    else if (t_1 <= 2d+266) then
        tmp = w0 * sqrt((1.0d0 - (((h / l) * t_2) * t_2)))
    else
        tmp = -((-(sqrt(((-0.25d0) * (h / l))) * d_m) * m_m) / d_m_1) * w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = D_m / (d_m + d_m);
	double t_1 = Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0);
	double t_2 = (M_m * D_m) / (d_m + d_m);
	double tmp;
	if (t_1 <= 4e-19) {
		tmp = Math.sqrt(((l - ((((M_m * t_0) * t_0) * M_m) * h)) / l)) * w0;
	} else if (t_1 <= 2e+266) {
		tmp = w0 * Math.sqrt((1.0 - (((h / l) * t_2) * t_2)));
	} else {
		tmp = -((-(Math.sqrt((-0.25 * (h / l))) * D_m) * M_m) / d_m) * w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = D_m / (d_m + d_m)
	t_1 = math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)
	t_2 = (M_m * D_m) / (d_m + d_m)
	tmp = 0
	if t_1 <= 4e-19:
		tmp = math.sqrt(((l - ((((M_m * t_0) * t_0) * M_m) * h)) / l)) * w0
	elif t_1 <= 2e+266:
		tmp = w0 * math.sqrt((1.0 - (((h / l) * t_2) * t_2)))
	else:
		tmp = -((-(math.sqrt((-0.25 * (h / l))) * D_m) * M_m) / d_m) * w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(D_m / Float64(d_m + d_m))
	t_1 = Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0
	t_2 = Float64(Float64(M_m * D_m) / Float64(d_m + d_m))
	tmp = 0.0
	if (t_1 <= 4e-19)
		tmp = Float64(sqrt(Float64(Float64(l - Float64(Float64(Float64(Float64(M_m * t_0) * t_0) * M_m) * h)) / l)) * w0);
	elseif (t_1 <= 2e+266)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h / l) * t_2) * t_2))));
	else
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(sqrt(Float64(-0.25 * Float64(h / l))) * D_m)) * M_m) / d_m)) * w0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = D_m / (d_m + d_m);
	t_1 = ((M_m * D_m) / (2.0 * d_m)) ^ 2.0;
	t_2 = (M_m * D_m) / (d_m + d_m);
	tmp = 0.0;
	if (t_1 <= 4e-19)
		tmp = sqrt(((l - ((((M_m * t_0) * t_0) * M_m) * h)) / l)) * w0;
	elseif (t_1 <= 2e+266)
		tmp = w0 * sqrt((1.0 - (((h / l) * t_2) * t_2)));
	else
		tmp = -((-(sqrt((-0.25 * (h / l))) * D_m) * M_m) / d_m) * w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(D$95$m / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-19], N[(N[Sqrt[N[(N[(l - N[(N[(N[(N[(M$95$m * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] * M$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], If[LessEqual[t$95$1, 2e+266], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h / l), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[(N[((-N[(N[Sqrt[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * D$95$m), $MachinePrecision]) * M$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]) * w0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 3.9999999999999999e-19

   1. Initial program 81.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{h}{\ell}}} \]

      3. associate-*r/ N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}} \]

   3. Applied rewrites 71.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d + d\right) \cdot \left(d + d\right)} \cdot h}{\ell}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d + d\right) \cdot \left(d + d\right)}} \cdot h}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{\left(d + d\right) \cdot \left(d + d\right)} \cdot h}{\ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\color{blue}{\left(d + d\right) \cdot \left(d + d\right)}} \cdot h}{\ell}} \]

      4. times-frac N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\frac{D \cdot M}{d + d} \cdot \frac{D \cdot M}{d + d}\right)} \cdot h}{\ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{\color{blue}{D \cdot M}}{d + d} \cdot \frac{D \cdot M}{d + d}\right) \cdot h}{\ell}} \]

      6. associate-*l/ N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot \frac{D \cdot M}{d + d}\right) \cdot h}{\ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\color{blue}{\frac{D}{d + d}} \cdot M\right) \cdot \frac{D \cdot M}{d + d}\right) \cdot h}{\ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot \frac{D \cdot M}{d + d}\right) \cdot h}{\ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \frac{\color{blue}{D \cdot M}}{d + d}\right) \cdot h}{\ell}} \]

      10. associate-*l/ N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \cdot h}{\ell}} \]

      11. lift-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\color{blue}{\frac{D}{d + d}} \cdot M\right)\right) \cdot h}{\ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \cdot h}{\ell}} \]

      13. lower-*.f64 85.7

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\frac{D}{d + d} \cdot M\right)\right)} \cdot h}{\ell}} \]

      14. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}} \]

      15. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(M \cdot \frac{D}{d + d}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}} \]

      16. lower-*.f64 85.7

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\color{blue}{\left(M \cdot \frac{D}{d + d}\right)} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot h}{\ell}} \]

      17. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \cdot h}{\ell}} \]

      18. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \cdot h}{\ell}} \]

      19. lower-*.f64 85.7

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \cdot h}{\ell}} \]

   5. Applied rewrites 85.7%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right)} \cdot h}{\ell}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot h}}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right)} \cdot h}{\ell}} \]

      3. associate-*l* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(M \cdot \frac{D}{d + d}\right) \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot h\right)}}{\ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(M \cdot \frac{D}{d + d}\right) \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot h\right)}}{\ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(M \cdot \frac{D}{d + d}\right)} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot h\right)}{\ell}} \]

      6. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot h\right)}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot h\right)}{\ell}} \]

      8. lower-*.f64 87.4

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot h\right)}}{\ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\color{blue}{\left(M \cdot \frac{D}{d + d}\right)} \cdot h\right)}{\ell}} \]

      10. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot h\right)}{\ell}} \]

      11. lower-*.f64 87.4

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\color{blue}{\left(\frac{D}{d + d} \cdot M\right)} \cdot h\right)}{\ell}} \]

   7. Applied rewrites 87.4%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot h\right)}}{\ell}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - \frac{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot h\right)}{\ell}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{1 - \frac{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot h\right)}{\ell}} \cdot w0} \]

      3. lower-*.f64 87.4

         \[\leadsto \color{blue}{\sqrt{1 - \frac{\left(\frac{D}{d + d} \cdot M\right) \cdot \left(\left(\frac{D}{d + d} \cdot M\right) \cdot h\right)}{\ell}} \cdot w0} \]

   9. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\sqrt{\frac{\ell - \left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \frac{D}{d + d}\right) \cdot M\right) \cdot h}{\ell}} \cdot w0} \]

   if 3.9999999999999999e-19 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 2.0000000000000001e266

   1. Initial program 81.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]

      2. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}} \]

      3. lift-pow.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}} \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}} \]

      7. lower-*.f64 83.4

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{2 \cdot d}} \]

      8. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \frac{M \cdot D}{2 \cdot d}} \]

      9. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}} \]

      10. associate-/l* N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}\right) \cdot \frac{M \cdot D}{2 \cdot d}} \]

      11. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}\right) \cdot \frac{M \cdot D}{2 \cdot d}} \]

      12. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}\right) \cdot \frac{M \cdot D}{2 \cdot d}} \]

      13. lower-/.f64 81.1

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\color{blue}{\frac{D}{2 \cdot d}} \cdot M\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}} \]

      14. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot M\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}} \]

      15. count-2-rev N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\frac{D}{\color{blue}{d + d}} \cdot M\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}} \]

      16. lower-+.f64 81.1

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\frac{D}{\color{blue}{d + d}} \cdot M\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}} \]

      17. lift-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}} \]

      18. lift-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}} \]

      19. associate-/l* N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}} \]

      20. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}} \]

      21. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \color{blue}{\left(\frac{D}{2 \cdot d} \cdot M\right)}} \]

   3. Applied rewrites 82.4%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \left(\frac{D}{d + d} \cdot M\right)\right) \cdot \left(\frac{D}{d + d} \cdot M\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\color{blue}{\frac{D}{d + d}} \cdot M\right)\right) \cdot \left(\frac{D}{d + d} \cdot M\right)} \]

      3. associate-*l/ N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \color{blue}{\frac{D \cdot M}{d + d}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{\color{blue}{D \cdot M}}{d + d}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)} \]

      5. lower-/.f64 81.1

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \color{blue}{\frac{D \cdot M}{d + d}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{\color{blue}{D \cdot M}}{d + d}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)} \]

      7. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{\color{blue}{M \cdot D}}{d + d}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)} \]

      8. lower-*.f64 81.1

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{\color{blue}{M \cdot D}}{d + d}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)} \]

   5. Applied rewrites 81.1%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \color{blue}{\frac{M \cdot D}{d + d}}\right) \cdot \left(\frac{D}{d + d} \cdot M\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d + d}\right) \cdot \color{blue}{\left(\frac{D}{d + d} \cdot M\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d + d}\right) \cdot \left(\color{blue}{\frac{D}{d + d}} \cdot M\right)} \]

      3. associate-*l/ N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d + d}\right) \cdot \color{blue}{\frac{D \cdot M}{d + d}}} \]

      4. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d + d}\right) \cdot \frac{\color{blue}{D \cdot M}}{d + d}} \]

      5. lower-/.f64 83.4

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d + d}\right) \cdot \color{blue}{\frac{D \cdot M}{d + d}}} \]

      6. lift-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d + d}\right) \cdot \frac{\color{blue}{D \cdot M}}{d + d}} \]

      7. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d + d}\right) \cdot \frac{\color{blue}{M \cdot D}}{d + d}} \]

      8. lower-*.f64 83.4

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d + d}\right) \cdot \frac{\color{blue}{M \cdot D}}{d + d}} \]

   7. Applied rewrites 83.4%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d + d}\right) \cdot \color{blue}{\frac{M \cdot D}{d + d}}} \]

   if 2.0000000000000001e266 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 81.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Taylor expanded in M around inf

      \[\leadsto w0 \cdot \color{blue}{\left(M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right) \]

      3. lower-neg.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      9. lower-pow.f64 17.0

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

   4. Applied rewrites 17.0%

      \[\leadsto w0 \cdot \color{blue}{\left(M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)} \]

   5. Taylor expanded in d around -inf

      \[\leadsto w0 \cdot \left(-1 \cdot \color{blue}{\frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{\color{blue}{d}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}\right) \]

      5. lower-neg.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      9. lower-pow.f64 1.3

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

   7. Applied rewrites 1.3%

      \[\leadsto w0 \cdot \left(-1 \cdot \color{blue}{\frac{M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}}\right) \]

   8. Taylor expanded in D around -inf

      \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-\frac{1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-\frac{1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]

      6. lower-/.f64 26.7

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]

   10. Applied rewrites 26.7%

       \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-\frac{1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d}\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-\frac{1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \cdot w0} \]

       3. lower-*.f64 26.7

          \[\leadsto \color{blue}{\left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \cdot w0} \]

   12. Applied rewrites 26.7%

       \[\leadsto \color{blue}{\left(-\frac{\left(-\sqrt{-0.25 \cdot \frac{h}{\ell}} \cdot D\right) \cdot M}{d}\right) \cdot w0} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 89.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+52}:\\ \;\;\;\;w0 \cdot \left(-1 \cdot \frac{-1 \cdot \left(D\_m \cdot \left(M\_m \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right)}{d\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -2e+52)
   (* w0 (* -1.0 (/ (* -1.0 (* D_m (* M_m (sqrt (- (* 0.25 (/ h l))))))) d_m)))
   (* w0 1.0)))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+52) {
		tmp = w0 * (-1.0 * ((-1.0 * (D_m * (M_m * sqrt(-(0.25 * (h / l)))))) / d_m));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
d_m =     private
NOTE: w0, M_m, D_m, h, l, 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(w0, m_m, d_m, h, l, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)) <= (-2d+52)) then
        tmp = w0 * ((-1.0d0) * (((-1.0d0) * (d_m * (m_m * sqrt(-(0.25d0 * (h / l)))))) / d_m_1))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+52) {
		tmp = w0 * (-1.0 * ((-1.0 * (D_m * (M_m * Math.sqrt(-(0.25 * (h / l)))))) / d_m));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+52:
		tmp = w0 * (-1.0 * ((-1.0 * (D_m * (M_m * math.sqrt(-(0.25 * (h / l)))))) / d_m))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e+52)
		tmp = Float64(w0 * Float64(-1.0 * Float64(Float64(-1.0 * Float64(D_m * Float64(M_m * sqrt(Float64(-Float64(0.25 * Float64(h / l))))))) / d_m)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -2e+52)
		tmp = w0 * (-1.0 * ((-1.0 * (D_m * (M_m * sqrt(-(0.25 * (h / l)))))) / d_m));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+52], N[(w0 * N[(-1.0 * N[(N[(-1.0 * N[(D$95$m * N[(M$95$m * N[Sqrt[(-N[(0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2e52

   1. Initial program 81.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Taylor expanded in M around inf

      \[\leadsto w0 \cdot \color{blue}{\left(M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right) \]

      3. lower-neg.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      9. lower-pow.f64 17.0

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

   4. Applied rewrites 17.0%

      \[\leadsto w0 \cdot \color{blue}{\left(M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)} \]

   5. Taylor expanded in d around -inf

      \[\leadsto w0 \cdot \left(-1 \cdot \color{blue}{\frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{\color{blue}{d}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}\right) \]

      5. lower-neg.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      9. lower-pow.f64 1.3

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

   7. Applied rewrites 1.3%

      \[\leadsto w0 \cdot \left(-1 \cdot \color{blue}{\frac{M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}}\right) \]

   8. Taylor expanded in D around -inf

      \[\leadsto w0 \cdot \left(-1 \cdot \frac{-1 \cdot \left(D \cdot \left(M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{-1 \cdot \left(D \cdot \left(M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{-1 \cdot \left(D \cdot \left(M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{-1 \cdot \left(D \cdot \left(M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{-1 \cdot \left(D \cdot \left(M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]

      5. lower-neg.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{-1 \cdot \left(D \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{-1 \cdot \left(D \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]

      7. lower-/.f64 27.1

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{-1 \cdot \left(D \cdot \left(M \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]

   10. Applied rewrites 27.1%

       \[\leadsto w0 \cdot \left(-1 \cdot \frac{-1 \cdot \left(D \cdot \left(M \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]

   if -2e52 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 81.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 67.2%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+52}:\\ \;\;\;\;\left(-\frac{\left(-\sqrt{-0.25 \cdot \frac{h}{\ell}} \cdot D\_m\right) \cdot M\_m}{d\_m}\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -2e+52)
   (* (- (/ (* (- (* (sqrt (* -0.25 (/ h l))) D_m)) M_m) d_m)) w0)
   (* w0 1.0)))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+52) {
		tmp = -((-(sqrt((-0.25 * (h / l))) * D_m) * M_m) / d_m) * w0;
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
d_m =     private
NOTE: w0, M_m, D_m, h, l, 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(w0, m_m, d_m, h, l, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)) <= (-2d+52)) then
        tmp = -((-(sqrt(((-0.25d0) * (h / l))) * d_m) * m_m) / d_m_1) * w0
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+52) {
		tmp = -((-(Math.sqrt((-0.25 * (h / l))) * D_m) * M_m) / d_m) * w0;
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+52:
		tmp = -((-(math.sqrt((-0.25 * (h / l))) * D_m) * M_m) / d_m) * w0
	else:
		tmp = w0 * 1.0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e+52)
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(sqrt(Float64(-0.25 * Float64(h / l))) * D_m)) * M_m) / d_m)) * w0);
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -2e+52)
		tmp = -((-(sqrt((-0.25 * (h / l))) * D_m) * M_m) / d_m) * w0;
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+52], N[((-N[(N[((-N[(N[Sqrt[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * D$95$m), $MachinePrecision]) * M$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]) * w0), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2e52

   1. Initial program 81.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Taylor expanded in M around inf

      \[\leadsto w0 \cdot \color{blue}{\left(M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right) \]

      3. lower-neg.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

      9. lower-pow.f64 17.0

         \[\leadsto w0 \cdot \left(M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right) \]

   4. Applied rewrites 17.0%

      \[\leadsto w0 \cdot \color{blue}{\left(M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)} \]

   5. Taylor expanded in d around -inf

      \[\leadsto w0 \cdot \left(-1 \cdot \color{blue}{\frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{\color{blue}{d}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}\right)}}{d}\right) \]

      5. lower-neg.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-\frac{1}{4} \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

      9. lower-pow.f64 1.3

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}\right) \]

   7. Applied rewrites 1.3%

      \[\leadsto w0 \cdot \left(-1 \cdot \color{blue}{\frac{M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{\ell}}}{d}}\right) \]

   8. Taylor expanded in D around -inf

      \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{\mathsf{neg}\left(\frac{1}{4} \cdot \frac{h}{\ell}\right)}\right)\right)}{d}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-\frac{1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-\frac{1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]

      6. lower-/.f64 26.7

         \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]

   10. Applied rewrites 26.7%

       \[\leadsto w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{w0 \cdot \left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-\frac{1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d}\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-\frac{1}{4} \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \cdot w0} \]

       3. lower-*.f64 26.7

          \[\leadsto \color{blue}{\left(-1 \cdot \frac{M \cdot \left(-1 \cdot \left(D \cdot \sqrt{-0.25 \cdot \frac{h}{\ell}}\right)\right)}{d}\right) \cdot w0} \]

   12. Applied rewrites 26.7%

       \[\leadsto \color{blue}{\left(-\frac{\left(-\sqrt{-0.25 \cdot \frac{h}{\ell}} \cdot D\right) \cdot M}{d}\right) \cdot w0} \]

   if -2e52 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 81.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 67.2%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 67.2% accurate, 10.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ w0 \cdot 1 \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m) :precision binary64 (* w0 1.0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0 * 1.0;
}

Fortran

M_m =     private
D_m =     private
d_m =     private
NOTE: w0, M_m, D_m, h, l, 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(w0, m_m, d_m, h, l, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0 * 1.0d0
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0 * 1.0;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	return w0 * 1.0

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	return Float64(w0 * 1.0)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp = code(w0, M_m, D_m, h, l, d_m)
	tmp = w0 * 1.0;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0 * 1.0), $MachinePrecision]

Derivation

1. Initial program 81.4%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

2. Taylor expanded in M around 0

   \[\leadsto w0 \cdot \color{blue}{1} \]
3. Step-by-step derivation

4. Applied rewrites 67.2%

   \[\leadsto w0 \cdot \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025148 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))