Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.6% → 88.8%

Time: 11.0s

Alternatives: 15

Speedup: 1.8×

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: 80.6% 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: 88.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 2 \cdot 10^{-171} \lor \neg \left(M\_m \cdot D\_m \leq 5 \cdot 10^{+180}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D\_m \cdot \frac{h \cdot M\_m}{4 \cdot d}\right) \cdot \frac{D\_m \cdot \frac{M\_m}{d}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(M\_m \cdot D\_m\right) \cdot \frac{h}{4 \cdot d}}{\ell \cdot d} \cdot \left(M\_m \cdot D\_m\right)}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (or (<= (* M_m D_m) 2e-171) (not (<= (* M_m D_m) 5e+180)))
   (*
    w0
    (sqrt (- 1.0 (* (* D_m (/ (* h M_m) (* 4.0 d))) (/ (* D_m (/ M_m d)) l)))))
   (*
    w0
    (sqrt
     (- 1.0 (* (/ (* (* M_m D_m) (/ h (* 4.0 d))) (* l d)) (* M_m D_m)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((M_m * D_m) <= 2e-171) || !((M_m * D_m) <= 5e+180)) {
		tmp = w0 * sqrt((1.0 - ((D_m * ((h * M_m) / (4.0 * d))) * ((D_m * (M_m / d)) / l))));
	} else {
		tmp = w0 * sqrt((1.0 - ((((M_m * D_m) * (h / (4.0 * d))) / (l * d)) * (M_m * D_m))));
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m_m, d_m, h, l, d)
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
    real(8) :: tmp
    if (((m_m * d_m) <= 2d-171) .or. (.not. ((m_m * d_m) <= 5d+180))) then
        tmp = w0 * sqrt((1.0d0 - ((d_m * ((h * m_m) / (4.0d0 * d))) * ((d_m * (m_m / d)) / l))))
    else
        tmp = w0 * sqrt((1.0d0 - ((((m_m * d_m) * (h / (4.0d0 * d))) / (l * d)) * (m_m * d_m))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((M_m * D_m) <= 2e-171) || !((M_m * D_m) <= 5e+180)) {
		tmp = w0 * Math.sqrt((1.0 - ((D_m * ((h * M_m) / (4.0 * d))) * ((D_m * (M_m / d)) / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((((M_m * D_m) * (h / (4.0 * d))) / (l * d)) * (M_m * D_m))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if ((M_m * D_m) <= 2e-171) or not ((M_m * D_m) <= 5e+180):
		tmp = w0 * math.sqrt((1.0 - ((D_m * ((h * M_m) / (4.0 * d))) * ((D_m * (M_m / d)) / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((((M_m * D_m) * (h / (4.0 * d))) / (l * d)) * (M_m * D_m))))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if ((Float64(M_m * D_m) <= 2e-171) || !(Float64(M_m * D_m) <= 5e+180))
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D_m * Float64(Float64(h * M_m) / Float64(4.0 * d))) * Float64(Float64(D_m * Float64(M_m / d)) / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M_m * D_m) * Float64(h / Float64(4.0 * d))) / Float64(l * d)) * Float64(M_m * D_m)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((M_m * D_m) <= 2e-171) || ~(((M_m * D_m) <= 5e+180)))
		tmp = w0 * sqrt((1.0 - ((D_m * ((h * M_m) / (4.0 * d))) * ((D_m * (M_m / d)) / l))));
	else
		tmp = w0 * sqrt((1.0 - ((((M_m * D_m) * (h / (4.0 * d))) / (l * d)) * (M_m * D_m))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[Or[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 2e-171], N[Not[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e+180]], $MachinePrecision]], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D$95$m * N[(N[(h * M$95$m), $MachinePrecision] / N[(4.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(h / N[(4.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2e-171 or 4.9999999999999996e180 < (*.f64 M D)

   1. Initial program 82.3%

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

   4. Applied rewrites 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 84.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 84.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 84.5

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

   6. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

   8. Applied rewrites 91.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 86.7

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

   10. Applied rewrites 86.7%

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

   if 2e-171 < (*.f64 M D) < 4.9999999999999996e180

   1. Initial program 83.7%

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

   4. Applied rewrites 87.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 91.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 91.4

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 91.4

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

   6. Applied rewrites 91.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

   8. Applied rewrites 86.5%

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

   10. Applied rewrites 92.7%

       \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(M \cdot D\right) \cdot \frac{h}{4 \cdot d}}{\ell \cdot d} \cdot \left(M \cdot D\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{-171} \lor \neg \left(M \cdot D \leq 5 \cdot 10^{+180}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D \cdot \frac{h \cdot M}{4 \cdot d}\right) \cdot \frac{D \cdot \frac{M}{d}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(M \cdot D\right) \cdot \frac{h}{4 \cdot d}}{\ell \cdot d} \cdot \left(M \cdot D\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(w0 \cdot \left(M\_m \cdot \left(\left(h \cdot M\_m\right) \cdot \left(\frac{D\_m}{d} \cdot \frac{\frac{D\_m}{\ell}}{d}\right)\right)\right), -0.125, w0\right)\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{+49}:\\ \;\;\;\;w0 \cdot \sqrt{\left(\left(\left(\frac{h}{d \cdot d} \cdot M\_m\right) \cdot \frac{M\_m}{\ell}\right) \cdot \left(-0.25 \cdot D\_m\right)\right) \cdot D\_m}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))))
   (if (<= t_0 -1e+304)
     (fma (* w0 (* M_m (* (* h M_m) (* (/ D_m d) (/ (/ D_m l) d))))) -0.125 w0)
     (if (<= t_0 -5e+49)
       (*
        w0
        (sqrt (* (* (* (* (/ h (* d d)) M_m) (/ M_m l)) (* -0.25 D_m)) D_m)))
       (* w0 1.0)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -1e+304) {
		tmp = fma((w0 * (M_m * ((h * M_m) * ((D_m / d) * ((D_m / l) / d))))), -0.125, w0);
	} else if (t_0 <= -5e+49) {
		tmp = w0 * sqrt((((((h / (d * d)) * M_m) * (M_m / l)) * (-0.25 * D_m)) * D_m));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= -1e+304)
		tmp = fma(Float64(w0 * Float64(M_m * Float64(Float64(h * M_m) * Float64(Float64(D_m / d) * Float64(Float64(D_m / l) / d))))), -0.125, w0);
	elseif (t_0 <= -5e+49)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(Float64(Float64(h / Float64(d * d)) * M_m) * Float64(M_m / l)) * Float64(-0.25 * D_m)) * D_m)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+304], N[(N[(w0 * N[(M$95$m * N[(N[(h * M$95$m), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(D$95$m / l), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + w0), $MachinePrecision], If[LessEqual[t$95$0, -5e+49], N[(w0 * N[Sqrt[N[(N[(N[(N[(N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -9.9999999999999994e303

   1. Initial program 67.1%

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

   4. Applied rewrites 66.7%

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

   5. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} + w0} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}} + w0 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}}, \frac{-1}{8}, w0\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot w0\right)} \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot w0\right)} \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\color{blue}{\left({M}^{2} \cdot h\right)} \cdot w0\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot w0\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot w0\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\color{blue}{{d}^{2} \cdot \ell}}, \frac{-1}{8}, w0\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      16. lower-*.f64 45.2

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}, -0.125, w0\right) \]

   7. Applied rewrites 45.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}, -0.125, w0\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 51.3%

      \[\leadsto \mathsf{fma}\left(w0 \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(\frac{D}{d \cdot d} \cdot \frac{D}{\ell}\right)\right), -0.125, w0\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 63.4%

       \[\leadsto \mathsf{fma}\left(w0 \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{\frac{D}{\ell}}{d}\right)\right)\right), -0.125, w0\right) \]

   if -9.9999999999999994e303 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000004e49

   1. Initial program 99.5%

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

   3. Taylor expanded in M around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 33.7

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

   5. Applied rewrites 33.7%

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

   7. Applied rewrites 41.8%

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

   if -5.0000000000000004e49 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 85.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 92.3%

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

Alternative 3: 81.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+280}:\\ \;\;\;\;\mathsf{fma}\left(w0 \cdot \left(M\_m \cdot \left(\left(h \cdot M\_m\right) \cdot \left(\frac{D\_m}{d} \cdot \frac{\frac{D\_m}{\ell}}{d}\right)\right)\right), -0.125, w0\right)\\ \mathbf{elif}\;t\_0 \leq -20:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{\left(\left(M\_m \cdot M\_m\right) \cdot D\_m\right) \cdot D\_m}{\left(d \cdot d\right) \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))))
   (if (<= t_0 -2e+280)
     (fma (* w0 (* M_m (* (* h M_m) (* (/ D_m d) (/ (/ D_m l) d))))) -0.125 w0)
     (if (<= t_0 -20.0)
       (*
        w0
        (sqrt
         (fma (* h -0.25) (/ (* (* (* M_m M_m) D_m) D_m) (* (* d d) l)) 1.0)))
       (* w0 1.0)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -2e+280) {
		tmp = fma((w0 * (M_m * ((h * M_m) * ((D_m / d) * ((D_m / l) / d))))), -0.125, w0);
	} else if (t_0 <= -20.0) {
		tmp = w0 * sqrt(fma((h * -0.25), ((((M_m * M_m) * D_m) * D_m) / ((d * d) * l)), 1.0));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= -2e+280)
		tmp = fma(Float64(w0 * Float64(M_m * Float64(Float64(h * M_m) * Float64(Float64(D_m / d) * Float64(Float64(D_m / l) / d))))), -0.125, w0);
	elseif (t_0 <= -20.0)
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(Float64(M_m * M_m) * D_m) * D_m) / Float64(Float64(d * d) * l)), 1.0)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+280], N[(N[(w0 * N[(M$95$m * N[(N[(h * M$95$m), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(D$95$m / l), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + w0), $MachinePrecision], If[LessEqual[t$95$0, -20.0], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.0000000000000001e280

   1. Initial program 68.3%

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

   4. Applied rewrites 64.3%

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

   5. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} + w0} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}} + w0 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}}, \frac{-1}{8}, w0\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot w0\right)} \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot w0\right)} \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\color{blue}{\left({M}^{2} \cdot h\right)} \cdot w0\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot w0\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot w0\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\color{blue}{{d}^{2} \cdot \ell}}, \frac{-1}{8}, w0\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      16. lower-*.f64 43.5

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}, -0.125, w0\right) \]

   7. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}, -0.125, w0\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.4%

      \[\leadsto \mathsf{fma}\left(w0 \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(\frac{D}{d \cdot d} \cdot \frac{D}{\ell}\right)\right), -0.125, w0\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 61.1%

       \[\leadsto \mathsf{fma}\left(w0 \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{\frac{D}{\ell}}{d}\right)\right)\right), -0.125, w0\right) \]

   if -2.0000000000000001e280 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -20

   1. Initial program 99.4%

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

   3. Taylor expanded in h around inf

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. rgt-mult-inverse N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 23.1%

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

   if -20 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 85.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 93.5%

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

Alternative 4: 74.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+280}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\left(D\_m \cdot M\_m\right) \cdot h\right) \cdot \left(\left(-D\_m\right) \cdot M\_m\right)}{\left(-2 \cdot d\right) \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq -20:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{\left(\left(M\_m \cdot M\_m\right) \cdot D\_m\right) \cdot D\_m}{\left(d \cdot d\right) \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))))
   (if (<= t_0 -2e+280)
     (*
      w0
      (sqrt
       (- 1.0 (/ (* (* (* D_m M_m) h) (* (- D_m) M_m)) (* (* -2.0 d) l)))))
     (if (<= t_0 -20.0)
       (*
        w0
        (sqrt
         (fma (* h -0.25) (/ (* (* (* M_m M_m) D_m) D_m) (* (* d d) l)) 1.0)))
       (* w0 1.0)))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -2e+280) {
		tmp = w0 * sqrt((1.0 - ((((D_m * M_m) * h) * (-D_m * M_m)) / ((-2.0 * d) * l))));
	} else if (t_0 <= -20.0) {
		tmp = w0 * sqrt(fma((h * -0.25), ((((M_m * M_m) * D_m) * D_m) / ((d * d) * l)), 1.0));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= -2e+280)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(D_m * M_m) * h) * Float64(Float64(-D_m) * M_m)) / Float64(Float64(-2.0 * d) * l)))));
	elseif (t_0 <= -20.0)
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(Float64(M_m * M_m) * D_m) * D_m) / Float64(Float64(d * d) * l)), 1.0)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+280], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * N[((-D$95$m) * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -20.0], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.0000000000000001e280

   1. Initial program 68.3%

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

   4. Applied rewrites 19.3%

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

   5. Taylor expanded in M around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 13.9

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

   7. Applied rewrites 13.9%

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

   9. Applied rewrites 19.4%

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

   if -2.0000000000000001e280 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -20

   1. Initial program 99.4%

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

   3. Taylor expanded in h around inf

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. rgt-mult-inverse N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 23.1%

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

   if -20 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 85.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 93.5%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+280}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\left(D \cdot M\right) \cdot h\right) \cdot \left(\left(-D\right) \cdot M\right)}{\left(-2 \cdot d\right) \cdot \ell}}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -20:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{\left(\left(M \cdot M\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := w0 \cdot \sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{+145}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\left(D\_m \cdot M\_m\right) \cdot h}{-2}}{d \cdot -2} \cdot \frac{D\_m \cdot \frac{M\_m}{d}}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0
         (*
          w0
          (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)))))))
   (if (<= t_0 4e+145)
     t_0
     (*
      w0
      (sqrt
       (-
        1.0
        (*
         (/ (/ (* (* D_m M_m) h) -2.0) (* d -2.0))
         (/ (* D_m (/ M_m d)) l))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = w0 * sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 4e+145) {
		tmp = t_0;
	} else {
		tmp = w0 * sqrt((1.0 - (((((D_m * M_m) * h) / -2.0) / (d * -2.0)) * ((D_m * (M_m / d)) / l))));
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m_m, d_m, h, l, d)
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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = w0 * sqrt((1.0d0 - ((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l))))
    if (t_0 <= 4d+145) then
        tmp = t_0
    else
        tmp = w0 * sqrt((1.0d0 - (((((d_m * m_m) * h) / (-2.0d0)) / (d * (-2.0d0))) * ((d_m * (m_m / d)) / l))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = w0 * Math.sqrt((1.0 - (Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 4e+145) {
		tmp = t_0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((((D_m * M_m) * h) / -2.0) / (d * -2.0)) * ((D_m * (M_m / d)) / l))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = w0 * math.sqrt((1.0 - (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))
	tmp = 0
	if t_0 <= 4e+145:
		tmp = t_0
	else:
		tmp = w0 * math.sqrt((1.0 - (((((D_m * M_m) * h) / -2.0) / (d * -2.0)) * ((D_m * (M_m / d)) / l))))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
	tmp = 0.0
	if (t_0 <= 4e+145)
		tmp = t_0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D_m * M_m) * h) / -2.0) / Float64(d * -2.0)) * Float64(Float64(D_m * Float64(M_m / d)) / l)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = w0 * sqrt((1.0 - ((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l))));
	tmp = 0.0;
	if (t_0 <= 4e+145)
		tmp = t_0;
	else
		tmp = w0 * sqrt((1.0 - (((((D_m * M_m) * h) / -2.0) / (d * -2.0)) * ((D_m * (M_m / d)) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e+145], t$95$0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / -2.0), $MachinePrecision] / N[(d * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))) < 4e145

   1. Initial program 94.0%

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

   if 4e145 < (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))))

   1. Initial program 50.4%

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

   4. Applied rewrites 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 85.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 85.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 85.3

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

   6. Applied rewrites 85.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

   8. Applied rewrites 82.3%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{\left(D \cdot M\right) \cdot h}{-2}}{d \cdot -2} \cdot \frac{D \cdot \frac{M}{d}}{\ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 86.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

D_m =     private
M_m =     private
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m_m, d_m, h, l, d)
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
    real(8) :: tmp
    if (sqrt((1.0d0 - ((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)))) <= 1.0d0) then
        tmp = w0 * 1.0d0
    else
        tmp = w0 * sqrt((1.0d0 - (((d_m * m_m) * (((d_m * m_m) * h) / (d * (-2.0d0)))) / (((-2.0d0) * d) * l))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 100.0%

      \[\leadsto w0 \cdot \color{blue}{1} \]

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

   1. Initial program 56.2%

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

   4. Applied rewrites 70.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 74.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.9

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

   6. Applied rewrites 74.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 74.9

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

   8. Applied rewrites 74.9%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(D \cdot M\right) \cdot \frac{\left(D \cdot M\right) \cdot h}{d \cdot -2}}}{\left(-2 \cdot d\right) \cdot \ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 80.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+49}:\\ \;\;\;\;w0 \cdot \sqrt{\left(-0.25 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \frac{\left(\frac{h}{d \cdot d} \cdot M\_m\right) \cdot M\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -5e+49)
   (* w0 (sqrt (* (* -0.25 (* D_m D_m)) (/ (* (* (/ h (* d d)) M_m) M_m) l))))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+49) {
		tmp = w0 * sqrt(((-0.25 * (D_m * D_m)) * ((((h / (d * d)) * M_m) * M_m) / l)));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m_m, d_m, h, l, d)
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
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-5d+49)) then
        tmp = w0 * sqrt((((-0.25d0) * (d_m * d_m)) * ((((h / (d * d)) * m_m) * m_m) / l)))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+49) {
		tmp = w0 * Math.sqrt(((-0.25 * (D_m * D_m)) * ((((h / (d * d)) * M_m) * M_m) / l)));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+49:
		tmp = w0 * math.sqrt(((-0.25 * (D_m * D_m)) * ((((h / (d * d)) * M_m) * M_m) / l)))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+49)
		tmp = Float64(w0 * sqrt(Float64(Float64(-0.25 * Float64(D_m * D_m)) * Float64(Float64(Float64(Float64(h / Float64(d * d)) * M_m) * M_m) / l))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e+49)
		tmp = w0 * sqrt(((-0.25 * (D_m * D_m)) * ((((h / (d * d)) * M_m) * M_m) / l)));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+49], N[(w0 * N[Sqrt[N[(N[(-0.25 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / l), $MachinePrecision]), $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)) < -5.0000000000000004e49

   1. Initial program 74.4%

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

   3. Taylor expanded in M around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 42.8

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 51.9%

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

   if -5.0000000000000004e49 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 85.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 92.3%

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

Alternative 8: 72.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(w0 * N[(-0.25 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $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)) < -inf.0

   1. Initial program 66.5%

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

   4. Applied rewrites 18.4%

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

   5. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 18.8

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

   7. Applied rewrites 18.8%

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

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

   1. Initial program 86.8%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 85.6%

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

Alternative 9: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+304}:\\ \;\;\;\;\frac{M\_m \cdot \left(\left(h \cdot M\_m\right) \cdot \left(\left(D\_m \cdot D\_m\right) \cdot w0\right)\right)}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -1e+304)
   (* (/ (* M_m (* (* h M_m) (* (* D_m D_m) w0))) (* (* d d) l)) -0.125)
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e+304) {
		tmp = ((M_m * ((h * M_m) * ((D_m * D_m) * w0))) / ((d * d) * l)) * -0.125;
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m_m, d_m, h, l, d)
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
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-1d+304)) then
        tmp = ((m_m * ((h * m_m) * ((d_m * d_m) * w0))) / ((d * d) * l)) * (-0.125d0)
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e+304) {
		tmp = ((M_m * ((h * M_m) * ((D_m * D_m) * w0))) / ((d * d) * l)) * -0.125;
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e+304:
		tmp = ((M_m * ((h * M_m) * ((D_m * D_m) * w0))) / ((d * d) * l)) * -0.125
	else:
		tmp = w0 * 1.0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+304)
		tmp = Float64(Float64(Float64(M_m * Float64(Float64(h * M_m) * Float64(Float64(D_m * D_m) * w0))) / Float64(Float64(d * d) * l)) * -0.125);
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -1e+304)
		tmp = ((M_m * ((h * M_m) * ((D_m * D_m) * w0))) / ((d * d) * l)) * -0.125;
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+304], N[(N[(N[(M$95$m * N[(N[(h * M$95$m), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $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)) < -9.9999999999999994e303

   1. Initial program 67.1%

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

   4. Applied rewrites 66.7%

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

   5. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} + w0} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}} + w0 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}}, \frac{-1}{8}, w0\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot w0\right)} \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot w0\right)} \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\color{blue}{\left({M}^{2} \cdot h\right)} \cdot w0\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot w0\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot w0\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\color{blue}{{d}^{2} \cdot \ell}}, \frac{-1}{8}, w0\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      16. lower-*.f64 45.2

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}, -0.125, w0\right) \]

   7. Applied rewrites 45.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}, -0.125, w0\right)} \]

   8. Taylor expanded in M around inf

      \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.0%

       \[\leadsto \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell} \cdot \color{blue}{-0.125} \]
   11. Step-by-step derivation

   12. Applied rewrites 51.3%

       \[\leadsto \frac{M \cdot \left(\left(h \cdot M\right) \cdot \left(\left(D \cdot D\right) \cdot w0\right)\right)}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125 \]

   if -9.9999999999999994e303 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 86.0%

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

Alternative 10: 87.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 2 \cdot 10^{-171}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 5 \cdot 10^{+278}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(M\_m \cdot D\_m\right) \cdot \frac{h}{4 \cdot d}}{\ell \cdot d} \cdot \left(M\_m \cdot D\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{-0.5 \cdot \left(\left(M\_m \cdot M\_m\right) \cdot h\right)}{d} \cdot \frac{D\_m}{\ell}, \frac{D\_m}{d}, 1\right)}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 2e-171)
   (* w0 1.0)
   (if (<= (* M_m D_m) 5e+278)
     (*
      w0
      (sqrt
       (- 1.0 (* (/ (* (* M_m D_m) (/ h (* 4.0 d))) (* l d)) (* M_m D_m)))))
     (*
      w0
      (sqrt
       (fma (* (/ (* -0.5 (* (* M_m M_m) h)) d) (/ D_m l)) (/ D_m d) 1.0))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 2e-171) {
		tmp = w0 * 1.0;
	} else if ((M_m * D_m) <= 5e+278) {
		tmp = w0 * sqrt((1.0 - ((((M_m * D_m) * (h / (4.0 * d))) / (l * d)) * (M_m * D_m))));
	} else {
		tmp = w0 * sqrt(fma((((-0.5 * ((M_m * M_m) * h)) / d) * (D_m / l)), (D_m / d), 1.0));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 2e-171)
		tmp = Float64(w0 * 1.0);
	elseif (Float64(M_m * D_m) <= 5e+278)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M_m * D_m) * Float64(h / Float64(4.0 * d))) / Float64(l * d)) * Float64(M_m * D_m)))));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(-0.5 * Float64(Float64(M_m * M_m) * h)) / d) * Float64(D_m / l)), Float64(D_m / d), 1.0)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 2e-171], N[(w0 * 1.0), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e+278], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(h / N[(4.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(N[(-0.5 * N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 2e-171

   1. Initial program 83.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 76.1%

      \[\leadsto w0 \cdot \color{blue}{1} \]

   if 2e-171 < (*.f64 M D) < 5.00000000000000029e278

   1. Initial program 82.1%

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

   4. Applied rewrites 83.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 88.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 88.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 88.2

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

   6. Applied rewrites 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

   8. Applied rewrites 84.8%

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

   10. Applied rewrites 90.8%

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

   if 5.00000000000000029e278 < (*.f64 M D)

   1. Initial program 79.3%

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

   4. Applied rewrites 79.3%

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

   5. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 65.0

          \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{-0.5 \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right)}{\color{blue}{\ell \cdot d}}, \frac{D}{d}, 1\right)} \]

   7. Applied rewrites 65.0%

      \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\frac{-0.5 \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right)}{\ell \cdot d}}, \frac{D}{d}, 1\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 79.3%

      \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(\frac{-0.5 \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d} \cdot \color{blue}{\frac{D}{\ell}}, \frac{D}{d}, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 87.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D\_m \cdot \frac{M\_m}{d}}{\ell}\\ \mathbf{if}\;d \leq 3 \cdot 10^{+83}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\left(D\_m \cdot M\_m\right) \cdot h}{-2}}{d \cdot -2} \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D\_m \cdot \frac{h \cdot M\_m}{4 \cdot d}\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (/ (* D_m (/ M_m d)) l)))
   (if (<= d 3e+83)
     (* w0 (sqrt (- 1.0 (* (/ (/ (* (* D_m M_m) h) -2.0) (* d -2.0)) t_0))))
     (* w0 (sqrt (- 1.0 (* (* D_m (/ (* h M_m) (* 4.0 d))) t_0)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (D_m * (M_m / d)) / l;
	double tmp;
	if (d <= 3e+83) {
		tmp = w0 * sqrt((1.0 - (((((D_m * M_m) * h) / -2.0) / (d * -2.0)) * t_0)));
	} else {
		tmp = w0 * sqrt((1.0 - ((D_m * ((h * M_m) / (4.0 * d))) * t_0)));
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m_m, d_m, h, l, d)
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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_m * (m_m / d)) / l
    if (d <= 3d+83) then
        tmp = w0 * sqrt((1.0d0 - (((((d_m * m_m) * h) / (-2.0d0)) / (d * (-2.0d0))) * t_0)))
    else
        tmp = w0 * sqrt((1.0d0 - ((d_m * ((h * m_m) / (4.0d0 * d))) * t_0)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (D_m * (M_m / d)) / l;
	double tmp;
	if (d <= 3e+83) {
		tmp = w0 * Math.sqrt((1.0 - (((((D_m * M_m) * h) / -2.0) / (d * -2.0)) * t_0)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((D_m * ((h * M_m) / (4.0 * d))) * t_0)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = (D_m * (M_m / d)) / l
	tmp = 0
	if d <= 3e+83:
		tmp = w0 * math.sqrt((1.0 - (((((D_m * M_m) * h) / -2.0) / (d * -2.0)) * t_0)))
	else:
		tmp = w0 * math.sqrt((1.0 - ((D_m * ((h * M_m) / (4.0 * d))) * t_0)))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(Float64(D_m * Float64(M_m / d)) / l)
	tmp = 0.0
	if (d <= 3e+83)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D_m * M_m) * h) / -2.0) / Float64(d * -2.0)) * t_0))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D_m * Float64(Float64(h * M_m) / Float64(4.0 * d))) * t_0))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = (D_m * (M_m / d)) / l;
	tmp = 0.0;
	if (d <= 3e+83)
		tmp = w0 * sqrt((1.0 - (((((D_m * M_m) * h) / -2.0) / (d * -2.0)) * t_0)));
	else
		tmp = w0 * sqrt((1.0 - ((D_m * ((h * M_m) / (4.0 * d))) * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[d, 3e+83], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / -2.0), $MachinePrecision] / N[(d * -2.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D$95$m * N[(N[(h * M$95$m), $MachinePrecision] / N[(4.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 3e83

   1. Initial program 83.2%

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

   4. Applied rewrites 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 87.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 87.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 87.9

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

   6. Applied rewrites 87.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

   8. Applied rewrites 90.8%

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

   if 3e83 < d

   1. Initial program 80.4%

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

   4. Applied rewrites 78.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 80.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 80.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 80.3

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

   6. Applied rewrites 80.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

   8. Applied rewrites 85.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 85.0

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

   10. Applied rewrites 85.0%

       \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(D \cdot \frac{h \cdot M}{4 \cdot d}\right)} \cdot \frac{D \cdot \frac{M}{d}}{\ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 87.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 2 \cdot 10^{-171}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 5 \cdot 10^{+278}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(M\_m \cdot D\_m\right) \cdot \frac{h}{4 \cdot d}}{\ell \cdot d} \cdot \left(M\_m \cdot D\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(w0 \cdot \left(M\_m \cdot \left(\left(h \cdot M\_m\right) \cdot \left(\frac{D\_m}{d} \cdot \frac{\frac{D\_m}{\ell}}{d}\right)\right)\right), -0.125, w0\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 2e-171)
   (* w0 1.0)
   (if (<= (* M_m D_m) 5e+278)
     (*
      w0
      (sqrt
       (- 1.0 (* (/ (* (* M_m D_m) (/ h (* 4.0 d))) (* l d)) (* M_m D_m)))))
     (fma
      (* w0 (* M_m (* (* h M_m) (* (/ D_m d) (/ (/ D_m l) d)))))
      -0.125
      w0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 2e-171) {
		tmp = w0 * 1.0;
	} else if ((M_m * D_m) <= 5e+278) {
		tmp = w0 * sqrt((1.0 - ((((M_m * D_m) * (h / (4.0 * d))) / (l * d)) * (M_m * D_m))));
	} else {
		tmp = fma((w0 * (M_m * ((h * M_m) * ((D_m / d) * ((D_m / l) / d))))), -0.125, w0);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 2e-171)
		tmp = Float64(w0 * 1.0);
	elseif (Float64(M_m * D_m) <= 5e+278)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M_m * D_m) * Float64(h / Float64(4.0 * d))) / Float64(l * d)) * Float64(M_m * D_m)))));
	else
		tmp = fma(Float64(w0 * Float64(M_m * Float64(Float64(h * M_m) * Float64(Float64(D_m / d) * Float64(Float64(D_m / l) / d))))), -0.125, w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 2e-171], N[(w0 * 1.0), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e+278], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(h / N[(4.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(w0 * N[(M$95$m * N[(N[(h * M$95$m), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(D$95$m / l), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + w0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 2e-171

   1. Initial program 83.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 76.1%

      \[\leadsto w0 \cdot \color{blue}{1} \]

   if 2e-171 < (*.f64 M D) < 5.00000000000000029e278

   1. Initial program 82.1%

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

   4. Applied rewrites 83.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 88.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 88.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 88.2

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

   6. Applied rewrites 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

   8. Applied rewrites 84.8%

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

   10. Applied rewrites 90.8%

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

   if 5.00000000000000029e278 < (*.f64 M D)

   1. Initial program 79.3%

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

   4. Applied rewrites 65.0%

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

   5. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} + w0} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}} + w0 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}}, \frac{-1}{8}, w0\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot w0\right)} \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot w0\right)} \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\color{blue}{\left({M}^{2} \cdot h\right)} \cdot w0\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot w0\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot w0\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\color{blue}{{d}^{2} \cdot \ell}}, \frac{-1}{8}, w0\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}, \frac{-1}{8}, w0\right) \]

      16. lower-*.f64 43.3

          \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}, -0.125, w0\right) \]

   7. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w0\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}, -0.125, w0\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 50.5%

      \[\leadsto \mathsf{fma}\left(w0 \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(\frac{D}{d \cdot d} \cdot \frac{D}{\ell}\right)\right), -0.125, w0\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 79.4%

       \[\leadsto \mathsf{fma}\left(w0 \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{\frac{D}{\ell}}{d}\right)\right)\right), -0.125, w0\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 73.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d} \leq 10^{+76}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\left(D\_m \cdot M\_m\right) \cdot h\right) \cdot \left(\left(-D\_m\right) \cdot M\_m\right)}{\left(-2 \cdot d\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (/ (* M_m D_m) (* 2.0 d)) 1e+76)
   (* w0 1.0)
   (*
    w0
    (sqrt
     (- 1.0 (/ (* (* (* D_m M_m) h) (* (- D_m) M_m)) (* (* -2.0 d) l)))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d)) <= 1e+76) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * sqrt((1.0 - ((((D_m * M_m) * h) * (-D_m * M_m)) / ((-2.0 * d) * l))));
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m_m, d_m, h, l, d)
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
    real(8) :: tmp
    if (((m_m * d_m) / (2.0d0 * d)) <= 1d+76) then
        tmp = w0 * 1.0d0
    else
        tmp = w0 * sqrt((1.0d0 - ((((d_m * m_m) * h) * (-d_m * m_m)) / (((-2.0d0) * d) * l))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d)) <= 1e+76) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((((D_m * M_m) * h) * (-D_m * M_m)) / ((-2.0 * d) * l))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if ((M_m * D_m) / (2.0 * d)) <= 1e+76:
		tmp = w0 * 1.0
	else:
		tmp = w0 * math.sqrt((1.0 - ((((D_m * M_m) * h) * (-D_m * M_m)) / ((-2.0 * d) * l))))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) <= 1e+76)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(D_m * M_m) * h) * Float64(Float64(-D_m) * M_m)) / Float64(Float64(-2.0 * d) * l)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((M_m * D_m) / (2.0 * d)) <= 1e+76)
		tmp = w0 * 1.0;
	else
		tmp = w0 * sqrt((1.0 - ((((D_m * M_m) * h) * (-D_m * M_m)) / ((-2.0 * d) * l))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 1e+76], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * N[((-D$95$m) * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1e76

   1. Initial program 87.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 79.1%

      \[\leadsto w0 \cdot \color{blue}{1} \]

   if 1e76 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 52.4%

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

   4. Applied rewrites 19.7%

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

   5. Taylor expanded in M around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 6.5

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

   7. Applied rewrites 6.5%

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

   9. Applied rewrites 19.7%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\left(D \cdot M\right) \cdot h\right) \cdot \color{blue}{\left(-D \cdot M\right)}}{\left(-2 \cdot d\right) \cdot \ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{+76}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\left(D \cdot M\right) \cdot h\right) \cdot \left(\left(-D\right) \cdot M\right)}{\left(-2 \cdot d\right) \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 87.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 2 \cdot 10^{-183}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D\_m \cdot \frac{h \cdot M\_m}{4 \cdot d}\right) \cdot \frac{D\_m \cdot \frac{M\_m}{d}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M\_m}{d} \cdot D\_m}{\ell} \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \frac{h}{4 \cdot d}\right)}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 2e-183)
   (*
    w0
    (sqrt (- 1.0 (* (* D_m (/ (* h M_m) (* 4.0 d))) (/ (* D_m (/ M_m d)) l)))))
   (*
    w0
    (sqrt
     (- 1.0 (* (/ (* (/ M_m d) D_m) l) (* (* M_m D_m) (/ h (* 4.0 d)))))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 2e-183) {
		tmp = w0 * sqrt((1.0 - ((D_m * ((h * M_m) / (4.0 * d))) * ((D_m * (M_m / d)) / l))));
	} else {
		tmp = w0 * sqrt((1.0 - ((((M_m / d) * D_m) / l) * ((M_m * D_m) * (h / (4.0 * d))))));
	}
	return tmp;
}

Fortran

D_m =     private
M_m =     private
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m_m, d_m, h, l, d)
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
    real(8) :: tmp
    if ((m_m * d_m) <= 2d-183) then
        tmp = w0 * sqrt((1.0d0 - ((d_m * ((h * m_m) / (4.0d0 * d))) * ((d_m * (m_m / d)) / l))))
    else
        tmp = w0 * sqrt((1.0d0 - ((((m_m / d) * d_m) / l) * ((m_m * d_m) * (h / (4.0d0 * d))))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 2e-183) {
		tmp = w0 * Math.sqrt((1.0 - ((D_m * ((h * M_m) / (4.0 * d))) * ((D_m * (M_m / d)) / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((((M_m / d) * D_m) / l) * ((M_m * D_m) * (h / (4.0 * d))))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (M_m * D_m) <= 2e-183:
		tmp = w0 * math.sqrt((1.0 - ((D_m * ((h * M_m) / (4.0 * d))) * ((D_m * (M_m / d)) / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((((M_m / d) * D_m) / l) * ((M_m * D_m) * (h / (4.0 * d))))))
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 2e-183)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D_m * Float64(Float64(h * M_m) / Float64(4.0 * d))) * Float64(Float64(D_m * Float64(M_m / d)) / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M_m / d) * D_m) / l) * Float64(Float64(M_m * D_m) * Float64(h / Float64(4.0 * d)))))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((M_m * D_m) <= 2e-183)
		tmp = w0 * sqrt((1.0 - ((D_m * ((h * M_m) / (4.0 * d))) * ((D_m * (M_m / d)) / l))));
	else
		tmp = w0 * sqrt((1.0 - ((((M_m / d) * D_m) / l) * ((M_m * D_m) * (h / (4.0 * d))))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 2e-183], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D$95$m * N[(N[(h * M$95$m), $MachinePrecision] / N[(4.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(h / N[(4.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.00000000000000001e-183

   1. Initial program 83.3%

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

   4. Applied rewrites 86.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 88.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 88.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 88.2

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

   6. Applied rewrites 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

   8. Applied rewrites 93.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 86.4

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

   10. Applied rewrites 86.4%

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

   if 2.00000000000000001e-183 < (*.f64 M D)

   1. Initial program 81.9%

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

   4. Applied rewrites 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 83.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 83.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.9

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

   6. Applied rewrites 83.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

   8. Applied rewrites 84.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 84.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 84.1

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 84.1

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

   10. Applied rewrites 84.1%

       \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M}{d} \cdot D}{\ell} \cdot \left(\left(M \cdot D\right) \cdot \frac{h}{4 \cdot d}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 67.5% accurate, 26.2× speedup

Math

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

FPCore

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

C

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

Fortran

D_m =     private
M_m =     private
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0, m_m, d_m, h, l, d)
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
    code = w0 * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 69.5%

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

Reproduce

herbie shell --seed 2024347 
(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))))))