Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 89.7%

Time: 10.5s

Alternatives: 9

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.1% 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: 89.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ w0 \cdot \sqrt{1 + \frac{h \cdot \left(\frac{M\_m}{d} \cdot D\_m\right)}{\ell \cdot -4} \cdot \left(D\_m \cdot \frac{M\_m}{d}\right)} \end{array} \]

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
  (sqrt (+ 1.0 (* (/ (* h (* (/ M_m d) D_m)) (* l -4.0)) (* D_m (/ M_m 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) {
	return w0 * sqrt((1.0 + (((h * ((M_m / d) * D_m)) / (l * -4.0)) * (D_m * (M_m / d)))));
}

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 * sqrt((1.0d0 + (((h * ((m_m / d) * d_m)) / (l * (-4.0d0))) * (d_m * (m_m / d)))))
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 * Math.sqrt((1.0 + (((h * ((M_m / d) * D_m)) / (l * -4.0)) * (D_m * (M_m / d)))));
}

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 * math.sqrt((1.0 + (((h * ((M_m / d) * D_m)) / (l * -4.0)) * (D_m * (M_m / d)))))

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 * sqrt(Float64(1.0 + Float64(Float64(Float64(h * Float64(Float64(M_m / d) * D_m)) / Float64(l * -4.0)) * Float64(D_m * Float64(M_m / d))))))
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 * sqrt((1.0 + (((h * ((M_m / d) * D_m)) / (l * -4.0)) * (D_m * (M_m / d)))));
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 * N[Sqrt[N[(1.0 + N[(N[(N[(h * N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * -4.0), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. lift-/.f64 N/A

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

   7. frac-2neg N/A

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

   8. associate-*r/ N/A

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

   9. lift-*.f64 N/A

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

   10. count-2-rev N/A

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

   11. flip-+ N/A

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

   12. distribute-neg-frac N/A

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

   13. +-inverses N/A

       \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\mathsf{neg}\left(M \cdot D\right)\right)}{\frac{\mathsf{neg}\left(\color{blue}{0}\right)}{d - d}}} \]

   14. metadata-eval N/A

       \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\mathsf{neg}\left(M \cdot D\right)\right)}{\frac{\color{blue}{0}}{d - d}}} \]

   15. +-inverses N/A

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

   16. flip-+ N/A

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

   17. count-2-rev N/A

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

   18. lift-*.f64 N/A

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

   19. lower-/.f64 N/A

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

4. Applied rewrites 80.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. frac-times N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 85.5

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

6. Applied rewrites 85.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f64 N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

8. Applied rewrites 83.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. associate-*l* N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

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

10. Applied rewrites 88.3%

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

11. Final simplification 88.3%

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

Alternative 2: 83.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 1000:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M\_m \cdot D\_m\right) \cdot \left(\left(M\_m \cdot D\_m\right) \cdot h\right)}{d \cdot \left(\left(-4 \cdot d\right) \cdot \ell\right)}}\\ \end{array} \end{array} \]

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 (<= (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)))) 1000.0)
   (* w0 1.0)
   (*
    w0
    (sqrt
     (+ 1.0 (/ (* (* M_m D_m) (* (* M_m D_m) h)) (* d (* (* -4.0 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)))) <= 1000.0) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * sqrt((1.0 + (((M_m * D_m) * ((M_m * D_m) * h)) / (d * ((-4.0 * 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)))) <= 1000.0d0) then
        tmp = w0 * 1.0d0
    else
        tmp = w0 * sqrt((1.0d0 + (((m_m * d_m) * ((m_m * d_m) * h)) / (d * (((-4.0d0) * 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)))) <= 1000.0) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * Math.sqrt((1.0 + (((M_m * D_m) * ((M_m * D_m) * h)) / (d * ((-4.0 * 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)))) <= 1000.0:
		tmp = w0 * 1.0
	else:
		tmp = w0 * math.sqrt((1.0 + (((M_m * D_m) * ((M_m * D_m) * h)) / (d * ((-4.0 * 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)))) <= 1000.0)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(M_m * D_m) * Float64(Float64(M_m * D_m) * h)) / Float64(d * Float64(Float64(-4.0 * 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)))) <= 1000.0)
		tmp = w0 * 1.0;
	else
		tmp = w0 * sqrt((1.0 + (((M_m * D_m) * ((M_m * D_m) * h)) / (d * ((-4.0 * 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], 1000.0], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(d * N[(N[(-4.0 * d), $MachinePrecision] * l), $MachinePrecision]), $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)))) < 1e3

   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 98.9%

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

   if 1e3 < (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 48.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. flip-+ N/A

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

      12. distribute-neg-frac N/A

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

      13. +-inverses N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\mathsf{neg}\left(M \cdot D\right)\right)}{\frac{\mathsf{neg}\left(\color{blue}{0}\right)}{d - d}}} \]

      14. metadata-eval N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\mathsf{neg}\left(M \cdot D\right)\right)}{\frac{\color{blue}{0}}{d - d}}} \]

      15. +-inverses N/A

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

      16. flip-+ N/A

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

      17. count-2-rev N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 52.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 67.4

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

   6. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 54.1

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   10. Applied rewrites 54.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 1000:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)}{d \cdot \left(\left(-4 \cdot d\right) \cdot \ell\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{-9}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot -0.125, \frac{M\_m}{d} \cdot \frac{h \cdot M\_m}{\ell \cdot d}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

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)) -5e-9)
   (* w0 (fma (* (* D_m D_m) -0.125) (* (/ M_m d) (/ (* h M_m) (* l d))) 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)) <= -5e-9) {
		tmp = w0 * fma(((D_m * D_m) * -0.125), ((M_m / d) * ((h * M_m) / (l * d))), 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)) <= -5e-9)
		tmp = Float64(w0 * fma(Float64(Float64(D_m * D_m) * -0.125), Float64(Float64(M_m / d) * Float64(Float64(h * M_m) / Float64(l * d))), 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], -5e-9], N[(w0 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(h * M$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $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)) < -5.0000000000000001e-9

   1. Initial program 61.0%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 29.4%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, \frac{h}{d \cdot d} \cdot \frac{M \cdot M}{\ell}, 1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 38.1%

      \[\leadsto w0 \cdot \mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, \frac{M}{d} \cdot \color{blue}{\frac{h \cdot M}{\ell \cdot d}}, 1\right) \]

   if -5.0000000000000001e-9 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.8%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.0%

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

Alternative 4: 79.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+195}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot -0.125, M\_m \cdot \frac{h \cdot M\_m}{\left(d \cdot d\right) \cdot \ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

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
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -5e+195)
   (* w0 (fma (* (* D_m D_m) -0.125) (* M_m (/ (* h M_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 tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -5e+195) {
		tmp = w0 * fma(((D_m * D_m) * -0.125), (M_m * ((h * M_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)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+195)
		tmp = Float64(w0 * fma(Float64(Float64(D_m * D_m) * -0.125), Float64(M_m * Float64(Float64(h * M_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_] := 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+195], N[(w0 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(M$95$m * N[(N[(h * M$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $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)) < -4.9999999999999998e195

   1. Initial program 52.3%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 34.1%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, \frac{h}{d \cdot d} \cdot \frac{M \cdot M}{\ell}, 1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.4%

      \[\leadsto w0 \cdot \mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, M \cdot \color{blue}{\frac{h \cdot M}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \]

   if -4.9999999999999998e195 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.4%

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

Alternative 5: 79.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+195}:\\ \;\;\;\;\frac{M\_m \cdot \left(\left(h \cdot M\_m\right) \cdot \left(\left(D\_m \cdot D\_m\right) \cdot w0\right)\right)}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

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)) -5e+195)
   (* (/ (* 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)) <= -5e+195) {
		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)) <= (-5d+195)) 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)) <= -5e+195) {
		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)) <= -5e+195:
		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)) <= -5e+195)
		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)) <= -5e+195)
		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], -5e+195], 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)) < -4.9999999999999998e195

   1. Initial program 52.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. flip-+ N/A

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

      12. distribute-neg-frac N/A

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

      13. +-inverses N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\mathsf{neg}\left(M \cdot D\right)\right)}{\frac{\mathsf{neg}\left(\color{blue}{0}\right)}{d - d}}} \]

      14. metadata-eval N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\mathsf{neg}\left(M \cdot D\right)\right)}{\frac{\color{blue}{0}}{d - d}}} \]

      15. +-inverses N/A

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

      16. flip-+ N/A

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

      17. count-2-rev N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 56.8%

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

   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 35.8

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

   7. Applied rewrites 35.8%

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

   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 35.8%

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

   12. Applied rewrites 39.2%

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

   if -4.9999999999999998e195 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.4%

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

Alternative 6: 86.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 1.25 \cdot 10^{-80}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M\_m}{d} \cdot \left(D\_m \cdot \left(\frac{M\_m \cdot h}{\ell} \cdot \frac{D\_m}{-4 \cdot d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + D\_m \cdot \frac{\left(\frac{M\_m}{d} \cdot h\right) \cdot \left(M\_m \cdot D\_m\right)}{\left(-4 \cdot d\right) \cdot \ell}}\\ \end{array} \end{array} \]

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 (<= D_m 1.25e-80)
   (*
    w0
    (sqrt
     (+ 1.0 (* (/ M_m d) (* D_m (* (/ (* M_m h) l) (/ D_m (* -4.0 d))))))))
   (*
    w0
    (sqrt
     (+ 1.0 (* D_m (/ (* (* (/ M_m d) h) (* M_m D_m)) (* (* -4.0 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 (D_m <= 1.25e-80) {
		tmp = w0 * sqrt((1.0 + ((M_m / d) * (D_m * (((M_m * h) / l) * (D_m / (-4.0 * d)))))));
	} else {
		tmp = w0 * sqrt((1.0 + (D_m * ((((M_m / d) * h) * (M_m * D_m)) / ((-4.0 * 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 (d_m <= 1.25d-80) then
        tmp = w0 * sqrt((1.0d0 + ((m_m / d) * (d_m * (((m_m * h) / l) * (d_m / ((-4.0d0) * d)))))))
    else
        tmp = w0 * sqrt((1.0d0 + (d_m * ((((m_m / d) * h) * (m_m * d_m)) / (((-4.0d0) * 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 (D_m <= 1.25e-80) {
		tmp = w0 * Math.sqrt((1.0 + ((M_m / d) * (D_m * (((M_m * h) / l) * (D_m / (-4.0 * d)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (D_m * ((((M_m / d) * h) * (M_m * D_m)) / ((-4.0 * 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 D_m <= 1.25e-80:
		tmp = w0 * math.sqrt((1.0 + ((M_m / d) * (D_m * (((M_m * h) / l) * (D_m / (-4.0 * d)))))))
	else:
		tmp = w0 * math.sqrt((1.0 + (D_m * ((((M_m / d) * h) * (M_m * D_m)) / ((-4.0 * 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 (D_m <= 1.25e-80)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(M_m / d) * Float64(D_m * Float64(Float64(Float64(M_m * h) / l) * Float64(D_m / Float64(-4.0 * d))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(D_m * Float64(Float64(Float64(Float64(M_m / d) * h) * Float64(M_m * D_m)) / Float64(Float64(-4.0 * 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 (D_m <= 1.25e-80)
		tmp = w0 * sqrt((1.0 + ((M_m / d) * (D_m * (((M_m * h) / l) * (D_m / (-4.0 * d)))))));
	else
		tmp = w0 * sqrt((1.0 + (D_m * ((((M_m / d) * h) * (M_m * D_m)) / ((-4.0 * 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[D$95$m, 1.25e-80], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(M$95$m / d), $MachinePrecision] * N[(D$95$m * N[(N[(N[(M$95$m * h), $MachinePrecision] / l), $MachinePrecision] * N[(D$95$m / N[(-4.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(D$95$m * N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(-4.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.25e-80

   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
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. flip-+ N/A

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

      12. distribute-neg-frac N/A

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

      13. +-inverses N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\mathsf{neg}\left(M \cdot D\right)\right)}{\frac{\mathsf{neg}\left(\color{blue}{0}\right)}{d - d}}} \]

      14. metadata-eval N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\mathsf{neg}\left(M \cdot D\right)\right)}{\frac{\color{blue}{0}}{d - d}}} \]

      15. +-inverses N/A

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

      16. flip-+ N/A

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

      17. count-2-rev N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 80.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 85.6

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

   6. Applied rewrites 85.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 82.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 79.3%

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

   if 1.25e-80 < D

   1. Initial program 77.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. flip-+ N/A

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

      12. distribute-neg-frac N/A

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

      13. +-inverses N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\mathsf{neg}\left(M \cdot D\right)\right)}{\frac{\mathsf{neg}\left(\color{blue}{0}\right)}{d - d}}} \]

      14. metadata-eval N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\mathsf{neg}\left(M \cdot D\right)\right)}{\frac{\color{blue}{0}}{d - d}}} \]

      15. +-inverses N/A

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

      16. flip-+ N/A

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

      17. count-2-rev N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 80.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 85.0

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

   6. Applied rewrites 85.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 87.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 87.6

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 87.6%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.25 \cdot 10^{-80}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M}{d} \cdot \left(D \cdot \left(\frac{M \cdot h}{\ell} \cdot \frac{D}{-4 \cdot d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + D \cdot \frac{\left(\frac{M}{d} \cdot h\right) \cdot \left(M \cdot D\right)}{\left(-4 \cdot d\right) \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 1.25 \cdot 10^{-80}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(\left(\frac{M\_m}{d} \cdot M\_m\right) \cdot D\_m\right) \cdot \left(-h\right)}{2 \cdot \ell}, \frac{D\_m}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + D\_m \cdot \frac{\left(\frac{M\_m}{d} \cdot h\right) \cdot \left(M\_m \cdot D\_m\right)}{\left(-4 \cdot d\right) \cdot \ell}}\\ \end{array} \end{array} \]

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 (<= D_m 1.25e-80)
   (*
    w0
    (sqrt
     (fma (/ (* (* (* (/ M_m d) M_m) D_m) (- h)) (* 2.0 l)) (/ D_m d) 1.0)))
   (*
    w0
    (sqrt
     (+ 1.0 (* D_m (/ (* (* (/ M_m d) h) (* M_m D_m)) (* (* -4.0 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 (D_m <= 1.25e-80) {
		tmp = w0 * sqrt(fma((((((M_m / d) * M_m) * D_m) * -h) / (2.0 * l)), (D_m / d), 1.0));
	} else {
		tmp = w0 * sqrt((1.0 + (D_m * ((((M_m / d) * h) * (M_m * D_m)) / ((-4.0 * 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 (D_m <= 1.25e-80)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(Float64(Float64(M_m / d) * M_m) * D_m) * Float64(-h)) / Float64(2.0 * l)), Float64(D_m / d), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(D_m * Float64(Float64(Float64(Float64(M_m / d) * h) * Float64(M_m * D_m)) / Float64(Float64(-4.0 * d) * l))))));
	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[D$95$m, 1.25e-80], N[(w0 * N[Sqrt[N[(N[(N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * (-h)), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(D$95$m * N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(-4.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.25e-80

   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 71.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 74.3

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

   6. Applied rewrites 74.3%

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

   if 1.25e-80 < D

   1. Initial program 77.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. flip-+ N/A

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

      12. distribute-neg-frac N/A

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

      13. +-inverses N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\mathsf{neg}\left(M \cdot D\right)\right)}{\frac{\mathsf{neg}\left(\color{blue}{0}\right)}{d - d}}} \]

      14. metadata-eval N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\mathsf{neg}\left(M \cdot D\right)\right)}{\frac{\color{blue}{0}}{d - d}}} \]

      15. +-inverses N/A

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

      16. flip-+ N/A

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

      17. count-2-rev N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 80.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 85.0

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

   6. Applied rewrites 85.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 87.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 87.6

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 87.6%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.25 \cdot 10^{-80}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot D\right) \cdot \left(-h\right)}{2 \cdot \ell}, \frac{D}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + D \cdot \frac{\left(\frac{M}{d} \cdot h\right) \cdot \left(M \cdot D\right)}{\left(-4 \cdot d\right) \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 2 \cdot 10^{-286}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + D\_m \cdot \frac{\left(\frac{M\_m}{d} \cdot h\right) \cdot \left(M\_m \cdot D\_m\right)}{\left(-4 \cdot d\right) \cdot \ell}}\\ \end{array} \end{array} \]

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) 2e-286)
   (* w0 1.0)
   (*
    w0
    (sqrt
     (+ 1.0 (* D_m (/ (* (* (/ M_m d) h) (* M_m D_m)) (* (* -4.0 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) <= 2e-286) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * sqrt((1.0 + (D_m * ((((M_m / d) * h) * (M_m * D_m)) / ((-4.0 * 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) <= 2d-286) then
        tmp = w0 * 1.0d0
    else
        tmp = w0 * sqrt((1.0d0 + (d_m * ((((m_m / d) * h) * (m_m * d_m)) / (((-4.0d0) * 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) <= 2e-286) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * Math.sqrt((1.0 + (D_m * ((((M_m / d) * h) * (M_m * D_m)) / ((-4.0 * 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) <= 2e-286:
		tmp = w0 * 1.0
	else:
		tmp = w0 * math.sqrt((1.0 + (D_m * ((((M_m / d) * h) * (M_m * D_m)) / ((-4.0 * 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(M_m * D_m) <= 2e-286)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(D_m * Float64(Float64(Float64(Float64(M_m / d) * h) * Float64(M_m * D_m)) / Float64(Float64(-4.0 * 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) <= 2e-286)
		tmp = w0 * 1.0;
	else
		tmp = w0 * sqrt((1.0 + (D_m * ((((M_m / d) * h) * (M_m * D_m)) / ((-4.0 * 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[(M$95$m * D$95$m), $MachinePrecision], 2e-286], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(D$95$m * N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(-4.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.0000000000000001e-286

   1. Initial program 81.8%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 76.7%

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

   if 2.0000000000000001e-286 < (*.f64 M D)

   1. Initial program 79.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. flip-+ N/A

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

      12. distribute-neg-frac N/A

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

      13. +-inverses N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\mathsf{neg}\left(M \cdot D\right)\right)}{\frac{\mathsf{neg}\left(\color{blue}{0}\right)}{d - d}}} \]

      14. metadata-eval N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\mathsf{neg}\left(M \cdot D\right)\right)}{\frac{\color{blue}{0}}{d - d}}} \]

      15. +-inverses N/A

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

      16. flip-+ N/A

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

      17. count-2-rev N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 82.1

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

   6. Applied rewrites 82.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 82.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 80.1

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 80.1%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{-286}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + D \cdot \frac{\left(\frac{M}{d} \cdot h\right) \cdot \left(M \cdot D\right)}{\left(-4 \cdot d\right) \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.1% 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 80.8%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 67.9%

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

Reproduce

herbie shell --seed 2024361 
(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))))))