Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.8% → 90.4%

Time: 10.1s

Alternatives: 17

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: 81.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.4% accurate, 1.8× speedup

Math

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

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
    (* (/ (* (* (/ M_m d) D_m) h) (* 2.0 l)) (* (/ D_m 2.0) (/ 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 - (((((M_m / d) * D_m) * h) / (2.0 * l)) * ((D_m / 2.0) * (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 - (((((m_m / d) * d_m) * h) / (2.0d0 * l)) * ((d_m / 2.0d0) * (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 - (((((M_m / d) * D_m) * h) / (2.0 * l)) * ((D_m / 2.0) * (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 - (((((M_m / d) * D_m) * h) / (2.0 * l)) * ((D_m / 2.0) * (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(Float64(Float64(M_m / d) * D_m) * h) / Float64(2.0 * l)) * Float64(Float64(D_m / 2.0) * 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 - (((((M_m / d) * D_m) * h) / (2.0 * l)) * ((D_m / 2.0) * (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[(N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.5%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
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. lower-*.f64 N/A

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

   7. lower-*.f64 85.9

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. times-frac N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 85.1

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

   16. lift-/.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. times-frac N/A

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

   21. lower-*.f64 N/A

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

   22. lower-/.f64 N/A

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

   23. lower-/.f64 86.2

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

4. Applied rewrites 86.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 92.5

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

6. Applied rewrites 92.5%

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

Alternative 2: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{elif}\;t\_0 \leq 10^{+110}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\left(-0.25 \cdot \left(\left(\left(M\_m \cdot D\_m\right) \cdot M\_m\right) \cdot D\_m\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(\frac{\frac{h}{d} \cdot M\_m}{d} \cdot \frac{M\_m}{\ell}\right) \cdot D\_m, -0.125 \cdot D\_m, 1\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))))))
   (if (<= t_0 2.0)
     (* w0 1.0)
     (if (<= t_0 1e+110)
       (*
        w0
        (sqrt (/ (* (* -0.25 (* (* (* M_m D_m) M_m) D_m)) h) (* (* d d) l))))
       (*
        w0
        (fma
         (* (* (/ (* (/ h d) M_m) d) (/ M_m l)) D_m)
         (* -0.125 D_m)
         1.0))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = w0 * 1.0;
	} else if (t_0 <= 1e+110) {
		tmp = w0 * sqrt((((-0.25 * (((M_m * D_m) * M_m) * D_m)) * h) / ((d * d) * l)));
	} else {
		tmp = w0 * fma((((((h / d) * M_m) / d) * (M_m / l)) * D_m), (-0.125 * D_m), 1.0);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = Float64(w0 * 1.0);
	elseif (t_0 <= 1e+110)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(-0.25 * Float64(Float64(Float64(M_m * D_m) * M_m) * D_m)) * h) / Float64(Float64(d * d) * l))));
	else
		tmp = Float64(w0 * fma(Float64(Float64(Float64(Float64(Float64(h / d) * M_m) / d) * Float64(M_m / l)) * D_m), Float64(-0.125 * D_m), 1.0));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[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]}, If[LessEqual[t$95$0, 2.0], N[(w0 * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e+110], N[(w0 * N[Sqrt[N[(N[(N[(-0.25 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[(N[(N[(N[(N[(h / d), $MachinePrecision] * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.125 * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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)))) < 2

   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 99.7%

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

   if 2 < (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)))) < 1e110

   1. Initial program 99.4%

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

   3. Taylor expanded in M around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 19.3

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 40.5%

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

   9. Applied rewrites 38.5%

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

   10. Taylor expanded in M around 0

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

   12. Applied rewrites 48.2%

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

   if 1e110 < (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 43.2%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 48.7%

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

   7. Applied rewrites 59.0%

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

   9. Applied rewrites 62.1%

      \[\leadsto w0 \cdot \mathsf{fma}\left(\left(\frac{\frac{h}{d} \cdot M}{d} \cdot \frac{M}{\ell}\right) \cdot D, -0.125 \cdot D, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 82.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{elif}\;t\_0 \leq 10^{+110}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\left(-0.25 \cdot \left(\left(\left(M\_m \cdot D\_m\right) \cdot M\_m\right) \cdot D\_m\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(h \cdot -0.125, \frac{D\_m}{d} \cdot \left(\frac{M\_m}{\ell} \cdot \left(\frac{M\_m}{d} \cdot D\_m\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))))))
   (if (<= t_0 2.0)
     (* w0 1.0)
     (if (<= t_0 1e+110)
       (*
        w0
        (sqrt (/ (* (* -0.25 (* (* (* M_m D_m) M_m) D_m)) h) (* (* d d) l))))
       (*
        w0
        (fma
         (* h -0.125)
         (* (/ D_m d) (* (/ M_m l) (* (/ M_m d) D_m)))
         1.0))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = w0 * 1.0;
	} else if (t_0 <= 1e+110) {
		tmp = w0 * sqrt((((-0.25 * (((M_m * D_m) * M_m) * D_m)) * h) / ((d * d) * l)));
	} else {
		tmp = w0 * fma((h * -0.125), ((D_m / d) * ((M_m / l) * ((M_m / d) * D_m))), 1.0);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = Float64(w0 * 1.0);
	elseif (t_0 <= 1e+110)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(-0.25 * Float64(Float64(Float64(M_m * D_m) * M_m) * D_m)) * h) / Float64(Float64(d * d) * l))));
	else
		tmp = Float64(w0 * fma(Float64(h * -0.125), Float64(Float64(D_m / d) * Float64(Float64(M_m / l) * Float64(Float64(M_m / d) * D_m))), 1.0));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[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]}, If[LessEqual[t$95$0, 2.0], N[(w0 * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e+110], N[(w0 * N[Sqrt[N[(N[(N[(-0.25 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[(h * -0.125), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(M$95$m / l), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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)))) < 2

   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 99.7%

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

   if 2 < (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)))) < 1e110

   1. Initial program 99.4%

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

   3. Taylor expanded in M around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 19.3

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 40.5%

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

   9. Applied rewrites 38.5%

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

   10. Taylor expanded in M around 0

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

   12. Applied rewrites 48.2%

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

   if 1e110 < (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 43.2%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 48.7%

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

   6. Taylor expanded in h around inf

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

   8. Applied rewrites 56.2%

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

   10. Applied rewrites 59.8%

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

   12. Applied rewrites 64.5%

       \[\leadsto w0 \cdot \mathsf{fma}\left(h \cdot -0.125, \frac{D}{d} \cdot \left(\frac{M}{\ell} \cdot \left(\frac{M}{d} \cdot \color{blue}{D}\right)\right), 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 82.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{elif}\;t\_0 \leq 10^{+110}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\left(-0.25 \cdot \left(\left(\left(M\_m \cdot D\_m\right) \cdot M\_m\right) \cdot D\_m\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(h \cdot -0.125, M\_m \cdot \left(\left(\frac{M\_m}{d} \cdot D\_m\right) \cdot \frac{\frac{D\_m}{d}}{\ell}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))))))
   (if (<= t_0 2.0)
     (* w0 1.0)
     (if (<= t_0 1e+110)
       (*
        w0
        (sqrt (/ (* (* -0.25 (* (* (* M_m D_m) M_m) D_m)) h) (* (* d d) l))))
       (*
        w0
        (fma
         (* h -0.125)
         (* M_m (* (* (/ M_m d) D_m) (/ (/ D_m d) l)))
         1.0))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = w0 * 1.0;
	} else if (t_0 <= 1e+110) {
		tmp = w0 * sqrt((((-0.25 * (((M_m * D_m) * M_m) * D_m)) * h) / ((d * d) * l)));
	} else {
		tmp = w0 * fma((h * -0.125), (M_m * (((M_m / d) * D_m) * ((D_m / d) / l))), 1.0);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = Float64(w0 * 1.0);
	elseif (t_0 <= 1e+110)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(-0.25 * Float64(Float64(Float64(M_m * D_m) * M_m) * D_m)) * h) / Float64(Float64(d * d) * l))));
	else
		tmp = Float64(w0 * fma(Float64(h * -0.125), Float64(M_m * Float64(Float64(Float64(M_m / d) * D_m) * Float64(Float64(D_m / d) / l))), 1.0));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[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]}, If[LessEqual[t$95$0, 2.0], N[(w0 * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e+110], N[(w0 * N[Sqrt[N[(N[(N[(-0.25 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[(h * -0.125), $MachinePrecision] * N[(M$95$m * N[(N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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)))) < 2

   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 99.7%

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

   if 2 < (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)))) < 1e110

   1. Initial program 99.4%

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

   3. Taylor expanded in M around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 19.3

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 40.5%

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

   9. Applied rewrites 38.5%

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

   10. Taylor expanded in M around 0

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

   12. Applied rewrites 48.2%

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

   if 1e110 < (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 43.2%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 48.7%

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

   6. Taylor expanded in h around inf

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

   8. Applied rewrites 56.2%

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

   10. Applied rewrites 61.0%

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

   12. Applied rewrites 66.5%

       \[\leadsto w0 \cdot \mathsf{fma}\left(h \cdot -0.125, M \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \frac{\frac{D}{d}}{\color{blue}{\ell}}\right), 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+219}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\left(-0.25 \cdot \left(\left(\left(M\_m \cdot D\_m\right) \cdot M\_m\right) \cdot D\_m\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(h \cdot -0.125, \frac{D\_m}{d} \cdot \left(M\_m \cdot \frac{M\_m \cdot D\_m}{\ell \cdot d}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)))))
   (if (<= t_0 2.0)
     (* w0 1.0)
     (if (<= t_0 5e+219)
       (*
        w0
        (sqrt (/ (* (* -0.25 (* (* (* M_m D_m) M_m) D_m)) h) (* (* d d) l))))
       (*
        w0
        (fma
         (* h -0.125)
         (* (/ D_m d) (* M_m (/ (* M_m D_m) (* l d))))
         1.0))))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = 1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = w0 * 1.0;
	} else if (t_0 <= 5e+219) {
		tmp = w0 * sqrt((((-0.25 * (((M_m * D_m) * M_m) * D_m)) * h) / ((d * d) * l)));
	} else {
		tmp = w0 * fma((h * -0.125), ((D_m / d) * (M_m * ((M_m * D_m) / (l * d)))), 1.0);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = Float64(w0 * 1.0);
	elseif (t_0 <= 5e+219)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(-0.25 * Float64(Float64(Float64(M_m * D_m) * M_m) * D_m)) * h) / Float64(Float64(d * d) * l))));
	else
		tmp = Float64(w0 * fma(Float64(h * -0.125), Float64(Float64(D_m / d) * Float64(M_m * Float64(Float64(M_m * D_m) / Float64(l * d)))), 1.0));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 2.0], N[(w0 * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 5e+219], N[(w0 * N[Sqrt[N[(N[(N[(-0.25 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[(h * -0.125), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.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))) < 2

   1. Initial program 99.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 99.7%

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

   if 2 < (-.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))) < 5e219

   1. Initial program 99.4%

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

   3. Taylor expanded in M around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 19.3

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 40.5%

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

   9. Applied rewrites 38.5%

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

   10. Taylor expanded in M around 0

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

   12. Applied rewrites 48.2%

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

   if 5e219 < (-.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 43.8%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   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 49.4%

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

   6. Taylor expanded in h around inf

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

   8. Applied rewrites 56.9%

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

   10. Applied rewrites 60.6%

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

   12. Applied rewrites 65.4%

       \[\leadsto w0 \cdot \mathsf{fma}\left(h \cdot -0.125, \frac{D}{d} \cdot \left(M \cdot \frac{M \cdot D}{\color{blue}{\ell \cdot d}}\right), 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 86.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.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))) < 2

   1. Initial program 99.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 99.7%

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

   if 2 < (-.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 55.3%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. lower-*.f64 N/A

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

      7. lower-*.f64 61.7

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 61.7

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

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. times-frac N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 N/A

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

      23. lower-/.f64 63.8

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

   4. Applied rewrites 63.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 81.1

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

   6. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 74.7

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

   8. Applied rewrites 74.7%

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

Alternative 7: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -500000:\\ \;\;\;\;w0 \cdot \sqrt{\left(\frac{h}{d} \cdot \left(\frac{M\_m}{d} \cdot \left(\left(\frac{M\_m}{\ell} \cdot -0.25\right) \cdot D\_m\right)\right)\right) \cdot D\_m}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -500000.0)
   (* w0 (sqrt (* (* (/ h d) (* (/ M_m d) (* (* (/ M_m l) -0.25) D_m))) D_m)))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -500000.0) {
		tmp = w0 * sqrt((((h / d) * ((M_m / d) * (((M_m / l) * -0.25) * D_m))) * D_m));
	} 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)) <= (-500000.0d0)) then
        tmp = w0 * sqrt((((h / d) * ((m_m / d) * (((m_m / l) * (-0.25d0)) * d_m))) * d_m))
    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)) <= -500000.0) {
		tmp = w0 * Math.sqrt((((h / d) * ((M_m / d) * (((M_m / l) * -0.25) * D_m))) * D_m));
	} 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)) <= -500000.0:
		tmp = w0 * math.sqrt((((h / d) * ((M_m / d) * (((M_m / l) * -0.25) * D_m))) * D_m))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -500000.0)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(h / d) * Float64(Float64(M_m / d) * Float64(Float64(Float64(M_m / l) * -0.25) * D_m))) * D_m)));
	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)) <= -500000.0)
		tmp = w0 * sqrt((((h / d) * ((M_m / d) * (((M_m / l) * -0.25) * D_m))) * D_m));
	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], -500000.0], N[(w0 * N[Sqrt[N[(N[(N[(h / d), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(N[(M$95$m / l), $MachinePrecision] * -0.25), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5e5

   1. Initial program 70.6%

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

   3. Taylor expanded in M around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 40.5

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

   5. Applied rewrites 40.5%

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

   7. Applied rewrites 54.6%

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

   9. Applied rewrites 65.4%

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

   11. Applied rewrites 66.8%

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

   if -5e5 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.6%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 97.1%

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

Alternative 8: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -500000:\\ \;\;\;\;w0 \cdot \sqrt{\left(\left(\left(-0.25 \cdot D\_m\right) \cdot \left(\frac{M\_m}{d} \cdot \frac{M\_m}{d}\right)\right) \cdot \frac{h}{\ell}\right) \cdot D\_m}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -500000.0)
   (* w0 (sqrt (* (* (* (* -0.25 D_m) (* (/ M_m d) (/ M_m d))) (/ h l)) D_m)))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -500000.0) {
		tmp = w0 * sqrt(((((-0.25 * D_m) * ((M_m / d) * (M_m / d))) * (h / l)) * D_m));
	} 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)) <= (-500000.0d0)) then
        tmp = w0 * sqrt((((((-0.25d0) * d_m) * ((m_m / d) * (m_m / d))) * (h / l)) * d_m))
    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)) <= -500000.0) {
		tmp = w0 * Math.sqrt(((((-0.25 * D_m) * ((M_m / d) * (M_m / d))) * (h / l)) * D_m));
	} 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)) <= -500000.0:
		tmp = w0 * math.sqrt(((((-0.25 * D_m) * ((M_m / d) * (M_m / d))) * (h / l)) * D_m))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -500000.0)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(Float64(-0.25 * D_m) * Float64(Float64(M_m / d) * Float64(M_m / d))) * Float64(h / l)) * D_m)));
	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)) <= -500000.0)
		tmp = w0 * sqrt(((((-0.25 * D_m) * ((M_m / d) * (M_m / d))) * (h / l)) * D_m));
	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], -500000.0], N[(w0 * N[Sqrt[N[(N[(N[(N[(-0.25 * D$95$m), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5e5

   1. Initial program 70.6%

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

   3. Taylor expanded in M around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 40.5

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

   5. Applied rewrites 40.5%

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

   7. Applied rewrites 54.6%

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

   9. Applied rewrites 65.4%

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

   11. Applied rewrites 66.3%

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

   if -5e5 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.6%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 97.1%

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

Alternative 9: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -500000:\\ \;\;\;\;w0 \cdot \sqrt{\left(\left(\frac{M\_m}{d} \cdot \frac{M\_m \cdot h}{\ell \cdot d}\right) \cdot \left(-0.25 \cdot D\_m\right)\right) \cdot D\_m}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -500000.0)
   (* w0 (sqrt (* (* (* (/ M_m d) (/ (* M_m h) (* l d))) (* -0.25 D_m)) D_m)))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -500000.0) {
		tmp = w0 * sqrt(((((M_m / d) * ((M_m * h) / (l * d))) * (-0.25 * D_m)) * D_m));
	} 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)) <= (-500000.0d0)) then
        tmp = w0 * sqrt(((((m_m / d) * ((m_m * h) / (l * d))) * ((-0.25d0) * d_m)) * d_m))
    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)) <= -500000.0) {
		tmp = w0 * Math.sqrt(((((M_m / d) * ((M_m * h) / (l * d))) * (-0.25 * D_m)) * D_m));
	} 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)) <= -500000.0:
		tmp = w0 * math.sqrt(((((M_m / d) * ((M_m * h) / (l * d))) * (-0.25 * D_m)) * D_m))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -500000.0)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(Float64(M_m / d) * Float64(Float64(M_m * h) / Float64(l * d))) * Float64(-0.25 * D_m)) * D_m)));
	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)) <= -500000.0)
		tmp = w0 * sqrt(((((M_m / d) * ((M_m * h) / (l * d))) * (-0.25 * D_m)) * D_m));
	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], -500000.0], N[(w0 * N[Sqrt[N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(M$95$m * h), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5e5

   1. Initial program 70.6%

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

   3. Taylor expanded in M around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 40.5

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

   5. Applied rewrites 40.5%

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

   7. Applied rewrites 54.6%

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

   9. Applied rewrites 47.9%

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

   11. Applied rewrites 68.7%

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

   if -5e5 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.6%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 97.1%

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

Alternative 10: 80.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(h \cdot -0.125, \frac{D\_m}{d} \cdot \left(M\_m \cdot \frac{M\_m \cdot D\_m}{\ell \cdot d}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))) 5e+153)
   (* w0 1.0)
   (*
    w0
    (fma (* h -0.125) (* (/ D_m d) (* M_m (/ (* M_m D_m) (* l d)))) 1.0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))) <= 5e+153) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * fma((h * -0.125), ((D_m / d) * (M_m * ((M_m * D_m) / (l * d)))), 1.0);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))) <= 5e+153)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * fma(Float64(h * -0.125), Float64(Float64(D_m / d) * Float64(M_m * Float64(Float64(M_m * D_m) / Float64(l * d)))), 1.0));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(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], 5e+153], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[(N[(h * -0.125), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.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))) < 5.00000000000000018e153

   1. Initial program 99.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 92.1%

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

   if 5.00000000000000018e153 < (-.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 46.7%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   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 47.0%

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

   6. Taylor expanded in h around inf

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

   8. Applied rewrites 54.3%

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

   10. Applied rewrites 57.8%

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

   12. Applied rewrites 62.3%

       \[\leadsto w0 \cdot \mathsf{fma}\left(h \cdot -0.125, \frac{D}{d} \cdot \left(M \cdot \frac{M \cdot D}{\color{blue}{\ell \cdot d}}\right), 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+219}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(h \cdot -0.125, M\_m \cdot \left(\left(D\_m \cdot M\_m\right) \cdot \frac{\frac{D\_m}{d}}{\ell \cdot d}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))) 5e+219)
   (* w0 1.0)
   (*
    w0
    (fma (* h -0.125) (* M_m (* (* D_m M_m) (/ (/ D_m d) (* l d)))) 1.0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))) <= 5e+219) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * fma((h * -0.125), (M_m * ((D_m * M_m) * ((D_m / d) / (l * d)))), 1.0);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))) <= 5e+219)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * fma(Float64(h * -0.125), Float64(M_m * Float64(Float64(D_m * M_m) * Float64(Float64(D_m / d) / Float64(l * d)))), 1.0));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(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], 5e+219], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[(N[(h * -0.125), $MachinePrecision] * N[(M$95$m * N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.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))) < 5e219

   1. Initial program 99.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 90.2%

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

   if 5e219 < (-.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 43.8%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   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 49.4%

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

   6. Taylor expanded in h around inf

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

   8. Applied rewrites 56.9%

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

   10. Applied rewrites 61.8%

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

   12. Applied rewrites 64.7%

       \[\leadsto w0 \cdot \mathsf{fma}\left(h \cdot -0.125, M \cdot \left(\left(D \cdot M\right) \cdot \frac{\frac{D}{d}}{\ell \cdot \color{blue}{d}}\right), 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 80.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+219}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(h \cdot -0.125, M\_m \cdot \left(\left(D\_m \cdot M\_m\right) \cdot \frac{D\_m}{\left(\ell \cdot d\right) \cdot d}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))) 5e+219)
   (* w0 1.0)
   (*
    w0
    (fma (* h -0.125) (* M_m (* (* D_m M_m) (/ D_m (* (* l d) d)))) 1.0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))) <= 5e+219) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * fma((h * -0.125), (M_m * ((D_m * M_m) * (D_m / ((l * d) * d)))), 1.0);
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))) <= 5e+219)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * fma(Float64(h * -0.125), Float64(M_m * Float64(Float64(D_m * M_m) * Float64(D_m / Float64(Float64(l * d) * d)))), 1.0));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(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], 5e+219], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[(N[(h * -0.125), $MachinePrecision] * N[(M$95$m * N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.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))) < 5e219

   1. Initial program 99.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 90.2%

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

   if 5e219 < (-.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 43.8%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   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 49.4%

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

   6. Taylor expanded in h around inf

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

   8. Applied rewrites 56.9%

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

   10. Applied rewrites 61.8%

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

   12. Applied rewrites 63.2%

       \[\leadsto w0 \cdot \mathsf{fma}\left(h \cdot -0.125, M \cdot \left(\left(D \cdot M\right) \cdot \frac{D}{\left(\ell \cdot d\right) \cdot d}\right), 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 79.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+219}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(h \cdot -0.125, M\_m \cdot \left(\left(D\_m \cdot M\_m\right) \cdot \frac{D\_m}{\left(d \cdot d\right) \cdot \ell}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))) 5e+219)
   (* w0 1.0)
   (*
    w0
    (fma (* h -0.125) (* M_m (* (* D_m M_m) (/ D_m (* (* d d) l)))) 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 ((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))) <= 5e+219) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * fma((h * -0.125), (M_m * ((D_m * M_m) * (D_m / ((d * d) * l)))), 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(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))) <= 5e+219)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * fma(Float64(h * -0.125), Float64(M_m * Float64(Float64(D_m * M_m) * Float64(D_m / Float64(Float64(d * d) * l)))), 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[(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], 5e+219], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[(N[(h * -0.125), $MachinePrecision] * N[(M$95$m * N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.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))) < 5e219

   1. Initial program 99.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 90.2%

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

   if 5e219 < (-.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 43.8%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   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 49.4%

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

   6. Taylor expanded in h around inf

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

   8. Applied rewrites 56.9%

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

   10. Applied rewrites 61.8%

       \[\leadsto w0 \cdot \mathsf{fma}\left(h \cdot -0.125, M \cdot \left(\left(D \cdot M\right) \cdot \color{blue}{\frac{D}{\left(d \cdot d\right) \cdot \ell}}\right), 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 79.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+219}:\\ \;\;\;\;w0 \cdot \frac{\left(-0.125 \cdot \left(\left(\left(M\_m \cdot D\_m\right) \cdot M\_m\right) \cdot D\_m\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -2e+219)
   (* w0 (/ (* (* -0.125 (* (* (* M_m D_m) M_m) D_m)) h) (* (* d d) l)))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+219) {
		tmp = w0 * (((-0.125 * (((M_m * D_m) * M_m) * D_m)) * h) / ((d * d) * l));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(w0, m_m, d_m, h, l, d)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-2d+219)) then
        tmp = w0 * ((((-0.125d0) * (((m_m * d_m) * m_m) * d_m)) * h) / ((d * d) * l))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+219) {
		tmp = w0 * (((-0.125 * (((M_m * D_m) * M_m) * D_m)) * h) / ((d * d) * l));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+219:
		tmp = w0 * (((-0.125 * (((M_m * D_m) * M_m) * D_m)) * h) / ((d * d) * l))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+219)
		tmp = Float64(w0 * Float64(Float64(Float64(-0.125 * Float64(Float64(Float64(M_m * D_m) * M_m) * D_m)) * h) / Float64(Float64(d * d) * l)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+219)
		tmp = w0 * (((-0.125 * (((M_m * D_m) * M_m) * D_m)) * h) / ((d * d) * l));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+219], N[(w0 * N[(N[(N[(-0.125 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.99999999999999993e219

   1. Initial program 61.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 45.4%

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

   6. Taylor expanded in h around inf

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

   8. Applied rewrites 49.1%

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

   10. Applied rewrites 55.6%

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

   11. Taylor expanded in M around inf

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

   13. Applied rewrites 53.0%

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

   if -1.99999999999999993e219 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.1%

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

Alternative 15: 89.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d}\\ w0 \cdot \sqrt{1 - \frac{\left(h \cdot t\_0\right) \cdot t\_0}{\ell \cdot 4}} \end{array} \end{array} \]

FPCore

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

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M_m / d);
	return w0 * sqrt((1.0 - (((h * t_0) * t_0) / (l * 4.0))));
}

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
    real(8) :: t_0
    t_0 = d_m * (m_m / d)
    code = w0 * sqrt((1.0d0 - (((h * t_0) * t_0) / (l * 4.0d0))))
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) {
	double t_0 = D_m * (M_m / d);
	return w0 * Math.sqrt((1.0 - (((h * t_0) * t_0) / (l * 4.0))));
}

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):
	t_0 = D_m * (M_m / d)
	return w0 * math.sqrt((1.0 - (((h * t_0) * t_0) / (l * 4.0))))

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)
	t_0 = Float64(D_m * Float64(M_m / d))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h * t_0) * t_0) / Float64(l * 4.0)))))
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)
	t_0 = D_m * (M_m / d);
	tmp = w0 * sqrt((1.0 - (((h * t_0) * t_0) / (l * 4.0))));
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_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(l * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 83.5%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
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. lower-*.f64 N/A

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

   7. lower-*.f64 85.9

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. times-frac N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 85.1

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

   16. lift-/.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. times-frac N/A

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

   21. lower-*.f64 N/A

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

   22. lower-/.f64 N/A

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

   23. lower-/.f64 86.2

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

4. Applied rewrites 86.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 92.5

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

6. Applied rewrites 92.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. frac-times N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. lower-*.f64 91.7

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

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lower-*.f64 91.7

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

8. Applied rewrites 91.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval 91.7

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

10. Applied rewrites 91.7%

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

Alternative 16: 87.2% accurate, 2.0× speedup

Math

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

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

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 - (((((m_m / d) * d_m) * h) * d_m) * ((m_m / d) / (4.0d0 * l)))))
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 - (((((M_m / d) * D_m) * h) * D_m) * ((M_m / d) / (4.0 * l)))));
}

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 - (((((M_m / d) * D_m) * h) * D_m) * ((M_m / d) / (4.0 * l)))))

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

Derivation

1. Initial program 83.5%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
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. lower-*.f64 N/A

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

   7. lower-*.f64 85.9

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. times-frac N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 85.1

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

   16. lift-/.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. times-frac N/A

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

   21. lower-*.f64 N/A

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

   22. lower-/.f64 N/A

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

   23. lower-/.f64 86.2

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

4. Applied rewrites 86.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 92.5

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

6. Applied rewrites 92.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. frac-times N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. lower-*.f64 91.7

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

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lower-*.f64 91.7

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

8. Applied rewrites 91.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 88.2

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. associate-*l* N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. metadata-eval 88.2

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

10. Applied rewrites 88.2%

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

Alternative 17: 68.6% 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 83.5%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 71.4%

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

Reproduce

herbie shell --seed 2025008 
(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))))))