Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 88.7%

Time: 16.0s

Alternatives: 17

Speedup: 1.9×

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: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

M_m =     private
NOTE: w0, M_m, D, 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, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    t_0 = d * (m_m / d_1)
    code = w0 * sqrt((1.0d0 - ((((h * t_0) * t_0) / 4.0d0) / l)))
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = D * (M_m / d);
	return w0 * Math.sqrt((1.0 - ((((h * t_0) * t_0) / 4.0) / l)));
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	t_0 = D * (M_m / d)
	return w0 * math.sqrt((1.0 - ((((h * t_0) * t_0) / 4.0) / l)))

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(D * Float64(M_m / d))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(h * t_0) * t_0) / 4.0) / l))))
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp = code(w0, M_m, D, h, l, d)
	t_0 = D * (M_m / d);
	tmp = w0 * sqrt((1.0 - ((((h * t_0) * t_0) / 4.0) / l)));
end

Wolfram

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

Derivation

1. Initial program 82.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 82.7

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

4. Applied rewrites 82.7%

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

6. Applied rewrites 87.2%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*r/ N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. *-commutative N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

   16. *-commutative N/A

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

   17. associate-/r* N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

8. Applied rewrites 88.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. metadata-eval 88.7

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

10. Applied rewrites 88.7%

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

Alternative 2: 80.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.125 \cdot \left(D \cdot D\right)}{d}, \left(\left(\frac{w0}{d} \cdot M\_m\right) \cdot M\_m\right) \cdot \frac{h}{\ell}, w0\right)\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{+27}:\\ \;\;\;\;w0 \cdot \sqrt{\left(-0.25 \cdot \left(D \cdot D\right)\right) \cdot \frac{\frac{h}{d} \cdot \left(M\_m \cdot M\_m\right)}{\ell \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l))))
   (if (<= t_0 (- INFINITY))
     (fma (/ (* -0.125 (* D D)) d) (* (* (* (/ w0 d) M_m) M_m) (/ h l)) w0)
     (if (<= t_0 -5e+27)
       (* w0 (sqrt (* (* -0.25 (* D D)) (/ (* (/ h d) (* M_m M_m)) (* l d)))))
       w0))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((-0.125 * (D * D)) / d), ((((w0 / d) * M_m) * M_m) * (h / l)), w0);
	} else if (t_0 <= -5e+27) {
		tmp = w0 * sqrt(((-0.25 * (D * D)) * (((h / d) * (M_m * M_m)) / (l * d))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	t_0 = Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(Float64(-0.125 * Float64(D * D)) / d), Float64(Float64(Float64(Float64(w0 / d) * M_m) * M_m) * Float64(h / l)), w0);
	elseif (t_0 <= -5e+27)
		tmp = Float64(w0 * sqrt(Float64(Float64(-0.25 * Float64(D * D)) * Float64(Float64(Float64(h / d) * Float64(M_m * M_m)) / Float64(l * d)))));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(N[(w0 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], If[LessEqual[t$95$0, -5e+27], N[(w0 * N[Sqrt[N[(N[(-0.25 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h / d), $MachinePrecision] * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 57.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 41.0%

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

   9. Applied rewrites 53.8%

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

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.99999999999999979e27

   1. Initial program 96.1%

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

   3. Taylor expanded in M around 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

   5. Applied rewrites 25.7%

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

   7. Applied rewrites 26.1%

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

   if -4.99999999999999979e27 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 94.7%

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

Alternative 3: 84.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-6}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{\left(h \cdot D\right) \cdot M\_m}{\ell \cdot \left(2 \cdot d\right)} \cdot \left(M\_m \cdot D\right)}{-2 \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -1e-6)
   (*
    w0
    (sqrt
     (+ 1.0 (/ (* (/ (* (* h D) M_m) (* l (* 2.0 d))) (* M_m D)) (* -2.0 d)))))
   w0))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e-6) {
		tmp = w0 * sqrt((1.0 + (((((h * D) * M_m) / (l * (2.0 * d))) * (M_m * D)) / (-2.0 * d))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: w0, M_m, D, 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, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m_m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-1d-6)) then
        tmp = w0 * sqrt((1.0d0 + (((((h * d) * m_m) / (l * (2.0d0 * d_1))) * (m_m * d)) / ((-2.0d0) * d_1))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e-6) {
		tmp = w0 * Math.sqrt((1.0 + (((((h * D) * M_m) / (l * (2.0 * d))) * (M_m * D)) / (-2.0 * d))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e-6:
		tmp = w0 * math.sqrt((1.0 + (((((h * D) * M_m) / (l * (2.0 * d))) * (M_m * D)) / (-2.0 * d))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e-6)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(Float64(h * D) * M_m) / Float64(l * Float64(2.0 * d))) * Float64(M_m * D)) / Float64(-2.0 * d)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (((((M_m * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -1e-6)
		tmp = w0 * sqrt((1.0 + (((((h * D) * M_m) / (l * (2.0 * d))) * (M_m * D)) / (-2.0 * d))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e-6], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(N[(N[(h * D), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(l * N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -9.99999999999999955e-7

   1. Initial program 64.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. flip-+ N/A

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

      12. distribute-neg-frac N/A

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

      13. +-inverses N/A

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

      14. metadata-eval N/A

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

      15. +-inverses N/A

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

      16. flip-+ N/A

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

      17. count-2-rev N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 66.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. *-commutative N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 65.3

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

   6. Applied rewrites 65.3%

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

   if -9.99999999999999955e-7 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.4%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-6}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{\left(h \cdot D\right) \cdot M}{\ell \cdot \left(2 \cdot d\right)} \cdot \left(M \cdot D\right)}{-2 \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-6}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M\_m \cdot \frac{h \cdot D}{\left(2 \cdot d\right) \cdot \ell}\right) \cdot \left(M\_m \cdot D\right)}{-2 \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -1e-6)
   (*
    w0
    (sqrt
     (+ 1.0 (/ (* (* M_m (/ (* h D) (* (* 2.0 d) l))) (* M_m D)) (* -2.0 d)))))
   w0))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e-6) {
		tmp = w0 * sqrt((1.0 + (((M_m * ((h * D) / ((2.0 * d) * l))) * (M_m * D)) / (-2.0 * d))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: w0, M_m, D, 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, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m_m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-1d-6)) then
        tmp = w0 * sqrt((1.0d0 + (((m_m * ((h * d) / ((2.0d0 * d_1) * l))) * (m_m * d)) / ((-2.0d0) * d_1))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e-6) {
		tmp = w0 * Math.sqrt((1.0 + (((M_m * ((h * D) / ((2.0 * d) * l))) * (M_m * D)) / (-2.0 * d))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e-6:
		tmp = w0 * math.sqrt((1.0 + (((M_m * ((h * D) / ((2.0 * d) * l))) * (M_m * D)) / (-2.0 * d))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e-6)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(M_m * Float64(Float64(h * D) / Float64(Float64(2.0 * d) * l))) * Float64(M_m * D)) / Float64(-2.0 * d)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (((((M_m * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -1e-6)
		tmp = w0 * sqrt((1.0 + (((M_m * ((h * D) / ((2.0 * d) * l))) * (M_m * D)) / (-2.0 * d))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e-6], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(M$95$m * N[(N[(h * D), $MachinePrecision] / N[(N[(2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -9.99999999999999955e-7

   1. Initial program 64.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. flip-+ N/A

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

      12. distribute-neg-frac N/A

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

      13. +-inverses N/A

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

      14. metadata-eval N/A

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

      15. +-inverses N/A

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

      16. flip-+ N/A

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

      17. count-2-rev N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 66.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. *-commutative N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 65.3

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

   6. Applied rewrites 65.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 64.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 64.8

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

   8. Applied rewrites 64.8%

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

   if -9.99999999999999955e-7 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.4%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-6}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M \cdot \frac{h \cdot D}{\left(2 \cdot d\right) \cdot \ell}\right) \cdot \left(M \cdot D\right)}{-2 \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-6}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\left(\left(h \cdot D\right) \cdot M\_m\right) \cdot \left(D \cdot M\_m\right)}{\left(2 \cdot d\right) \cdot \ell}}{2 \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -1e-6)
   (*
    w0
    (sqrt
     (- 1.0 (/ (/ (* (* (* h D) M_m) (* D M_m)) (* (* 2.0 d) l)) (* 2.0 d)))))
   w0))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e-6) {
		tmp = w0 * sqrt((1.0 - (((((h * D) * M_m) * (D * M_m)) / ((2.0 * d) * l)) / (2.0 * d))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: w0, M_m, D, 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, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m_m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-1d-6)) then
        tmp = w0 * sqrt((1.0d0 - (((((h * d) * m_m) * (d * m_m)) / ((2.0d0 * d_1) * l)) / (2.0d0 * d_1))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e-6) {
		tmp = w0 * Math.sqrt((1.0 - (((((h * D) * M_m) * (D * M_m)) / ((2.0 * d) * l)) / (2.0 * d))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e-6:
		tmp = w0 * math.sqrt((1.0 - (((((h * D) * M_m) * (D * M_m)) / ((2.0 * d) * l)) / (2.0 * d))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e-6)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(h * D) * M_m) * Float64(D * M_m)) / Float64(Float64(2.0 * d) * l)) / Float64(2.0 * d)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (((((M_m * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -1e-6)
		tmp = w0 * sqrt((1.0 - (((((h * D) * M_m) * (D * M_m)) / ((2.0 * d) * l)) / (2.0 * d))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e-6], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(N[(h * D), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -9.99999999999999955e-7

   1. Initial program 64.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. flip-+ N/A

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

      12. distribute-neg-frac N/A

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

      13. +-inverses N/A

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

      14. metadata-eval N/A

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

      15. +-inverses N/A

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

      16. flip-+ N/A

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

      17. count-2-rev N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 66.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. *-commutative N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 65.3

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

   6. Applied rewrites 65.3%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 60.9%

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

   if -9.99999999999999955e-7 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.4%

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

Alternative 6: 81.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1000000000000:\\ \;\;\;\;w0 \cdot \sqrt{\left(\left(\left(\frac{h}{d \cdot d} \cdot M\_m\right) \cdot \frac{M\_m}{\ell}\right) \cdot \left(-0.25 \cdot D\right)\right) \cdot D}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -1000000000000.0)
   (* w0 (sqrt (* (* (* (* (/ h (* d d)) M_m) (/ M_m l)) (* -0.25 D)) D)))
   w0))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1000000000000.0) {
		tmp = w0 * sqrt((((((h / (d * d)) * M_m) * (M_m / l)) * (-0.25 * D)) * D));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m =     private
NOTE: w0, M_m, D, 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, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m_m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-1000000000000.0d0)) then
        tmp = w0 * sqrt((((((h / (d_1 * d_1)) * m_m) * (m_m / l)) * ((-0.25d0) * d)) * d))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1000000000000.0) {
		tmp = w0 * Math.sqrt((((((h / (d * d)) * M_m) * (M_m / l)) * (-0.25 * D)) * D));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1000000000000.0:
		tmp = w0 * math.sqrt((((((h / (d * d)) * M_m) * (M_m / l)) * (-0.25 * D)) * D))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1000000000000.0)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(Float64(Float64(h / Float64(d * d)) * M_m) * Float64(M_m / l)) * Float64(-0.25 * D)) * D)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (((((M_m * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -1000000000000.0)
		tmp = w0 * sqrt((((((h / (d * d)) * M_m) * (M_m / l)) * (-0.25 * D)) * D));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1000000000000.0], N[(w0 * N[Sqrt[N[(N[(N[(N[(N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(M$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * D), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

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)) < -1e12

   1. Initial program 63.9%

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

   3. Taylor expanded in M around 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

   5. Applied rewrites 39.5%

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

   7. Applied rewrites 46.6%

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

   if -1e12 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.1%

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

Alternative 7: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.125 \cdot \left(D \cdot D\right)}{d}, \left(\left(\frac{w0}{d} \cdot M\_m\right) \cdot M\_m\right) \cdot \frac{h}{\ell}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -1e+256)
   (fma (/ (* -0.125 (* D D)) d) (* (* (* (/ w0 d) M_m) M_m) (/ h l)) w0)
   w0))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+256) {
		tmp = fma(((-0.125 * (D * D)) / d), ((((w0 / d) * M_m) * M_m) * (h / l)), w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+256)
		tmp = fma(Float64(Float64(-0.125 * Float64(D * D)) / d), Float64(Float64(Float64(Float64(w0 / d) * M_m) * M_m) * Float64(h / l)), w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+256], N[(N[(N[(-0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(N[(w0 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], w0]

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)) < -1e256

   1. Initial program 58.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 42.2%

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

   7. Applied rewrites 40.5%

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

   9. Applied rewrites 53.1%

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

   if -1e256 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.6%

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

Alternative 8: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, \frac{\left(\frac{w0}{d} \cdot M\_m\right) \cdot \left(h \cdot M\_m\right)}{\ell \cdot d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -1e+256)
   (fma (* (* D D) -0.125) (/ (* (* (/ w0 d) M_m) (* h M_m)) (* l d)) w0)
   w0))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+256) {
		tmp = fma(((D * D) * -0.125), ((((w0 / d) * M_m) * (h * M_m)) / (l * d)), w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+256)
		tmp = fma(Float64(Float64(D * D) * -0.125), Float64(Float64(Float64(Float64(w0 / d) * M_m) * Float64(h * M_m)) / Float64(l * d)), w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+256], N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(N[(w0 / d), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], w0]

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)) < -1e256

   1. Initial program 58.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 42.2%

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

   7. Applied rewrites 49.7%

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

   9. Applied rewrites 53.0%

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

   if -1e256 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.6%

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

Alternative 9: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, \frac{\frac{\left(M\_m \cdot M\_m\right) \cdot w0}{d} \cdot h}{\ell \cdot d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -1e+256)
   (fma (* (* D D) -0.125) (/ (* (/ (* (* M_m M_m) w0) d) h) (* l d)) w0)
   w0))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+256) {
		tmp = fma(((D * D) * -0.125), (((((M_m * M_m) * w0) / d) * h) / (l * d)), w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+256)
		tmp = fma(Float64(Float64(D * D) * -0.125), Float64(Float64(Float64(Float64(Float64(M_m * M_m) * w0) / d) * h) / Float64(l * d)), w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+256], N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * w0), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], w0]

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)) < -1e256

   1. Initial program 58.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 42.2%

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

   7. Applied rewrites 49.7%

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

   8. Taylor expanded in w0 around 0

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

   10. Applied rewrites 51.3%

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

   if -1e256 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.6%

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

Alternative 10: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, \frac{\left(M\_m \cdot M\_m\right) \cdot h}{d} \cdot \frac{w0}{\ell \cdot d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -1e+256)
   (fma (* (* D D) -0.125) (* (/ (* (* M_m M_m) h) d) (/ w0 (* l d))) w0)
   w0))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+256) {
		tmp = fma(((D * D) * -0.125), ((((M_m * M_m) * h) / d) * (w0 / (l * d))), w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+256)
		tmp = fma(Float64(Float64(D * D) * -0.125), Float64(Float64(Float64(Float64(M_m * M_m) * h) / d) * Float64(w0 / Float64(l * d))), w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+256], N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] * N[(w0 / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], w0]

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)) < -1e256

   1. Initial program 58.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 42.2%

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

   7. Applied rewrites 50.0%

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

   if -1e256 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.6%

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

Alternative 11: 79.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+205}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\frac{w0}{\left(d \cdot d\right) \cdot \ell} \cdot M\_m\right) \cdot \left(h \cdot M\_m\right)\right) \cdot D, -0.125 \cdot D, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -5e+205)
   (fma (* (* (* (/ w0 (* (* d d) l)) M_m) (* h M_m)) D) (* -0.125 D) w0)
   w0))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+205) {
		tmp = fma(((((w0 / ((d * d) * l)) * M_m) * (h * M_m)) * D), (-0.125 * D), w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+205)
		tmp = fma(Float64(Float64(Float64(Float64(w0 / Float64(Float64(d * d) * l)) * M_m) * Float64(h * M_m)) * D), Float64(-0.125 * D), w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+205], N[(N[(N[(N[(N[(w0 / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * N[(-0.125 * D), $MachinePrecision] + w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000002e205

   1. Initial program 58.6%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 41.6%

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

   7. Applied rewrites 39.9%

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

   9. Applied rewrites 47.1%

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

   if -5.0000000000000002e205 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 90.1%

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

Alternative 12: 77.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 41.0%

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

   9. Applied rewrites 44.2%

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

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

   1. Initial program 90.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.2%

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

Alternative 13: 77.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 42.8%

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

   6. Taylor expanded in h around inf

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

   8. Applied rewrites 45.8%

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

   9. Taylor expanded in M around inf

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

   11. Applied rewrites 45.7%

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

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

   1. Initial program 90.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.2%

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

Alternative 14: 79.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 5.9 \cdot 10^{+51}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\left(\left(\frac{h}{\ell} \cdot D\right) \cdot M\_m\right) \cdot \left(M\_m \cdot D\right)}{2 \cdot d}}{2 \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{\left(\frac{M\_m \cdot M\_m}{d} \cdot \frac{D}{d}\right) \cdot D}{\ell}, 1\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= d 5.9e+51)
   (*
    w0
    (sqrt
     (- 1.0 (/ (/ (* (* (* (/ h l) D) M_m) (* M_m D)) (* 2.0 d)) (* 2.0 d)))))
   (*
    w0
    (sqrt (fma (* h -0.25) (/ (* (* (/ (* M_m M_m) d) (/ D d)) D) l) 1.0)))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (d <= 5.9e+51) {
		tmp = w0 * sqrt((1.0 - ((((((h / l) * D) * M_m) * (M_m * D)) / (2.0 * d)) / (2.0 * d))));
	} else {
		tmp = w0 * sqrt(fma((h * -0.25), (((((M_m * M_m) / d) * (D / d)) * D) / l), 1.0));
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (d <= 5.9e+51)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(h / l) * D) * M_m) * Float64(M_m * D)) / Float64(2.0 * d)) / Float64(2.0 * d)))));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(Float64(Float64(M_m * M_m) / d) * Float64(D / d)) * D) / l), 1.0)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[d, 5.9e+51], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(N[(N[(h / l), $MachinePrecision] * D), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 5.89999999999999983e51

   1. Initial program 81.6%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 81.5

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

   4. Applied rewrites 81.5%

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

   6. Applied rewrites 86.4%

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

   8. Applied rewrites 81.5%

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

   if 5.89999999999999983e51 < d

   1. Initial program 83.3%

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

   3. Taylor expanded in h around inf

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

   5. Applied rewrites 70.4%

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

   7. Applied rewrites 80.4%

      \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{\left(\frac{M \cdot M}{d} \cdot \frac{D}{d}\right) \cdot D}{\color{blue}{\ell}}, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 74.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 8.2 \cdot 10^{-159}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{\left(\frac{M\_m \cdot M\_m}{d} \cdot \frac{D}{d}\right) \cdot D}{\ell}, 1\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 8.2e-159)
   w0
   (*
    w0
    (sqrt (fma (* h -0.25) (/ (* (* (/ (* M_m M_m) d) (/ D d)) D) l) 1.0)))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 8.2e-159) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt(fma((h * -0.25), (((((M_m * M_m) / d) * (D / d)) * D) / l), 1.0));
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 8.2e-159)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(Float64(Float64(M_m * M_m) / d) * Float64(D / d)) * D) / l), 1.0)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 8.2e-159], w0, N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 8.20000000000000029e-159

   1. Initial program 81.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 70.4%

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

   if 8.20000000000000029e-159 < M

   1. Initial program 83.4%

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

   3. Taylor expanded in h around inf

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

   5. Applied rewrites 57.4%

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

   7. Applied rewrites 69.5%

      \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{\left(\frac{M \cdot M}{d} \cdot \frac{D}{d}\right) \cdot D}{\color{blue}{\ell}}, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 74.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 4.5 \cdot 10^{-166}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{\left(M\_m \cdot M\_m\right) \cdot D}{d} \cdot \frac{D}{\ell \cdot d}, 1\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 4.5e-166)
   w0
   (*
    w0
    (sqrt (fma (* h -0.25) (* (/ (* (* M_m M_m) D) d) (/ D (* l d))) 1.0)))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 4.5e-166) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt(fma((h * -0.25), ((((M_m * M_m) * D) / d) * (D / (l * d))), 1.0));
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 4.5e-166)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(Float64(M_m * M_m) * D) / d) * Float64(D / Float64(l * d))), 1.0)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 4.5e-166], w0, N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] * N[(D / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.4999999999999998e-166

   1. Initial program 81.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 70.0%

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

   if 4.4999999999999998e-166 < M

   1. Initial program 83.8%

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

   3. Taylor expanded in h around inf

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 69.7%

      \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{\left(M \cdot M\right) \cdot D}{d} \cdot \color{blue}{\frac{D}{\ell \cdot d}}, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 67.3% accurate, 157.0× speedup

Math

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

FPCore

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

C

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

Fortran

M_m =     private
NOTE: w0, M_m, D, 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, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.0%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 67.5%

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

Reproduce

herbie shell --seed 2025020 
(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))))))