Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.7% → 89.0%

Time: 6.5s

Alternatives: 8

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot \frac{M}{2}\\ w0 \cdot \sqrt{1 - t\_0 \cdot \frac{t\_0 \cdot h}{\ell}} \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (/ D d) (/ M 2.0))))
   (* w0 (sqrt (- 1.0 (* t_0 (/ (* t_0 h) l)))))))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (D / d) * (M / 2.0);
	return w0 * sqrt((1.0 - (t_0 * ((t_0 * h) / l))));
}

Fortran

NOTE: w0, 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, 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
    real(8) :: t_0
    t_0 = (d / d_1) * (m / 2.0d0)
    code = w0 * sqrt((1.0d0 - (t_0 * ((t_0 * h) / l))))
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (D / d) * (M / 2.0);
	return w0 * Math.sqrt((1.0 - (t_0 * ((t_0 * h) / l))));
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	t_0 = (D / d) * (M / 2.0)
	return w0 * math.sqrt((1.0 - (t_0 * ((t_0 * h) / l))))

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(D / d) * Float64(M / 2.0))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(t_0 * h) / l)))))
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp = code(w0, M, D, h, l, d)
	t_0 = (D / d) * (M / 2.0);
	tmp = w0 * sqrt((1.0 - (t_0 * ((t_0 * h) / l))));
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(t$95$0 * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.0%

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

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 87.2

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

4. Applied rewrites 87.2%

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

   1. lift-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. unpow1 N/A

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

   5. metadata-eval N/A

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

   6. pow-neg N/A

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

   7. lower-/.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-/.f64 87.2

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

6. Applied rewrites 87.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-flip N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. unpow1 N/A

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

   15. pow2 N/A

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

   16. associate-*l* N/A

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

8. Applied rewrites 83.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 91.7

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

10. Applied rewrites 91.7%

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

Alternative 2: 87.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e-12)
   (*
    w0
    (sqrt (- 1.0 (* (* (/ D d) (/ M 2.0)) (* (* (/ D d) (* 0.5 M)) (/ h l))))))
   w0))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-12) {
		tmp = w0 * sqrt((1.0 - (((D / d) * (M / 2.0)) * (((D / d) * (0.5 * M)) * (h / l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: w0, 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, 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
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d-12)) then
        tmp = w0 * sqrt((1.0d0 - (((d / d_1) * (m / 2.0d0)) * (((d / d_1) * (0.5d0 * m)) * (h / l)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-12) {
		tmp = w0 * Math.sqrt((1.0 - (((D / d) * (M / 2.0)) * (((D / d) * (0.5 * M)) * (h / l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-12:
		tmp = w0 * math.sqrt((1.0 - (((D / d) * (M / 2.0)) * (((D / d) * (0.5 * M)) * (h / l)))))
	else:
		tmp = w0
	return tmp

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e-12)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D / d) * Float64(M / 2.0)) * Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * Float64(h / l))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e-12)
		tmp = w0 * sqrt((1.0 - (((D / d) * (M / 2.0)) * (((D / d) * (0.5 * M)) * (h / l)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-12], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $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)) < -1.99999999999999996e-12

   1. Initial program 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 72.4

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

   4. Applied rewrites 72.4%

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

      1. lift-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. pow-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 72.4

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

   6. Applied rewrites 72.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. unpow1 N/A

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

      15. pow2 N/A

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

      16. associate-*l* N/A

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

   8. Applied rewrites 78.2%

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

   9. Taylor expanded in M around 0

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

       1. lower-*.f64 78.2

          \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\left(\frac{D}{d} \cdot \left(0.5 \cdot \color{blue}{M}\right)\right) \cdot \frac{h}{\ell}\right)} \]

   11. Applied rewrites 78.2%

       \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\left(\frac{D}{d} \cdot \color{blue}{\left(0.5 \cdot M\right)}\right) \cdot \frac{h}{\ell}\right)} \]

   if -1.99999999999999996e-12 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 83.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 94.0%

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

Alternative 3: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)\right) \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e-12)
   (*
    w0
    (sqrt (- 1.0 (* (* (* (/ M 2.0) (/ D d)) (* (* 0.5 M) (/ D d))) (/ h l)))))
   w0))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-12) {
		tmp = w0 * sqrt((1.0 - ((((M / 2.0) * (D / d)) * ((0.5 * M) * (D / d))) * (h / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: w0, 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, 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
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d-12)) then
        tmp = w0 * sqrt((1.0d0 - ((((m / 2.0d0) * (d / d_1)) * ((0.5d0 * m) * (d / d_1))) * (h / l))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-12) {
		tmp = w0 * Math.sqrt((1.0 - ((((M / 2.0) * (D / d)) * ((0.5 * M) * (D / d))) * (h / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-12:
		tmp = w0 * math.sqrt((1.0 - ((((M / 2.0) * (D / d)) * ((0.5 * M) * (D / d))) * (h / l))))
	else:
		tmp = w0
	return tmp

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e-12)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M / 2.0) * Float64(D / d)) * Float64(Float64(0.5 * M) * Float64(D / d))) * Float64(h / l)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e-12)
		tmp = w0 * sqrt((1.0 - ((((M / 2.0) * (D / d)) * ((0.5 * M) * (D / d))) * (h / l))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-12], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $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)) < -1.99999999999999996e-12

   1. Initial program 73.5%

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

      1. lift-pow.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. lift-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 72.5

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in M around 0

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

      1. lower-*.f64 72.5

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\left(0.5 \cdot \color{blue}{M}\right) \cdot \frac{D}{d}\right)\right) \cdot \frac{h}{\ell}} \]

   7. Applied rewrites 72.5%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\color{blue}{\left(0.5 \cdot M\right)} \cdot \frac{D}{d}\right)\right) \cdot \frac{h}{\ell}} \]

   if -1.99999999999999996e-12 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 83.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 94.0%

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

Alternative 4: 79.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+133}:\\ \;\;\;\;w0 \cdot \sqrt{\left(\left(\frac{-0.25}{d \cdot d} \cdot \frac{\left(D \cdot D\right) \cdot h}{\ell}\right) \cdot M\right) \cdot M}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+133)
   (* w0 (sqrt (* (* (* (/ -0.25 (* d d)) (/ (* (* D D) h) l)) M) M)))
   w0))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+133) {
		tmp = w0 * sqrt(((((-0.25 / (d * d)) * (((D * D) * h) / l)) * M) * M));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: w0, 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, 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
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d+133)) then
        tmp = w0 * sqrt((((((-0.25d0) / (d_1 * d_1)) * (((d * d) * h) / l)) * m) * m))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+133) {
		tmp = w0 * Math.sqrt(((((-0.25 / (d * d)) * (((D * D) * h) / l)) * M) * M));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+133:
		tmp = w0 * math.sqrt(((((-0.25 / (d * d)) * (((D * D) * h) / l)) * M) * M))
	else:
		tmp = w0
	return tmp

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+133)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(Float64(-0.25 / Float64(d * d)) * Float64(Float64(Float64(D * D) * h) / l)) * M) * M)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+133)
		tmp = w0 * sqrt(((((-0.25 / (d * d)) * (((D * D) * h) / l)) * M) * M));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+133], N[(w0 * N[Sqrt[N[(N[(N[(N[(-0.25 / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * M), $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)) < -2e133

   1. Initial program 70.1%

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

   3. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 53.6

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

   8. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   10. Applied rewrites 57.1%

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

   if -2e133 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 84.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.7%

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

Alternative 5: 79.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+147}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{d \cdot \left(d \cdot \ell\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+147)
   (* w0 (fma -0.125 (/ (* (* (* D M) (* D M)) h) (* d (* d l))) 1.0))
   w0))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+147) {
		tmp = w0 * fma(-0.125, ((((D * M) * (D * M)) * h) / (d * (d * l))), 1.0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+147)
		tmp = Float64(w0 * fma(-0.125, Float64(Float64(Float64(Float64(D * M) * Float64(D * M)) * h) / Float64(d * Float64(d * l))), 1.0));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+147], N[(w0 * N[(-0.125 * N[(N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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)) < -5.0000000000000002e147

   1. Initial program 69.7%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 54.9

          \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right) \]

   5. Applied rewrites 54.9%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 54.9

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

   7. Applied rewrites 54.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 57.6

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

   9. Applied rewrites 57.6%

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

   if -5.0000000000000002e147 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 84.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.3%

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

Alternative 6: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+147}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+147)
   (* w0 (fma -0.125 (/ (* (* M D) (* (* M D) h)) (* (* d d) l)) 1.0))
   w0))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+147) {
		tmp = w0 * fma(-0.125, (((M * D) * ((M * D) * h)) / ((d * d) * l)), 1.0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+147)
		tmp = Float64(w0 * fma(-0.125, Float64(Float64(Float64(M * D) * Float64(Float64(M * D) * h)) / Float64(Float64(d * d) * l)), 1.0));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+147], N[(w0 * N[(-0.125 * N[(N[(N[(M * D), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + 1.0), $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)) < -5.0000000000000002e147

   1. Initial program 69.7%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 54.9

          \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right) \]

   5. Applied rewrites 54.9%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 54.9

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

   7. Applied rewrites 54.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 56.3

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

   9. Applied rewrites 56.3%

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

   if -5.0000000000000002e147 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 84.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.3%

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

Alternative 7: 82.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{h \cdot M}{\ell}\right) \cdot 0.5\right)} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (*
  w0
  (sqrt (- 1.0 (* (* (/ D d) (/ M 2.0)) (* (* (/ D d) (/ (* h M) l)) 0.5))))))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (((D / d) * (M / 2.0)) * (((D / d) * ((h * M) / l)) * 0.5))));
}

Fortran

NOTE: w0, 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, 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 - (((d / d_1) * (m / 2.0d0)) * (((d / d_1) * ((h * m) / l)) * 0.5d0))))
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (((D / d) * (M / 2.0)) * (((D / d) * ((h * M) / l)) * 0.5))));
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (((D / d) * (M / 2.0)) * (((D / d) * ((h * M) / l)) * 0.5))))

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D / d) * Float64(M / 2.0)) * Float64(Float64(Float64(D / d) * Float64(Float64(h * M) / l)) * 0.5)))))
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - (((D / d) * (M / 2.0)) * (((D / d) * ((h * M) / l)) * 0.5))));
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D / d), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 87.2

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

4. Applied rewrites 87.2%

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

   1. lift-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. unpow1 N/A

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

   5. metadata-eval N/A

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

   6. pow-neg N/A

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

   7. lower-/.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-/.f64 87.2

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

6. Applied rewrites 87.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-flip N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. unpow1 N/A

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

   15. pow2 N/A

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

   16. associate-*l* N/A

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

8. Applied rewrites 83.6%

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

9. Taylor expanded in M around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. times-frac N/A

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

    4. lower-*.f64 N/A

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

    5. lift-/.f64 N/A

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

    6. lower-/.f64 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 84.0

       \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{h \cdot M}{\ell}\right) \cdot 0.5\right)} \]

11. Applied rewrites 84.0%

    \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \color{blue}{\left(\left(\frac{D}{d} \cdot \frac{h \cdot M}{\ell}\right) \cdot 0.5\right)}} \]
12. Add Preprocessing

Alternative 8: 67.8% accurate, 157.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: w0, 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, 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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 80.0%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 65.1%

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

Reproduce

herbie shell --seed 2025075 
(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))))))