Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.2% → 87.8%

Time: 7.4s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.2% 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: 87.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{d + d}\\ t_1 := t\_0 \cdot M\_m\\ \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1.4 \cdot 10^{+38}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_1 \cdot \left(t\_1 \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_1 \cdot \frac{t\_0 \cdot \left(h \cdot M\_m\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (/ D_m (+ d d))) (t_1 (* t_0 M_m)))
   (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -1.4e+38)
     (* w0 (sqrt (- 1.0 (* t_1 (* t_1 (/ h l))))))
     (* w0 (sqrt (- 1.0 (* t_1 (/ (* t_0 (* h M_m)) l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m / (d + d);
	double t_1 = t_0 * M_m;
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1.4e+38) {
		tmp = w0 * sqrt((1.0 - (t_1 * (t_1 * (h / l)))));
	} else {
		tmp = w0 * sqrt((1.0 - (t_1 * ((t_0 * (h * M_m)) / l))));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(w0, m_m, d_m, h, l, d)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = d_m / (d + d)
    t_1 = t_0 * m_m
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-1.4d+38)) then
        tmp = w0 * sqrt((1.0d0 - (t_1 * (t_1 * (h / l)))))
    else
        tmp = w0 * sqrt((1.0d0 - (t_1 * ((t_0 * (h * m_m)) / l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m / (d + d);
	double t_1 = t_0 * M_m;
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1.4e+38) {
		tmp = w0 * Math.sqrt((1.0 - (t_1 * (t_1 * (h / l)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_1 * ((t_0 * (h * M_m)) / l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = D_m / (d + d)
	t_1 = t_0 * M_m
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1.4e+38:
		tmp = w0 * math.sqrt((1.0 - (t_1 * (t_1 * (h / l)))))
	else:
		tmp = w0 * math.sqrt((1.0 - (t_1 * ((t_0 * (h * M_m)) / l))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(D_m / Float64(d + d))
	t_1 = Float64(t_0 * M_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1.4e+38)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_1 * Float64(t_1 * Float64(h / l))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_1 * Float64(Float64(t_0 * Float64(h * M_m)) / l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = D_m / (d + d);
	t_1 = t_0 * M_m;
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -1.4e+38)
		tmp = w0 * sqrt((1.0 - (t_1 * (t_1 * (h / l)))));
	else
		tmp = w0 * sqrt((1.0 - (t_1 * ((t_0 * (h * M_m)) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(D$95$m / N[(d + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * M$95$m), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1.4e+38], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$1 * N[(t$95$1 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$1 * N[(N[(t$95$0 * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.4e38

   1. Initial program 63.7%

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

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

   3. Applied rewrites 64.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 67.8%

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

   if -1.4e38 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.5%

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

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

   3. Applied rewrites 95.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-*.f64 96.7

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-+.f64 96.7

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

   5. Applied rewrites 96.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 96.8

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.2

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

   7. Applied rewrites 96.2%

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

Alternative 2: 87.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{d + d} \cdot M\_m\\ \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq \infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(t\_0 \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \frac{0.5 \cdot \left(\left(M\_m \cdot D\_m\right) \cdot h\right)}{\ell \cdot d}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* (/ D_m (+ d d)) M_m)))
   (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) INFINITY)
     (* w0 (sqrt (- 1.0 (* t_0 (* t_0 (/ h l))))))
     (* w0 (sqrt (- 1.0 (* t_0 (/ (* 0.5 (* (* M_m D_m) h)) (* l d)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (D_m / (d + d)) * M_m;
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= ((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * ((0.5 * ((M_m * D_m) * h)) / (l * d)))));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (D_m / (d + d)) * M_m;
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * ((0.5 * ((M_m * D_m) * h)) / (l * d)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = (D_m / (d + d)) * M_m
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= math.inf:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))))
	else:
		tmp = w0 * math.sqrt((1.0 - (t_0 * ((0.5 * ((M_m * D_m) * h)) / (l * d)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(Float64(D_m / Float64(d + d)) * M_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= Inf)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(h / l))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(0.5 * Float64(Float64(M_m * D_m) * h)) / Float64(l * d))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = (D_m / (d + d)) * M_m;
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= Inf)
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	else
		tmp = w0 * sqrt((1.0 - (t_0 * ((0.5 * ((M_m * D_m) * h)) / (l * d)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[(D$95$m / N[(d + d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], Infinity], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.2%

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

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

   3. Applied rewrites 87.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 88.7%

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

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

   1. Initial program 0.0%

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

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

   3. Applied rewrites 67.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-*.f64 82.2

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-+.f64 82.2

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

   5. Applied rewrites 82.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 83.0

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 81.6

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

   7. Applied rewrites 81.6%

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

   8. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 70.2

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

   10. Applied rewrites 70.2%

       \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\frac{0.5 \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\ell \cdot d}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(w0, m_m, d_m, h, l, d)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    t_0 = (d_m / (d + d)) * m_m
    code = w0 * sqrt((1.0d0 - ((t_0 * (t_0 * h)) / l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

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

3. Applied rewrites 85.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. lower-*.f64 87.7

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

   13. lift-*.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. lift-+.f64 87.7

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

5. Applied rewrites 87.7%

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

Alternative 4: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{d + d}\\ t_1 := M\_m \cdot t\_0\\ \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq \infty:\\ \;\;\;\;\sqrt{1 - \left(t\_1 \cdot t\_1\right) \cdot \frac{h}{\ell}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(t\_0 \cdot M\_m\right) \cdot \frac{0.5 \cdot \left(\left(M\_m \cdot D\_m\right) \cdot h\right)}{\ell \cdot d}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (/ D_m (+ d d))) (t_1 (* M_m t_0)))
   (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) INFINITY)
     (* (sqrt (- 1.0 (* (* t_1 t_1) (/ h l)))) w0)
     (*
      w0
      (sqrt (- 1.0 (* (* t_0 M_m) (/ (* 0.5 (* (* M_m D_m) h)) (* l d)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m / (d + d);
	double t_1 = M_m * t_0;
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= ((double) INFINITY)) {
		tmp = sqrt((1.0 - ((t_1 * t_1) * (h / l)))) * w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((t_0 * M_m) * ((0.5 * ((M_m * D_m) * h)) / (l * d)))));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m / (d + d);
	double t_1 = M_m * t_0;
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((1.0 - ((t_1 * t_1) * (h / l)))) * w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((t_0 * M_m) * ((0.5 * ((M_m * D_m) * h)) / (l * d)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = D_m / (d + d)
	t_1 = M_m * t_0
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= math.inf:
		tmp = math.sqrt((1.0 - ((t_1 * t_1) * (h / l)))) * w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((t_0 * M_m) * ((0.5 * ((M_m * D_m) * h)) / (l * d)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(D_m / Float64(d + d))
	t_1 = Float64(M_m * t_0)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= Inf)
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(t_1 * t_1) * Float64(h / l)))) * w0);
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_0 * M_m) * Float64(Float64(0.5 * Float64(Float64(M_m * D_m) * h)) / Float64(l * d))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = D_m / (d + d);
	t_1 = M_m * t_0;
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= Inf)
		tmp = sqrt((1.0 - ((t_1 * t_1) * (h / l)))) * w0;
	else
		tmp = w0 * sqrt((1.0 - ((t_0 * M_m) * ((0.5 * ((M_m * D_m) * h)) / (l * d)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(D$95$m / N[(d + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(M$95$m * t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[Sqrt[N[(1.0 - N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(t$95$0 * M$95$m), $MachinePrecision] * N[(N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      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. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 87.4%

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

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

   1. Initial program 0.0%

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

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

   3. Applied rewrites 67.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-*.f64 82.2

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-+.f64 82.2

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

   5. Applied rewrites 82.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 83.0

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 81.6

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

   7. Applied rewrites 81.6%

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

   8. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 70.2

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

   10. Applied rewrites 70.2%

       \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\frac{0.5 \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\ell \cdot d}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 85.8% accurate, 0.5× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))) 1.0)
   w0
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* (/ D_m (+ d d)) M_m) (/ (* 0.5 (* (* M_m D_m) h)) (* l d))))))))

C

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

Fortran

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

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

real(8) function code(w0, m_m, d_m, h, l, d)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((1.0d0 - ((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l))) <= 1.0d0) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - (((d_m / (d + d)) * m_m) * ((0.5d0 * ((m_m * d_m) * h)) / (l * d)))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D$95$m / N[(d + d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 99.7%

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

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

   1. Initial program 52.2%

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

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

   3. Applied rewrites 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-*.f64 70.3

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-+.f64 70.3

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

   5. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 72.3

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 63.9

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

   10. Applied rewrites 63.9%

       \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \color{blue}{\frac{0.5 \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\ell \cdot d}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 84.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -200000000:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\left(\frac{D\_m}{d} \cdot \left(\left(\left(M\_m \cdot D\_m\right) \cdot M\_m\right) \cdot \frac{h}{d}\right)\right) \cdot -0.25}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -200000000.0)
   (* w0 (sqrt (/ (* (* (/ D_m d) (* (* (* M_m D_m) M_m) (/ h d))) -0.25) l)))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -200000000.0) {
		tmp = w0 * sqrt(((((D_m / d) * (((M_m * D_m) * M_m) * (h / d))) * -0.25) / l));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(w0, m_m, d_m, h, l, d)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-200000000.0d0)) then
        tmp = w0 * sqrt(((((d_m / d) * (((m_m * d_m) * m_m) * (h / d))) * (-0.25d0)) / l))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -200000000.0) {
		tmp = w0 * Math.sqrt(((((D_m / d) * (((M_m * D_m) * M_m) * (h / d))) * -0.25) / l));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -200000000.0:
		tmp = w0 * math.sqrt(((((D_m / d) * (((M_m * D_m) * M_m) * (h / d))) * -0.25) / l))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -200000000.0)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(Float64(D_m / d) * Float64(Float64(Float64(M_m * D_m) * M_m) * Float64(h / d))) * -0.25) / l)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -200000000.0)
		tmp = w0 * sqrt(((((D_m / d) * (((M_m * D_m) * M_m) * (h / d))) * -0.25) / l));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -200000000.0], N[(w0 * N[Sqrt[N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] / l), $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)) < -2e8

   1. Initial program 64.7%

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

   2. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 38.8

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

   4. Applied rewrites 38.8%

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

   5. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 40.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 48.6

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

   9. Applied rewrites 48.6%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lift-/.f64 48.6

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

       6. lift-/.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-/l* N/A

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

       9. lower-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. lower-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 57.0

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

   11. Applied rewrites 57.0%

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

   if -2e8 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.4%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 95.7%

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

Alternative 7: 82.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -200000000:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\left(M\_m \cdot D\_m\right) \cdot \left(\left(M\_m \cdot D\_m\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -200000000.0)
   (* w0 (sqrt (* (/ (* (* M_m D_m) (* (* M_m D_m) h)) (* (* d d) l)) -0.25)))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -200000000.0) {
		tmp = w0 * sqrt(((((M_m * D_m) * ((M_m * D_m) * h)) / ((d * d) * l)) * -0.25));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(w0, m_m, d_m, h, l, d)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-200000000.0d0)) then
        tmp = w0 * sqrt(((((m_m * d_m) * ((m_m * d_m) * h)) / ((d * d) * l)) * (-0.25d0)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -200000000.0) {
		tmp = w0 * Math.sqrt(((((M_m * D_m) * ((M_m * D_m) * h)) / ((d * d) * l)) * -0.25));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -200000000.0:
		tmp = w0 * math.sqrt(((((M_m * D_m) * ((M_m * D_m) * h)) / ((d * d) * l)) * -0.25))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -200000000.0)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(Float64(M_m * D_m) * Float64(Float64(M_m * D_m) * h)) / Float64(Float64(d * d) * l)) * -0.25)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -200000000.0)
		tmp = w0 * sqrt(((((M_m * D_m) * ((M_m * D_m) * h)) / ((d * d) * l)) * -0.25));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -200000000.0], N[(w0 * N[Sqrt[N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.25), $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)) < -2e8

   1. Initial program 64.7%

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

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

   3. Applied rewrites 64.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-*.f64 66.7

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-+.f64 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in M around inf

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

   8. Applied rewrites 52.2%

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

   if -2e8 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.4%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 95.7%

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

Alternative 8: 79.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+301}:\\ \;\;\;\;\left(\left(h \cdot M\_m\right) \cdot \left(M\_m \cdot \left(\left(w0 \cdot D\_m\right) \cdot \frac{D\_m}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right) \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -1e+301)
   (* (* (* h M_m) (* M_m (* (* w0 D_m) (/ D_m (* (* d d) l))))) -0.125)
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e+301) {
		tmp = ((h * M_m) * (M_m * ((w0 * D_m) * (D_m / ((d * d) * l))))) * -0.125;
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(w0, m_m, d_m, h, l, d)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-1d+301)) then
        tmp = ((h * m_m) * (m_m * ((w0 * d_m) * (d_m / ((d * d) * l))))) * (-0.125d0)
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e+301) {
		tmp = ((h * M_m) * (M_m * ((w0 * D_m) * (D_m / ((d * d) * l))))) * -0.125;
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e+301:
		tmp = ((h * M_m) * (M_m * ((w0 * D_m) * (D_m / ((d * d) * l))))) * -0.125
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+301)
		tmp = Float64(Float64(Float64(h * M_m) * Float64(M_m * Float64(Float64(w0 * D_m) * Float64(D_m / Float64(Float64(d * d) * l))))) * -0.125);
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -1e+301)
		tmp = ((h * M_m) * (M_m * ((w0 * D_m) * (D_m / ((d * d) * l))))) * -0.125;
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+301], N[(N[(N[(h * M$95$m), $MachinePrecision] * N[(M$95$m * N[(N[(w0 * D$95$m), $MachinePrecision] * N[(D$95$m / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $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)) < -1.00000000000000005e301

   1. Initial program 55.3%

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

   2. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 41.8%

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

   5. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

   7. Applied rewrites 41.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. pow2 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. pow2 N/A

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

   9. Applied rewrites 42.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*r/ N/A

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

       11. pow2 N/A

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

       12. *-commutative N/A

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

       13. associate-*r* N/A

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

       14. pow2 N/A

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

       15. *-commutative N/A

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

       16. associate-/l* N/A

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

   11. Applied rewrites 49.8%

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

   if -1.00000000000000005e301 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.3%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 88.2%

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

Alternative 9: 78.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+301}:\\ \;\;\;\;\left(\left(M\_m \cdot \left(\left(h \cdot M\_m\right) \cdot w0\right)\right) \cdot \frac{D\_m \cdot D\_m}{\left(d \cdot d\right) \cdot \ell}\right) \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -1e+301)
   (* (* (* M_m (* (* h M_m) w0)) (/ (* D_m D_m) (* (* d d) l))) -0.125)
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e+301) {
		tmp = ((M_m * ((h * M_m) * w0)) * ((D_m * D_m) / ((d * d) * l))) * -0.125;
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(w0, m_m, d_m, h, l, d)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-1d+301)) then
        tmp = ((m_m * ((h * m_m) * w0)) * ((d_m * d_m) / ((d * d) * l))) * (-0.125d0)
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e+301) {
		tmp = ((M_m * ((h * M_m) * w0)) * ((D_m * D_m) / ((d * d) * l))) * -0.125;
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e+301:
		tmp = ((M_m * ((h * M_m) * w0)) * ((D_m * D_m) / ((d * d) * l))) * -0.125
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+301)
		tmp = Float64(Float64(Float64(M_m * Float64(Float64(h * M_m) * w0)) * Float64(Float64(D_m * D_m) / Float64(Float64(d * d) * l))) * -0.125);
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -1e+301)
		tmp = ((M_m * ((h * M_m) * w0)) * ((D_m * D_m) / ((d * d) * l))) * -0.125;
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+301], N[(N[(N[(M$95$m * N[(N[(h * M$95$m), $MachinePrecision] * w0), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $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)) < -1.00000000000000005e301

   1. Initial program 55.3%

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

   2. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 41.8%

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

   5. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

   7. Applied rewrites 41.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 48.2

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

   9. Applied rewrites 48.2%

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

   if -1.00000000000000005e301 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.3%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 88.2%

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

Alternative 10: 77.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.1%

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

   2. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 41.9%

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

   5. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

   7. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. pow2 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. pow2 N/A

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

   9. Applied rewrites 42.2%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-/l* N/A

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

       9. pow2 N/A

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

       10. lower-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 N/A

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

       15. pow2 N/A

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

       16. lift-*.f64 N/A

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

       17. lift-*.f64 42.6

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

   11. Applied rewrites 42.6%

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

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

   1. Initial program 89.3%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 88.1%

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

Alternative 11: 77.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.1%

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

   2. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 41.9%

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

   5. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

   7. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. pow2 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. pow2 N/A

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

   9. Applied rewrites 42.2%

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

   11. Applied rewrites 42.3%

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

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

   1. Initial program 89.3%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 88.1%

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

Alternative 12: 68.4% accurate, 39.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(w0, m_m, d_m, h, l, d)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

2. Taylor expanded in M around 0

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

4. Applied rewrites 68.4%

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

Reproduce

herbie shell --seed 2025120 
(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))))))