Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 88.0%

Time: 7.5s

Alternatives: 11

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{d}\\ \mathbf{if}\;\sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 4 \cdot 10^{+102}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(t\_0 \cdot t\_0\right) \cdot 0.25\right) \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right) \cdot \left(\frac{\left(h \cdot M\_m\right) \cdot D\_m}{d} \cdot 0.5\right)}{\ell}}\\ \end{array} \end{array} \]

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 (/ (* M_m D_m) d)))
   (if (<=
        (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))))
        4e+102)
     (* w0 (sqrt (- 1.0 (* (* (* t_0 t_0) 0.25) (/ h l)))))
     (*
      w0
      (sqrt
       (-
        1.0
        (/
         (* (* (/ D_m d) (/ M_m 2.0)) (* (/ (* (* h M_m) D_m) d) 0.5))
         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 = (M_m * D_m) / d;
	double tmp;
	if (sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)))) <= 4e+102) {
		tmp = w0 * sqrt((1.0 - (((t_0 * t_0) * 0.25) * (h / l))));
	} else {
		tmp = w0 * sqrt((1.0 - ((((D_m / d) * (M_m / 2.0)) * ((((h * M_m) * D_m) / d) * 0.5)) / 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) :: tmp
    t_0 = (m_m * d_m) / d
    if (sqrt((1.0d0 - ((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)))) <= 4d+102) then
        tmp = w0 * sqrt((1.0d0 - (((t_0 * t_0) * 0.25d0) * (h / l))))
    else
        tmp = w0 * sqrt((1.0d0 - ((((d_m / d) * (m_m / 2.0d0)) * ((((h * m_m) * d_m) / d) * 0.5d0)) / 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 = (M_m * D_m) / d;
	double tmp;
	if (Math.sqrt((1.0 - (Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)))) <= 4e+102) {
		tmp = w0 * Math.sqrt((1.0 - (((t_0 * t_0) * 0.25) * (h / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((((D_m / d) * (M_m / 2.0)) * ((((h * M_m) * D_m) / d) * 0.5)) / 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 = (M_m * D_m) / d
	tmp = 0
	if math.sqrt((1.0 - (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)))) <= 4e+102:
		tmp = w0 * math.sqrt((1.0 - (((t_0 * t_0) * 0.25) * (h / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((((D_m / d) * (M_m / 2.0)) * ((((h * M_m) * D_m) / d) * 0.5)) / 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(Float64(M_m * D_m) / d)
	tmp = 0.0
	if (sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))) <= 4e+102)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(t_0 * t_0) * 0.25) * Float64(h / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(D_m / d) * Float64(M_m / 2.0)) * Float64(Float64(Float64(Float64(h * M_m) * D_m) / d) * 0.5)) / 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 = (M_m * D_m) / d;
	tmp = 0.0;
	if (sqrt((1.0 - ((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)))) <= 4e+102)
		tmp = w0 * sqrt((1.0 - (((t_0 * t_0) * 0.25) * (h / l))));
	else
		tmp = w0 * sqrt((1.0 - ((((D_m / d) * (M_m / 2.0)) * ((((h * M_m) * D_m) / d) * 0.5)) / 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[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4e+102], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.25), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(h * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))) < 3.99999999999999991e102

   1. Initial program 99.4%

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

   3. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 74.7

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   5. Applied rewrites 74.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot d} \cdot 0.25\right)} \cdot \frac{h}{\ell}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 74.7

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   7. Applied rewrites 74.7%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 99.4

          \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right) \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   9. Applied rewrites 99.4%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right) \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   if 3.99999999999999991e102 < (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))

   1. Initial program 41.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.3

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

   4. Applied rewrites 64.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 64.3

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

   6. Applied rewrites 64.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 71.5

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

   8. Applied rewrites 71.5%

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

   9. Taylor expanded in M around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 71.5

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

   11. Applied rewrites 71.5%

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

Alternative 2: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{2 \cdot d}\\ t_1 := \frac{M\_m \cdot D\_m}{d}\\ \mathbf{if}\;\sqrt{1 - {t\_0}^{2} \cdot \frac{h}{\ell}} \leq 4 \cdot 10^{+102}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(t\_1 \cdot t\_1\right) \cdot 0.25\right) \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(t\_0 \cdot \left(\frac{D\_m}{d} \cdot \left(0.5 \cdot M\_m\right)\right)\right) \cdot h}{\ell}}\\ \end{array} \end{array} \]

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 (/ (* M_m D_m) (* 2.0 d))) (t_1 (/ (* M_m D_m) d)))
   (if (<= (sqrt (- 1.0 (* (pow t_0 2.0) (/ h l)))) 4e+102)
     (* w0 (sqrt (- 1.0 (* (* (* t_1 t_1) 0.25) (/ h l)))))
     (* w0 (sqrt (- 1.0 (/ (* (* t_0 (* (/ D_m d) (* 0.5 M_m))) 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 = (M_m * D_m) / (2.0 * d);
	double t_1 = (M_m * D_m) / d;
	double tmp;
	if (sqrt((1.0 - (pow(t_0, 2.0) * (h / l)))) <= 4e+102) {
		tmp = w0 * sqrt((1.0 - (((t_1 * t_1) * 0.25) * (h / l))));
	} else {
		tmp = w0 * sqrt((1.0 - (((t_0 * ((D_m / d) * (0.5 * M_m))) * h) / 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 = (m_m * d_m) / (2.0d0 * d)
    t_1 = (m_m * d_m) / d
    if (sqrt((1.0d0 - ((t_0 ** 2.0d0) * (h / l)))) <= 4d+102) then
        tmp = w0 * sqrt((1.0d0 - (((t_1 * t_1) * 0.25d0) * (h / l))))
    else
        tmp = w0 * sqrt((1.0d0 - (((t_0 * ((d_m / d) * (0.5d0 * m_m))) * h) / 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 = (M_m * D_m) / (2.0 * d);
	double t_1 = (M_m * D_m) / d;
	double tmp;
	if (Math.sqrt((1.0 - (Math.pow(t_0, 2.0) * (h / l)))) <= 4e+102) {
		tmp = w0 * Math.sqrt((1.0 - (((t_1 * t_1) * 0.25) * (h / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((t_0 * ((D_m / d) * (0.5 * M_m))) * h) / 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 = (M_m * D_m) / (2.0 * d)
	t_1 = (M_m * D_m) / d
	tmp = 0
	if math.sqrt((1.0 - (math.pow(t_0, 2.0) * (h / l)))) <= 4e+102:
		tmp = w0 * math.sqrt((1.0 - (((t_1 * t_1) * 0.25) * (h / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - (((t_0 * ((D_m / d) * (0.5 * M_m))) * h) / 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(Float64(M_m * D_m) / Float64(2.0 * d))
	t_1 = Float64(Float64(M_m * D_m) / d)
	tmp = 0.0
	if (sqrt(Float64(1.0 - Float64((t_0 ^ 2.0) * Float64(h / l)))) <= 4e+102)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(t_1 * t_1) * 0.25) * Float64(h / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(t_0 * Float64(Float64(D_m / d) * Float64(0.5 * M_m))) * h) / 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 = (M_m * D_m) / (2.0 * d);
	t_1 = (M_m * D_m) / d;
	tmp = 0.0;
	if (sqrt((1.0 - ((t_0 ^ 2.0) * (h / l)))) <= 4e+102)
		tmp = w0 * sqrt((1.0 - (((t_1 * t_1) * 0.25) * (h / l))));
	else
		tmp = w0 * sqrt((1.0 - (((t_0 * ((D_m / d) * (0.5 * M_m))) * h) / 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[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(1.0 - N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4e+102], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * 0.25), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(t$95$0 * N[(N[(D$95$m / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))) < 3.99999999999999991e102

   1. Initial program 99.4%

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

   3. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 74.7

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   5. Applied rewrites 74.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot d} \cdot 0.25\right)} \cdot \frac{h}{\ell}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 74.7

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   7. Applied rewrites 74.7%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 99.4

          \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right) \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   9. Applied rewrites 99.4%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right) \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   if 3.99999999999999991e102 < (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))

   1. Initial program 41.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.3

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

   4. Applied rewrites 64.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 64.3

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

   6. Applied rewrites 64.3%

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

      1. lift-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. count-2-rev N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-*.f64 61.9

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

   8. Applied rewrites 61.9%

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

   9. Taylor expanded in M around 0

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

       1. lower-*.f64 61.9

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

   11. Applied rewrites 61.9%

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

Alternative 3: 86.5% accurate, 0.7× speedup

Math

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

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
 (let* ((t_0 (/ (* M_m D_m) d)))
   (if (<=
        (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))))
        INFINITY)
     (* w0 (sqrt (- 1.0 (* (* (* t_0 t_0) 0.25) (/ h 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 t_0 = (M_m * D_m) / d;
	double tmp;
	if (sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)))) <= ((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - (((t_0 * t_0) * 0.25) * (h / l))));
	} 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 t_0 = (M_m * D_m) / d;
	double tmp;
	if (Math.sqrt((1.0 - (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) * 0.25) * (h / 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):
	t_0 = (M_m * D_m) / d
	tmp = 0
	if math.sqrt((1.0 - (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) * 0.25) * (h / 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)
	t_0 = Float64(Float64(M_m * D_m) / d)
	tmp = 0.0
	if (sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))) <= Inf)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(t_0 * t_0) * 0.25) * Float64(h / 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)
	t_0 = (M_m * D_m) / d;
	tmp = 0.0;
	if (sqrt((1.0 - ((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)))) <= Inf)
		tmp = w0 * sqrt((1.0 - (((t_0 * t_0) * 0.25) * (h / 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_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], Infinity], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.25), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))) < +inf.0

   1. Initial program 87.5%

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

   3. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 68.4

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   5. Applied rewrites 68.4%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot d} \cdot 0.25\right)} \cdot \frac{h}{\ell}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 68.4

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   7. Applied rewrites 68.4%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 87.5

          \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right) \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   9. Applied rewrites 87.5%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right) \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   if +inf.0 < (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))

   1. Initial program 0.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 79.9%

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

Alternative 4: 83.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.0005:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(\left(M\_m \cdot D\_m\right) \cdot \frac{M\_m \cdot D\_m}{d \cdot d}\right) \cdot 0.25\right) \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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)) -0.0005)
   (*
    w0
    (sqrt
     (- 1.0 (* (* (* (* M_m D_m) (/ (* M_m D_m) (* d d))) 0.25) (/ h 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)) <= -0.0005) {
		tmp = w0 * sqrt((1.0 - ((((M_m * D_m) * ((M_m * D_m) / (d * d))) * 0.25) * (h / 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)) <= (-0.0005d0)) then
        tmp = w0 * sqrt((1.0d0 - ((((m_m * d_m) * ((m_m * d_m) / (d * d))) * 0.25d0) * (h / 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)) <= -0.0005) {
		tmp = w0 * Math.sqrt((1.0 - ((((M_m * D_m) * ((M_m * D_m) / (d * d))) * 0.25) * (h / 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)) <= -0.0005:
		tmp = w0 * math.sqrt((1.0 - ((((M_m * D_m) * ((M_m * D_m) / (d * d))) * 0.25) * (h / 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)) <= -0.0005)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M_m * D_m) * Float64(Float64(M_m * D_m) / Float64(d * d))) * 0.25) * Float64(h / 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)) <= -0.0005)
		tmp = w0 * sqrt((1.0 - ((((M_m * D_m) * ((M_m * D_m) / (d * d))) * 0.25) * (h / 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], -0.0005], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.0000000000000001e-4

   1. Initial program 68.5%

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

   3. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 47.7

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   5. Applied rewrites 47.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot d} \cdot 0.25\right)} \cdot \frac{h}{\ell}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 47.7

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   7. Applied rewrites 47.7%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d} \cdot 0.25\right) \cdot \frac{h}{\ell}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 55.9

          \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\left(M \cdot D\right) \cdot \frac{M \cdot D}{d \cdot d}\right) \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   9. Applied rewrites 55.9%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\left(M \cdot D\right) \cdot \frac{M \cdot D}{d \cdot d}\right) \cdot 0.25\right) \cdot \frac{h}{\ell}} \]

   if -5.0000000000000001e-4 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 97.0%

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

Alternative 5: 79.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.02:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \left(D\_m \cdot \frac{D\_m}{d \cdot d}\right) \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot h}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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)) -0.02)
   (*
    w0
    (sqrt (fma -0.25 (* (* D_m (/ D_m (* d d))) (/ (* (* M_m M_m) h) l)) 1.0)))
   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)) <= -0.02) {
		tmp = w0 * sqrt(fma(-0.25, ((D_m * (D_m / (d * d))) * (((M_m * M_m) * h) / l)), 1.0));
	} 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)) <= -0.02)
		tmp = Float64(w0 * sqrt(fma(-0.25, Float64(Float64(D_m * Float64(D_m / Float64(d * d))) * Float64(Float64(Float64(M_m * M_m) * h) / l)), 1.0)));
	else
		tmp = w0;
	end
	return 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], -0.02], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(D$95$m * N[(D$95$m / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $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)) < -0.0200000000000000004

   1. Initial program 68.1%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 44.8

          \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \]

   5. Applied rewrites 44.8%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow-prod-down N/A

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

      9. associate-*r* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 39.0

          \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}, 1\right)} \]

   7. Applied rewrites 39.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 46.0

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \left(D \cdot \frac{D}{d \cdot d}\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}, 1\right)} \]

   9. Applied rewrites 46.0%

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

   if -0.0200000000000000004 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.6%

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

Alternative 6: 81.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -50:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{\left(\left(D\_m \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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)) -50.0)
   (*
    w0
    (sqrt (fma -0.25 (/ (* (* (* D_m M_m) (* D_m M_m)) h) (* (* d d) l)) 1.0)))
   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)) <= -50.0) {
		tmp = w0 * sqrt(fma(-0.25, ((((D_m * M_m) * (D_m * M_m)) * h) / ((d * d) * l)), 1.0));
	} 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)) <= -50.0)
		tmp = Float64(w0 * sqrt(fma(-0.25, Float64(Float64(Float64(Float64(D_m * M_m) * Float64(D_m * M_m)) * h) / Float64(Float64(d * d) * l)), 1.0)));
	else
		tmp = w0;
	end
	return 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], -50.0], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + 1.0), $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)) < -50

   1. Initial program 67.8%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 45.3

          \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \]

   5. Applied rewrites 45.3%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 45.3

         \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \]

   7. Applied rewrites 45.3%

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

   if -50 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.2%

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

Alternative 7: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -50:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, D\_m \cdot \left(D\_m \cdot \left(\frac{h}{\left(d \cdot d\right) \cdot \ell} \cdot \left(M\_m \cdot M\_m\right)\right)\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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)) -50.0)
   (*
    w0
    (sqrt (fma -0.25 (* D_m (* D_m (* (/ h (* (* d d) l)) (* M_m M_m)))) 1.0)))
   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)) <= -50.0) {
		tmp = w0 * sqrt(fma(-0.25, (D_m * (D_m * ((h / ((d * d) * l)) * (M_m * M_m)))), 1.0));
	} 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)) <= -50.0)
		tmp = Float64(w0 * sqrt(fma(-0.25, Float64(D_m * Float64(D_m * Float64(Float64(h / Float64(Float64(d * d) * l)) * Float64(M_m * M_m)))), 1.0)));
	else
		tmp = w0;
	end
	return 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], -50.0], N[(w0 * N[Sqrt[N[(-0.25 * N[(D$95$m * N[(D$95$m * N[(N[(h / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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)) < -50

   1. Initial program 67.8%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 45.3

          \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)} \]

   5. Applied rewrites 45.3%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-0.25, \frac{{\left(D \cdot M\right)}^{2} \cdot h}{\left(d \cdot d\right) \cdot \ell}, 1\right)}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow-prod-down N/A

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

      9. associate-*r* N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 40.8

          \[\leadsto w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{\left(d \cdot d\right) \cdot \ell}\right), 1\right)} \]

   7. Applied rewrites 40.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 43.6

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

   9. Applied rewrites 43.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 50.1

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. associate-*r* N/A

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

       15. pow2 N/A

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

       16. lower-/.f64 N/A

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

       17. pow2 N/A

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

       18. lift-*.f64 N/A

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

       19. lift-*.f64 N/A

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

       20. pow2 N/A

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

       21. lift-*.f64 47.2

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

   11. Applied rewrites 47.2%

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

   if -50 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.2%

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

Alternative 8: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(D\_m \cdot \left(D\_m \cdot \left(\left(M\_m \cdot \left(h \cdot M\_m\right)\right) \cdot \frac{w0}{\left(d \cdot d\right) \cdot \ell}\right)\right), -0.125, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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+189)
   (fma (* D_m (* D_m (* (* M_m (* h M_m)) (/ w0 (* (* d d) l))))) -0.125 w0)
   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+189) {
		tmp = fma((D_m * (D_m * ((M_m * (h * M_m)) * (w0 / ((d * d) * l))))), -0.125, w0);
	} 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+189)
		tmp = fma(Float64(D_m * Float64(D_m * Float64(Float64(M_m * Float64(h * M_m)) * Float64(w0 / Float64(Float64(d * d) * l))))), -0.125, w0);
	else
		tmp = w0;
	end
	return 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+189], N[(N[(D$95$m * N[(D$95$m * N[(N[(M$95$m * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(w0 / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1e189

   1. Initial program 56.8%

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

   3. Taylor expanded in M around 0

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

      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. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. pow-prod-down N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 45.9

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

   5. Applied rewrites 45.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 N/A

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

   7. Applied rewrites 42.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 47.8

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. pow2 N/A

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

      19. lower-/.f64 N/A

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

   9. Applied rewrites 47.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 49.3

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

   11. Applied rewrites 49.3%

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

   if -1e189 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 85.7%

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

Alternative 9: 87.5% accurate, 1.7× speedup

Math

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

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) (/ M_m 2.0))))
   (* 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) * (M_m / 2.0);
	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) * (m_m / 2.0d0)
    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) * (M_m / 2.0);
	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) * (M_m / 2.0)
	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 / d) * Float64(M_m / 2.0))
	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) * (M_m / 2.0);
	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 / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $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 80.6%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 86.8

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

4. Applied rewrites 86.8%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-/.f64 86.8

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

6. Applied rewrites 86.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 89.2

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

8. Applied rewrites 89.2%

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

Alternative 10: 85.7% accurate, 1.8× speedup

Math

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

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
  (sqrt
   (-
    1.0
    (/ (* (* (* (/ D_m d) (/ M_m 2.0)) (* (/ D_m d) (* 0.5 M_m))) 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) {
	return w0 * sqrt((1.0 - (((((D_m / d) * (M_m / 2.0)) * ((D_m / d) * (0.5 * M_m))) * 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
    code = w0 * sqrt((1.0d0 - (((((d_m / d) * (m_m / 2.0d0)) * ((d_m / d) * (0.5d0 * m_m))) * 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) {
	return w0 * Math.sqrt((1.0 - (((((D_m / d) * (M_m / 2.0)) * ((D_m / d) * (0.5 * M_m))) * 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):
	return w0 * math.sqrt((1.0 - (((((D_m / d) * (M_m / 2.0)) * ((D_m / d) * (0.5 * M_m))) * 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)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D_m / d) * Float64(M_m / 2.0)) * Float64(Float64(D_m / d) * Float64(0.5 * M_m))) * 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)
	tmp = w0 * sqrt((1.0 - (((((D_m / d) * (M_m / 2.0)) * ((D_m / d) * (0.5 * M_m))) * 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_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.6%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 86.8

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

4. Applied rewrites 86.8%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-/.f64 86.8

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

6. Applied rewrites 86.8%

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

7. Taylor expanded in M around 0

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

   1. lower-*.f64 86.8

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

9. Applied rewrites 86.8%

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

Alternative 11: 68.4% accurate, 157.0× 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 80.6%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 65.3%

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

Reproduce

herbie shell --seed 2025037 
(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))))))