Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 87.7%

Time: 6.8s

Alternatives: 11

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\frac{-0.5}{d} \cdot \left(D\_m \cdot M\_m\right)\right), \frac{M\_m}{d + d} \cdot D\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\left(D\_m \cdot M\_m\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \left(D\_m \cdot M\_m\right)}{d + d}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) 0.0)
   (*
    w0
    (sqrt
     (fma (* (/ h l) (* (/ -0.5 d) (* D_m M_m))) (* (/ M_m (+ d d)) D_m) 1.0)))
   (*
    w0
    (sqrt
     (-
      1.0
      (/ (* (/ (* (* D_m M_m) h) (* (+ d d) l)) (* D_m M_m)) (+ d d)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= 0.0) {
		tmp = w0 * sqrt(fma(((h / l) * ((-0.5 / d) * (D_m * M_m))), ((M_m / (d + d)) * D_m), 1.0));
	} else {
		tmp = w0 * sqrt((1.0 - (((((D_m * M_m) * h) / ((d + d) * l)) * (D_m * M_m)) / (d + d))));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= 0.0)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(h / l) * Float64(Float64(-0.5 / d) * Float64(D_m * M_m))), Float64(Float64(M_m / Float64(d + d)) * D_m), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D_m * M_m) * h) / Float64(Float64(d + d) * l)) * Float64(D_m * M_m)) / Float64(d + d)))));
	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], 0.0], N[(w0 * N[Sqrt[N[(N[(N[(h / l), $MachinePrecision] * N[(N[(-0.5 / d), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / N[(d + d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 0.0

   1. Initial program 81.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 83.2%

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

   if 0.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 81.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 84.3%

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

Alternative 2: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{d}\\ \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 5 \cdot 10^{+260}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t\_0 \cdot t\_0}{4} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(h \cdot \left(M\_m \cdot D\_m\right)\right) \cdot -0.5}{\left(\ell \cdot \left(d + d\right)\right) \cdot d} \cdot M\_m, D\_m, 1\right)} \cdot w0\\ \end{array} \end{array} \]

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 (<=
        (* w0 (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)))))
        5e+260)
     (* w0 (sqrt (- 1.0 (* (/ (* t_0 t_0) 4.0) (/ h l)))))
     (*
      (sqrt
       (fma
        (* (/ (* (* h (* M_m D_m)) -0.5) (* (* l (+ d d)) d)) M_m)
        D_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 t_0 = (M_m * D_m) / d;
	double tmp;
	if ((w0 * sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 5e+260) {
		tmp = w0 * sqrt((1.0 - (((t_0 * t_0) / 4.0) * (h / l))));
	} else {
		tmp = sqrt(fma(((((h * (M_m * D_m)) * -0.5) / ((l * (d + d)) * d)) * M_m), D_m, 1.0)) * 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 (Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))) <= 5e+260)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(t_0 * t_0) / 4.0) * Float64(h / l)))));
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(h * Float64(M_m * D_m)) * -0.5) / Float64(Float64(l * Float64(d + d)) * d)) * M_m), D_m, 1.0)) * 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_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[N[(w0 * 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]), $MachinePrecision], 5e+260], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / N[(N[(l * N[(d + d), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (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))))) < 4.9999999999999996e260

   1. Initial program 81.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. cosh-0-rev N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 80.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 80.0

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

   5. Applied rewrites 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 81.3

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 81.3

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

   7. Applied rewrites 81.3%

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

   if 4.9999999999999996e260 < (*.f64 w0 (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 81.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 84.6%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}, \frac{-0.5}{d} \cdot \left(D \cdot M\right), 1\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 84.6

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

   5. Applied rewrites 79.2%

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

Alternative 3: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{d}\\ \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t\_0 \cdot t\_0}{4} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\left(D\_m \cdot M\_m\right) \cdot \left(0.25 \cdot M\_m\right)\right) \cdot \left(h \cdot D\_m\right)}{d \cdot \left(\ell \cdot d\right)}}\\ \end{array} \end{array} \]

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 (<=
        (* w0 (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)))))
        2e+302)
     (* w0 (sqrt (- 1.0 (* (/ (* t_0 t_0) 4.0) (/ h l)))))
     (*
      w0
      (sqrt
       (-
        1.0
        (/ (* (* (* D_m M_m) (* 0.25 M_m)) (* h D_m)) (* d (* l d)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (M_m * D_m) / d;
	double tmp;
	if ((w0 * sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 2e+302) {
		tmp = w0 * sqrt((1.0 - (((t_0 * t_0) / 4.0) * (h / l))));
	} else {
		tmp = w0 * sqrt((1.0 - ((((D_m * M_m) * (0.25 * M_m)) * (h * D_m)) / (d * (l * d)))));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(w0, m_m, d_m, h, l, d)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m_m * d_m) / d
    if ((w0 * sqrt((1.0d0 - ((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l))))) <= 2d+302) then
        tmp = w0 * sqrt((1.0d0 - (((t_0 * t_0) / 4.0d0) * (h / l))))
    else
        tmp = w0 * sqrt((1.0d0 - ((((d_m * m_m) * (0.25d0 * m_m)) * (h * d_m)) / (d * (l * d)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (M_m * D_m) / d;
	double tmp;
	if ((w0 * Math.sqrt((1.0 - (Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 2e+302) {
		tmp = w0 * Math.sqrt((1.0 - (((t_0 * t_0) / 4.0) * (h / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((((D_m * M_m) * (0.25 * M_m)) * (h * D_m)) / (d * (l * d)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = (M_m * D_m) / d
	tmp = 0
	if (w0 * math.sqrt((1.0 - (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 2e+302:
		tmp = w0 * math.sqrt((1.0 - (((t_0 * t_0) / 4.0) * (h / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((((D_m * M_m) * (0.25 * M_m)) * (h * D_m)) / (d * (l * d)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(Float64(M_m * D_m) / d)
	tmp = 0.0
	if (Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))) <= 2e+302)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(t_0 * t_0) / 4.0) * Float64(h / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(D_m * M_m) * Float64(0.25 * M_m)) * Float64(h * D_m)) / Float64(d * Float64(l * d))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = (M_m * D_m) / d;
	tmp = 0.0;
	if ((w0 * sqrt((1.0 - ((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l))))) <= 2e+302)
		tmp = w0 * sqrt((1.0 - (((t_0 * t_0) / 4.0) * (h / l))));
	else
		tmp = w0 * sqrt((1.0 - ((((D_m * M_m) * (0.25 * M_m)) * (h * D_m)) / (d * (l * d)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[N[(w0 * 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]), $MachinePrecision], 2e+302], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(0.25 * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))) < 2.0000000000000002e302

   1. Initial program 81.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. cosh-0-rev N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 80.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 80.0

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

   5. Applied rewrites 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 81.3

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 81.3

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

   7. Applied rewrites 81.3%

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

   if 2.0000000000000002e302 < (*.f64 w0 (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 81.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 77.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 71.6

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

   7. Applied rewrites 71.6%

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

Alternative 4: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{d} \cdot M\_m\\ \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t\_0 \cdot t\_0}{4} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\left(D\_m \cdot M\_m\right) \cdot \left(0.25 \cdot M\_m\right)\right) \cdot \left(h \cdot D\_m\right)}{d \cdot \left(\ell \cdot d\right)}}\\ \end{array} \end{array} \]

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)))
   (if (<=
        (* w0 (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)))))
        2e+302)
     (* w0 (sqrt (- 1.0 (* (/ (* t_0 t_0) 4.0) (/ h l)))))
     (*
      w0
      (sqrt
       (-
        1.0
        (/ (* (* (* D_m M_m) (* 0.25 M_m)) (* h D_m)) (* d (* l d)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (D_m / d) * M_m;
	double tmp;
	if ((w0 * sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 2e+302) {
		tmp = w0 * sqrt((1.0 - (((t_0 * t_0) / 4.0) * (h / l))));
	} else {
		tmp = w0 * sqrt((1.0 - ((((D_m * M_m) * (0.25 * M_m)) * (h * D_m)) / (d * (l * d)))));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(w0, m_m, d_m, h, l, d)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_m / d) * m_m
    if ((w0 * sqrt((1.0d0 - ((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l))))) <= 2d+302) then
        tmp = w0 * sqrt((1.0d0 - (((t_0 * t_0) / 4.0d0) * (h / l))))
    else
        tmp = w0 * sqrt((1.0d0 - ((((d_m * m_m) * (0.25d0 * m_m)) * (h * d_m)) / (d * (l * d)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (D_m / d) * M_m;
	double tmp;
	if ((w0 * Math.sqrt((1.0 - (Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 2e+302) {
		tmp = w0 * Math.sqrt((1.0 - (((t_0 * t_0) / 4.0) * (h / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((((D_m * M_m) * (0.25 * M_m)) * (h * D_m)) / (d * (l * d)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = (D_m / d) * M_m
	tmp = 0
	if (w0 * math.sqrt((1.0 - (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 2e+302:
		tmp = w0 * math.sqrt((1.0 - (((t_0 * t_0) / 4.0) * (h / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((((D_m * M_m) * (0.25 * M_m)) * (h * D_m)) / (d * (l * d)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(Float64(D_m / d) * M_m)
	tmp = 0.0
	if (Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))) <= 2e+302)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(t_0 * t_0) / 4.0) * Float64(h / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(D_m * M_m) * Float64(0.25 * M_m)) * Float64(h * D_m)) / Float64(d * Float64(l * d))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = (D_m / d) * M_m;
	tmp = 0.0;
	if ((w0 * sqrt((1.0 - ((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l))))) <= 2e+302)
		tmp = w0 * sqrt((1.0 - (((t_0 * t_0) / 4.0) * (h / l))));
	else
		tmp = w0 * sqrt((1.0 - ((((D_m * M_m) * (0.25 * M_m)) * (h * D_m)) / (d * (l * d)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[N[(w0 * 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]), $MachinePrecision], 2e+302], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(0.25 * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))) < 2.0000000000000002e302

   1. Initial program 81.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. cosh-0-rev N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 81.1%

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

   if 2.0000000000000002e302 < (*.f64 w0 (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 81.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 77.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 71.6

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

   7. Applied rewrites 71.6%

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

Alternative 5: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(\left(\left(\frac{D\_m}{d} \cdot M\_m\right) \cdot M\_m\right) \cdot -0.25\right) \cdot \frac{h}{\ell}, \frac{D\_m}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\left(D\_m \cdot M\_m\right) \cdot \left(0.25 \cdot M\_m\right)\right) \cdot \left(h \cdot D\_m\right)}{d \cdot \left(\ell \cdot d\right)}}\\ \end{array} \end{array} \]

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 (<=
      (* w0 (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)))))
      2e+302)
   (*
    w0
    (sqrt (fma (* (* (* (* (/ D_m d) M_m) M_m) -0.25) (/ h l)) (/ D_m d) 1.0)))
   (*
    w0
    (sqrt
     (- 1.0 (/ (* (* (* D_m M_m) (* 0.25 M_m)) (* h D_m)) (* d (* l d))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((w0 * sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 2e+302) {
		tmp = w0 * sqrt(fma((((((D_m / d) * M_m) * M_m) * -0.25) * (h / l)), (D_m / d), 1.0));
	} else {
		tmp = w0 * sqrt((1.0 - ((((D_m * M_m) * (0.25 * M_m)) * (h * D_m)) / (d * (l * d)))));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))) <= 2e+302)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(Float64(Float64(D_m / d) * M_m) * M_m) * -0.25) * Float64(h / l)), Float64(D_m / d), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(D_m * M_m) * Float64(0.25 * M_m)) * Float64(h * D_m)) / Float64(d * Float64(l * d))))));
	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[(w0 * 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]), $MachinePrecision], 2e+302], N[(w0 * N[Sqrt[N[(N[(N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * -0.25), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(0.25 * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))) < 2.0000000000000002e302

   1. Initial program 81.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 77.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 82.0%

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

   if 2.0000000000000002e302 < (*.f64 w0 (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 81.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 77.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 71.6

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

   7. Applied rewrites 71.6%

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

Alternative 6: 83.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{if}\;t\_0 \leq 1:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(\frac{D\_m}{d} \cdot M\_m\right) \cdot M\_m\right) \cdot -0.25, D\_m \cdot \frac{h}{\ell \cdot d}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\left(D\_m \cdot M\_m\right) \cdot \left(0.25 \cdot M\_m\right)\right) \cdot \left(h \cdot D\_m\right)}{d \cdot \left(\ell \cdot d\right)}}\\ \end{array} \end{array} \]

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 (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))))))
   (if (<= t_0 1.0)
     (* w0 1.0)
     (if (<= t_0 2e+51)
       (*
        (sqrt
         (fma (* (* (* (/ D_m d) M_m) M_m) -0.25) (* D_m (/ h (* l d))) 1.0))
        w0)
       (*
        w0
        (sqrt
         (-
          1.0
          (/ (* (* (* D_m M_m) (* 0.25 M_m)) (* h D_m)) (* d (* l d))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 1.0) {
		tmp = w0 * 1.0;
	} else if (t_0 <= 2e+51) {
		tmp = sqrt(fma(((((D_m / d) * M_m) * M_m) * -0.25), (D_m * (h / (l * d))), 1.0)) * w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((((D_m * M_m) * (0.25 * M_m)) * (h * D_m)) / (d * (l * d)))));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 1.0)
		tmp = Float64(w0 * 1.0);
	elseif (t_0 <= 2e+51)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(D_m / d) * M_m) * M_m) * -0.25), Float64(D_m * Float64(h / Float64(l * d))), 1.0)) * w0);
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(D_m * M_m) * Float64(0.25 * M_m)) * Float64(h * D_m)) / Float64(d * Float64(l * d))))));
	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_] := Block[{t$95$0 = N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 1.0], N[(w0 * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+51], N[(N[Sqrt[N[(N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * -0.25), $MachinePrecision] * N[(D$95$m * N[(h / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(0.25 * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 81.3%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 67.9%

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

   if 1 < (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)))) < 2e51

   1. Initial program 81.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 77.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 77.9

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

   5. Applied rewrites 79.9%

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

   if 2e51 < (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 81.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 77.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 71.6

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

   7. Applied rewrites 71.6%

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

Alternative 7: 83.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \leq 4 \cdot 10^{-237}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(\frac{D\_m}{d} \cdot M\_m\right) \cdot M\_m\right) \cdot -0.25, D\_m \cdot \frac{h}{\ell \cdot d}, 1\right)} \cdot w0\\ \end{array} \end{array} \]

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) 4e-237)
   (* w0 1.0)
   (*
    (sqrt (fma (* (* (* (/ D_m d) M_m) M_m) -0.25) (* D_m (/ h (* l d))) 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) <= 4e-237) {
		tmp = w0 * 1.0;
	} else {
		tmp = sqrt(fma(((((D_m / d) * M_m) * M_m) * -0.25), (D_m * (h / (l * d))), 1.0)) * 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(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) <= 4e-237)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(D_m / d) * M_m) * M_m) * -0.25), Float64(D_m * Float64(h / Float64(l * d))), 1.0)) * 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[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 4e-237], N[(w0 * 1.0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * -0.25), $MachinePrecision] * N[(D$95$m * N[(h / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 4e-237

   1. Initial program 81.3%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 67.9%

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

   if 4e-237 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 81.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 77.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 77.9

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

   5. Applied rewrites 79.9%

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

Alternative 8: 82.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M\_m}{d \cdot d}, \left(\left(\frac{h}{\ell} \cdot D\_m\right) \cdot \left(D\_m \cdot M\_m\right)\right) \cdot -0.25, 1\right)}\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(D\_m \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)}{d \cdot d} \cdot -0.25, \frac{h}{\ell}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

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 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))))
   (if (<= t_0 (- INFINITY))
     (*
      w0
      (sqrt
       (fma (/ M_m (* d d)) (* (* (* (/ h l) D_m) (* D_m M_m)) -0.25) 1.0)))
     (if (<= t_0 -2e-13)
       (*
        (sqrt
         (fma (* (/ (* (* D_m M_m) (* D_m M_m)) (* d d)) -0.25) (/ h l) 1.0))
        w0)
       (* w0 1.0)))))

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 = pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = w0 * sqrt(fma((M_m / (d * d)), ((((h / l) * D_m) * (D_m * M_m)) * -0.25), 1.0));
	} else if (t_0 <= -2e-13) {
		tmp = sqrt(fma(((((D_m * M_m) * (D_m * M_m)) / (d * d)) * -0.25), (h / l), 1.0)) * w0;
	} else {
		tmp = w0 * 1.0;
	}
	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(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(fma(Float64(M_m / Float64(d * d)), Float64(Float64(Float64(Float64(h / l) * D_m) * Float64(D_m * M_m)) * -0.25), 1.0)));
	elseif (t_0 <= -2e-13)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(D_m * M_m) * Float64(D_m * M_m)) / Float64(d * d)) * -0.25), Float64(h / l), 1.0)) * w0);
	else
		tmp = Float64(w0 * 1.0);
	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_] := Block[{t$95$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]}, If[LessEqual[t$95$0, (-Infinity)], N[(w0 * N[Sqrt[N[(N[(M$95$m / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(h / l), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -2e-13], N[(N[Sqrt[N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 81.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. swap-sqr N/A

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

      18. metadata-eval N/A

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

      19. cosh-0-rev N/A

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

      20. lower-*.f64 N/A

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

      21. cosh-0-rev N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

      24. lower-*.f64 64.8

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

   3. Applied rewrites 64.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. mult-flip N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 63.3

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

   5. Applied rewrites 63.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 70.2

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

   7. Applied rewrites 70.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 67.2%

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

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.0000000000000001e-13

   1. Initial program 81.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. swap-sqr N/A

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

      18. metadata-eval N/A

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

      19. cosh-0-rev N/A

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

      20. lower-*.f64 N/A

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

      21. cosh-0-rev N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

      24. lower-*.f64 64.8

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

   3. Applied rewrites 64.8%

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

   5. Applied rewrites 64.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 67.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 67.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

   if -2.0000000000000001e-13 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 81.3%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 67.9%

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

Alternative 9: 82.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(D\_m \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)}{d \cdot d} \cdot -0.25, \frac{h}{\ell}, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

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)) -2e-13)
   (*
    (sqrt (fma (* (/ (* (* D_m M_m) (* D_m M_m)) (* d d)) -0.25) (/ h l) 1.0))
    w0)
   (* w0 1.0)))

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)) <= -2e-13) {
		tmp = sqrt(fma(((((D_m * M_m) * (D_m * M_m)) / (d * d)) * -0.25), (h / l), 1.0)) * w0;
	} else {
		tmp = w0 * 1.0;
	}
	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)) <= -2e-13)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(D_m * M_m) * Float64(D_m * M_m)) / Float64(d * d)) * -0.25), Float64(h / l), 1.0)) * w0);
	else
		tmp = Float64(w0 * 1.0);
	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], -2e-13], N[(N[Sqrt[N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.0000000000000001e-13

   1. Initial program 81.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. swap-sqr N/A

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

      18. metadata-eval N/A

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

      19. cosh-0-rev N/A

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

      20. lower-*.f64 N/A

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

      21. cosh-0-rev N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

      24. lower-*.f64 64.8

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

   3. Applied rewrites 64.8%

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

   5. Applied rewrites 64.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 67.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 67.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

   if -2.0000000000000001e-13 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 81.3%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 67.9%

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

Alternative 10: 82.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\left(\frac{M\_m}{d \cdot d} \cdot \left(\left(D\_m \cdot M\_m\right) \cdot D\_m\right)\right) \cdot -0.25, h, \ell\right)}{\ell}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

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)) -2e-13)
   (*
    (sqrt (/ (fma (* (* (/ M_m (* d d)) (* (* D_m M_m) D_m)) -0.25) h l) l))
    w0)
   (* w0 1.0)))

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)) <= -2e-13) {
		tmp = sqrt((fma((((M_m / (d * d)) * ((D_m * M_m) * D_m)) * -0.25), h, l) / l)) * w0;
	} else {
		tmp = w0 * 1.0;
	}
	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)) <= -2e-13)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(Float64(M_m / Float64(d * d)) * Float64(Float64(D_m * M_m) * D_m)) * -0.25), h, l) / l)) * w0);
	else
		tmp = Float64(w0 * 1.0);
	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], -2e-13], N[(N[Sqrt[N[(N[(N[(N[(N[(M$95$m / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] * h + l), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.0000000000000001e-13

   1. Initial program 81.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. swap-sqr N/A

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

      18. metadata-eval N/A

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

      19. cosh-0-rev N/A

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

      20. lower-*.f64 N/A

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

      21. cosh-0-rev N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

      24. lower-*.f64 64.8

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

   3. Applied rewrites 64.8%

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

   5. Applied rewrites 64.8%

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

   7. Applied rewrites 69.9%

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

   if -2.0000000000000001e-13 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 81.3%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 67.9%

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

Alternative 11: 67.9% accurate, 10.1× speedup

Math

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

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 1.0))

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 * 1.0;
}

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 * 1.0d0
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 * 1.0;
}

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 * 1.0

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 * 1.0)
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 * 1.0;
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 * 1.0), $MachinePrecision]

Derivation

1. Initial program 81.3%

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

2. Taylor expanded in M around 0

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

4. Applied rewrites 67.9%

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

Reproduce

herbie shell --seed 2025159 
(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))))))