Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.2% → 85.0%

Time: 10.6s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+84}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{D\_m}{\ell} \cdot \left(\frac{D\_m}{d} \cdot \frac{M\_m \cdot M\_m}{d}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\left(h \cdot \left(M\_m \cdot \frac{\frac{D\_m}{2}}{d}\right)\right) \cdot \left(M\_m \cdot D\_m\right)}{d \cdot 2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= l -4e+84)
   (*
    w0
    (sqrt (fma (* h -0.25) (* (/ D_m l) (* (/ D_m d) (/ (* M_m M_m) d))) 1.0)))
   (*
    w0
    (sqrt
     (-
      1.0
      (/ (/ (* (* h (* M_m (/ (/ D_m 2.0) d))) (* M_m D_m)) (* d 2.0)) l))))))

C

D_m = fabs(D);
M_m = fabs(M);
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 (l <= -4e+84) {
		tmp = w0 * sqrt(fma((h * -0.25), ((D_m / l) * ((D_m / d) * ((M_m * M_m) / d))), 1.0));
	} else {
		tmp = w0 * sqrt((1.0 - ((((h * (M_m * ((D_m / 2.0) / d))) * (M_m * D_m)) / (d * 2.0)) / l)));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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 (l <= -4e+84)
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(D_m / l) * Float64(Float64(D_m / d) * Float64(Float64(M_m * M_m) / d))), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(h * Float64(M_m * Float64(Float64(D_m / 2.0) / d))) * Float64(M_m * D_m)) / Float64(d * 2.0)) / l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[l, -4e+84], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(D$95$m / l), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(h * N[(M$95$m * N[(N[(D$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.00000000000000023e84

   1. Initial program 93.7%

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

   3. Taylor expanded in h around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. rgt-mult-inverse N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 66.8%

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

   7. Applied rewrites 77.9%

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

   if -4.00000000000000023e84 < l

   1. Initial program 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. 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. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. times-frac N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

   4. Applied rewrites 73.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

   6. Applied rewrites 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. *-commutative N/A

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

      6. unpow-prod-down N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. unpow2 N/A

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

      12. frac-times N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. frac-times N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*l* N/A

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

   8. Applied rewrites 92.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 92.1

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

   10. Applied rewrites 92.1%

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

Alternative 2: 81.3% accurate, 0.7× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -50.0)
   (*
    w0
    (sqrt (fma (* h -0.25) (* (* (/ (* M_m M_m) d) D_m) (/ D_m (* l d))) 1.0)))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -50.0) {
		tmp = w0 * sqrt(fma((h * -0.25), ((((M_m * M_m) / d) * D_m) * (D_m / (l * d))), 1.0));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -50.0)
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(Float64(M_m * M_m) / d) * D_m) * Float64(D_m / Float64(l * d))), 1.0)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -50.0], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $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)) < -50

   1. Initial program 78.4%

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

   3. Taylor expanded in h around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. rgt-mult-inverse N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 47.9%

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

   7. Applied rewrites 62.8%

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

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

   1. Initial program 87.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 97.9%

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

Alternative 3: 81.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1000000000000:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)}{\left(d \cdot d\right) \cdot \ell}, -0.25 \cdot h, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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)) -1000000000000.0)
   (*
    w0
    (sqrt (fma (/ (* (* M_m D_m) (* M_m D_m)) (* (* d d) l)) (* -0.25 h) 1.0)))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
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)) <= -1000000000000.0) {
		tmp = w0 * sqrt(fma((((M_m * D_m) * (M_m * D_m)) / ((d * d) * l)), (-0.25 * h), 1.0));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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)) <= -1000000000000.0)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) / Float64(Float64(d * d) * l)), Float64(-0.25 * h), 1.0)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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], -1000000000000.0], N[(w0 * N[Sqrt[N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * h), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $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)) < -1e12

   1. Initial program 78.1%

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

   4. Applied rewrites 68.2%

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

   5. Taylor expanded in h around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 53.6%

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

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

   1. Initial program 87.6%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 97.4%

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

Alternative 4: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1000000000000:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, M\_m \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \frac{D\_m}{\left(d \cdot d\right) \cdot \ell}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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)) -1000000000000.0)
   (*
    w0
    (sqrt (fma (* h -0.25) (* M_m (* (* M_m D_m) (/ D_m (* (* d d) l)))) 1.0)))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
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)) <= -1000000000000.0) {
		tmp = w0 * sqrt(fma((h * -0.25), (M_m * ((M_m * D_m) * (D_m / ((d * d) * l)))), 1.0));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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)) <= -1000000000000.0)
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(M_m * Float64(Float64(M_m * D_m) * Float64(D_m / Float64(Float64(d * d) * l)))), 1.0)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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], -1000000000000.0], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(M$95$m * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(D$95$m / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $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)) < -1e12

   1. Initial program 78.1%

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

   3. Taylor expanded in h around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. rgt-mult-inverse N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 48.5%

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

   7. Applied rewrites 54.1%

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

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

   1. Initial program 87.6%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 97.4%

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

Alternative 5: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+48}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot -0.125, M\_m \cdot \frac{\frac{M\_m}{d} \cdot h}{\ell \cdot d}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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)) -5e+48)
   (* w0 (fma (* (* D_m D_m) -0.125) (* M_m (/ (* (/ M_m d) h) (* l d))) 1.0))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
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)) <= -5e+48) {
		tmp = w0 * fma(((D_m * D_m) * -0.125), (M_m * (((M_m / d) * h) / (l * d))), 1.0);
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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)) <= -5e+48)
		tmp = Float64(w0 * fma(Float64(Float64(D_m * D_m) * -0.125), Float64(M_m * Float64(Float64(Float64(M_m / d) * h) / Float64(l * d))), 1.0));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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], -5e+48], N[(w0 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(M$95$m * N[(N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $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)) < -4.99999999999999973e48

   1. Initial program 77.0%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 43.2%

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

   7. Applied rewrites 45.7%

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

   9. Applied rewrites 51.5%

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

   if -4.99999999999999973e48 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.8%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.5%

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

Alternative 6: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+48}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot -0.125, M\_m \cdot \left(\frac{M\_m}{d} \cdot \frac{h}{\ell \cdot d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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)) -5e+48)
   (* w0 (fma (* (* D_m D_m) -0.125) (* M_m (* (/ M_m d) (/ h (* l d)))) 1.0))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
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)) <= -5e+48) {
		tmp = w0 * fma(((D_m * D_m) * -0.125), (M_m * ((M_m / d) * (h / (l * d)))), 1.0);
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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)) <= -5e+48)
		tmp = Float64(w0 * fma(Float64(Float64(D_m * D_m) * -0.125), Float64(M_m * Float64(Float64(M_m / d) * Float64(h / Float64(l * d)))), 1.0));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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], -5e+48], N[(w0 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(M$95$m * N[(N[(M$95$m / d), $MachinePrecision] * N[(h / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $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)) < -4.99999999999999973e48

   1. Initial program 77.0%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 43.2%

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

   7. Applied rewrites 45.7%

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

   9. Applied rewrites 50.2%

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

   if -4.99999999999999973e48 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.8%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.5%

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

Alternative 7: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+69}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot -0.125, M\_m \cdot \left(\frac{M\_m}{\left(d \cdot d\right) \cdot \ell} \cdot h\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -1e+69)
   (* w0 (fma (* (* D_m D_m) -0.125) (* M_m (* (/ M_m (* (* d d) l)) h)) 1.0))
   (* w0 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e+69) {
		tmp = w0 * fma(((D_m * D_m) * -0.125), (M_m * ((M_m / ((d * d) * l)) * h)), 1.0);
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+69)
		tmp = Float64(w0 * fma(Float64(Float64(D_m * D_m) * -0.125), Float64(M_m * Float64(Float64(M_m / Float64(Float64(d * d) * l)) * h)), 1.0));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+69], N[(w0 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(M$95$m * N[(N[(M$95$m / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $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)) < -1.0000000000000001e69

   1. Initial program 76.4%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 44.2%

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

   7. Applied rewrites 46.8%

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

   9. Applied rewrites 47.2%

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

   if -1.0000000000000001e69 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 94.5%

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

Alternative 8: 85.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 5 \cdot 10^{-193}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{M\_m}{d}, \left(\frac{M\_m}{d} \cdot h\right) \cdot \left(-0.25 \cdot \left(D\_m \cdot D\_m\right)\right), \ell\right)}{\ell}}\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 5 \cdot 10^{+68}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\left(h \cdot \left(M\_m \cdot D\_m\right)\right) \cdot \left(M\_m \cdot D\_m\right)}{-2 \cdot d}}{\left(-2 \cdot d\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \left(\frac{M\_m \cdot M\_m}{d} \cdot D\_m\right) \cdot \frac{D\_m}{\ell \cdot d}, 1\right)}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= (* M_m D_m) 5e-193)
   (*
    w0
    (sqrt (/ (fma (/ M_m d) (* (* (/ M_m d) h) (* -0.25 (* D_m D_m))) l) l)))
   (if (<= (* M_m D_m) 5e+68)
     (*
      w0
      (sqrt
       (-
        1.0
        (/
         (/ (* (* h (* M_m D_m)) (* M_m D_m)) (* -2.0 d))
         (* (* -2.0 d) l)))))
     (*
      w0
      (sqrt
       (fma (* h -0.25) (* (* (/ (* M_m M_m) d) D_m) (/ D_m (* l d))) 1.0))))))

C

D_m = fabs(D);
M_m = fabs(M);
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 ((M_m * D_m) <= 5e-193) {
		tmp = w0 * sqrt((fma((M_m / d), (((M_m / d) * h) * (-0.25 * (D_m * D_m))), l) / l));
	} else if ((M_m * D_m) <= 5e+68) {
		tmp = w0 * sqrt((1.0 - ((((h * (M_m * D_m)) * (M_m * D_m)) / (-2.0 * d)) / ((-2.0 * d) * l))));
	} else {
		tmp = w0 * sqrt(fma((h * -0.25), ((((M_m * M_m) / d) * D_m) * (D_m / (l * d))), 1.0));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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(M_m * D_m) <= 5e-193)
		tmp = Float64(w0 * sqrt(Float64(fma(Float64(M_m / d), Float64(Float64(Float64(M_m / d) * h) * Float64(-0.25 * Float64(D_m * D_m))), l) / l)));
	elseif (Float64(M_m * D_m) <= 5e+68)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(h * Float64(M_m * D_m)) * Float64(M_m * D_m)) / Float64(-2.0 * d)) / Float64(Float64(-2.0 * d) * l)))));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(Float64(M_m * M_m) / d) * D_m) * Float64(D_m / Float64(l * d))), 1.0)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(M$95$m * D$95$m), $MachinePrecision], 5e-193], N[(w0 * N[Sqrt[N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision] * N[(-0.25 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e+68], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(h * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * d), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 5.0000000000000005e-193

   1. Initial program 83.3%

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

   3. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 71.6

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

   5. Applied rewrites 71.6%

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

   7. Applied rewrites 77.2%

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

   9. Applied rewrites 80.5%

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

   11. Applied rewrites 84.5%

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

   if 5.0000000000000005e-193 < (*.f64 M D) < 5.0000000000000004e68

   1. Initial program 89.9%

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

   4. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 98.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 98.0

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

   6. Applied rewrites 98.0%

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

   if 5.0000000000000004e68 < (*.f64 M D)

   1. Initial program 84.0%

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

   3. Taylor expanded in h around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. rgt-mult-inverse N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 48.1%

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

   7. Applied rewrites 67.8%

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

Alternative 9: 85.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+84}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{D\_m}{\ell} \cdot \left(\frac{D\_m}{d} \cdot \frac{M\_m \cdot M\_m}{d}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{D\_m \cdot \left(\left(h \cdot \left(M\_m \cdot \frac{\frac{D\_m}{2}}{d}\right)\right) \cdot M\_m\right)}{d \cdot 2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= l -4e+84)
   (*
    w0
    (sqrt (fma (* h -0.25) (* (/ D_m l) (* (/ D_m d) (/ (* M_m M_m) d))) 1.0)))
   (*
    w0
    (sqrt
     (-
      1.0
      (/ (/ (* D_m (* (* h (* M_m (/ (/ D_m 2.0) d))) M_m)) (* d 2.0)) l))))))

C

D_m = fabs(D);
M_m = fabs(M);
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 (l <= -4e+84) {
		tmp = w0 * sqrt(fma((h * -0.25), ((D_m / l) * ((D_m / d) * ((M_m * M_m) / d))), 1.0));
	} else {
		tmp = w0 * sqrt((1.0 - (((D_m * ((h * (M_m * ((D_m / 2.0) / d))) * M_m)) / (d * 2.0)) / l)));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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 (l <= -4e+84)
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(D_m / l) * Float64(Float64(D_m / d) * Float64(Float64(M_m * M_m) / d))), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D_m * Float64(Float64(h * Float64(M_m * Float64(Float64(D_m / 2.0) / d))) * M_m)) / Float64(d * 2.0)) / l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[l, -4e+84], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(D$95$m / l), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D$95$m * N[(N[(h * N[(M$95$m * N[(N[(D$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.00000000000000023e84

   1. Initial program 93.7%

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

   3. Taylor expanded in h around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. rgt-mult-inverse N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 66.8%

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

   7. Applied rewrites 77.9%

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

   if -4.00000000000000023e84 < l

   1. Initial program 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. 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. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. times-frac N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

   4. Applied rewrites 73.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

   6. Applied rewrites 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. *-commutative N/A

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

      6. unpow-prod-down N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. unpow2 N/A

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

      12. frac-times N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. frac-times N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*l* N/A

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

   8. Applied rewrites 92.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 91.1

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

   10. Applied rewrites 91.1%

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

Alternative 10: 87.0% accurate, 1.7× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 2.0) d) M_m)))
   (* w0 (sqrt (- 1.0 (/ (* t_0 (* t_0 h)) l))))))

C

D_m = fabs(D);
M_m = fabs(M);
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 / 2.0) / d) * M_m;
	return w0 * sqrt((1.0 - ((t_0 * (t_0 * h)) / l)));
}

Fortran

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

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

real(8) function code(w0, m_m, d_m, h, l, d)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    t_0 = ((d_m / 2.0d0) / d) * m_m
    code = w0 * sqrt((1.0d0 - ((t_0 * (t_0 * h)) / l)))
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
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 / 2.0) / d) * M_m;
	return w0 * Math.sqrt((1.0 - ((t_0 * (t_0 * h)) / l)));
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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 / 2.0) / d) * M_m
	return w0 * math.sqrt((1.0 - ((t_0 * (t_0 * h)) / l)))

Julia

D_m = abs(D)
M_m = abs(M)
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(D_m / 2.0) / d) * M_m)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * h)) / l))))
end

MATLAB

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

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[(D$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 84.7%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. 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. associate-/l* N/A

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

   7. associate-*r* N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. times-frac N/A

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

   15. associate-*l* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. lower-*.f64 N/A

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

4. Applied rewrites 75.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

6. Applied rewrites 91.4%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. unpow-prod-down N/A

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

   5. *-commutative N/A

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

   6. unpow-prod-down N/A

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

   7. *-commutative N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. frac-times N/A

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

   11. unpow2 N/A

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

   12. frac-times N/A

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

   13. lift-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. frac-times N/A

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

   17. lift-/.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. associate-*l* N/A

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

8. Applied rewrites 92.5%

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

Alternative 11: 83.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 6.2 \cdot 10^{+130}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\frac{M\_m}{d}, \left(\frac{M\_m}{d} \cdot h\right) \cdot \left(-0.25 \cdot \left(D\_m \cdot D\_m\right)\right), \ell\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \left(\frac{M\_m \cdot M\_m}{d} \cdot D\_m\right) \cdot \frac{D\_m}{\ell \cdot d}, 1\right)}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= D_m 6.2e+130)
   (*
    w0
    (sqrt (/ (fma (/ M_m d) (* (* (/ M_m d) h) (* -0.25 (* D_m D_m))) l) l)))
   (*
    w0
    (sqrt
     (fma (* h -0.25) (* (* (/ (* M_m M_m) d) D_m) (/ D_m (* l d))) 1.0)))))

C

D_m = fabs(D);
M_m = fabs(M);
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 (D_m <= 6.2e+130) {
		tmp = w0 * sqrt((fma((M_m / d), (((M_m / d) * h) * (-0.25 * (D_m * D_m))), l) / l));
	} else {
		tmp = w0 * sqrt(fma((h * -0.25), ((((M_m * M_m) / d) * D_m) * (D_m / (l * d))), 1.0));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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 (D_m <= 6.2e+130)
		tmp = Float64(w0 * sqrt(Float64(fma(Float64(M_m / d), Float64(Float64(Float64(M_m / d) * h) * Float64(-0.25 * Float64(D_m * D_m))), l) / l)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(Float64(M_m * M_m) / d) * D_m) * Float64(D_m / Float64(l * d))), 1.0)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[D$95$m, 6.2e+130], N[(w0 * N[Sqrt[N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision] * N[(-0.25 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 6.1999999999999999e130

   1. Initial program 84.9%

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

   3. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 80.6%

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

   9. Applied rewrites 83.2%

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

   11. Applied rewrites 86.4%

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

   if 6.1999999999999999e130 < D

   1. Initial program 82.9%

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

   3. Taylor expanded in h around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. rgt-mult-inverse N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 59.7%

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

   7. Applied rewrites 80.2%

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

Alternative 12: 82.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 2.8 \cdot 10^{+129}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(h \cdot \frac{M\_m}{d}\right) \cdot \frac{M\_m}{d}, \left(D\_m \cdot D\_m\right) \cdot -0.25, \ell\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \left(\frac{M\_m \cdot M\_m}{d} \cdot D\_m\right) \cdot \frac{D\_m}{\ell \cdot d}, 1\right)}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= D_m 2.8e+129)
   (*
    w0
    (sqrt (/ (fma (* (* h (/ M_m d)) (/ M_m d)) (* (* D_m D_m) -0.25) l) l)))
   (*
    w0
    (sqrt
     (fma (* h -0.25) (* (* (/ (* M_m M_m) d) D_m) (/ D_m (* l d))) 1.0)))))

C

D_m = fabs(D);
M_m = fabs(M);
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 (D_m <= 2.8e+129) {
		tmp = w0 * sqrt((fma(((h * (M_m / d)) * (M_m / d)), ((D_m * D_m) * -0.25), l) / l));
	} else {
		tmp = w0 * sqrt(fma((h * -0.25), ((((M_m * M_m) / d) * D_m) * (D_m / (l * d))), 1.0));
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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 (D_m <= 2.8e+129)
		tmp = Float64(w0 * sqrt(Float64(fma(Float64(Float64(h * Float64(M_m / d)) * Float64(M_m / d)), Float64(Float64(D_m * D_m) * -0.25), l) / l)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(Float64(M_m * M_m) / d) * D_m) * Float64(D_m / Float64(l * d))), 1.0)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[D$95$m, 2.8e+129], N[(w0 * N[Sqrt[N[(N[(N[(N[(h * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.25), $MachinePrecision] + l), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.79999999999999975e129

   1. Initial program 84.9%

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

   3. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 80.6%

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

   9. Applied rewrites 83.2%

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

   if 2.79999999999999975e129 < D

   1. Initial program 82.9%

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

   3. Taylor expanded in h around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. rgt-mult-inverse N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 59.7%

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

   7. Applied rewrites 80.2%

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

Alternative 13: 67.2% accurate, 26.2× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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

D_m = fabs(D);
M_m = fabs(M);
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

D_m =     private
M_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

D_m = Math.abs(D);
M_m = Math.abs(M);
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

D_m = math.fabs(D)
M_m = math.fabs(M)
[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

D_m = abs(D)
M_m = abs(M)
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

D_m = abs(D);
M_m = abs(M);
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

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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 84.7%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 69.0%

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

Reproduce

herbie shell --seed 2024363 
(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))))))