Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 89.3%

Time: 10.9s

Alternatives: 18

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: 80.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. 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. associate-*l* N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. times-frac N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. associate-/l* N/A

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

   20. associate-*r* N/A

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

   21. lower-*.f64 N/A

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

4. Applied rewrites 77.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. frac-times N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 85.3

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

6. Applied rewrites 85.3%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

8. Applied rewrites 89.6%

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

Alternative 2: 85.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(w0, m_m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((1.0d0 - ((((m_m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))) <= 1.0d0) then
        tmp = w0 * 1.0d0
    else
        tmp = w0 * sqrt((1.0d0 - ((((h * (m_m * d)) * (m_m * d)) / ((-2.0d0) * d_1)) / (((-2.0d0) * d_1) * l))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 99.5%

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

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

   1. Initial program 48.8%

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

   4. Applied rewrites 65.5%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{h \cdot {\left(D \cdot M\right)}^{2}}{-2 \cdot d}}{\left(-2 \cdot d\right) \cdot \ell}}} \]
   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 66.5

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

      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 66.5

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

      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 66.5

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

   6. Applied rewrites 66.5%

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

Alternative 3: 86.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+102}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\ell - \frac{\left(h \cdot M\_m\right) \cdot \left(\left(\frac{M\_m}{d} \cdot D\right) \cdot 0.25\right)}{d} \cdot D}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{2} \cdot \left(\frac{M\_m}{d} \cdot \frac{\frac{M\_m}{d} \cdot \left(D \cdot h\right)}{2 \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (pow (/ (* M_m D) (* 2.0 d)) 2.0) 5e+102)
   (* w0 (sqrt (/ (- l (* (/ (* (* h M_m) (* (* (/ M_m d) D) 0.25)) d) D)) l)))
   (*
    w0
    (sqrt
     (-
      1.0
      (* (/ D 2.0) (* (/ M_m d) (/ (* (/ M_m d) (* D h)) (* 2.0 l)))))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (pow(((M_m * D) / (2.0 * d)), 2.0) <= 5e+102) {
		tmp = w0 * sqrt(((l - ((((h * M_m) * (((M_m / d) * D) * 0.25)) / d) * D)) / l));
	} else {
		tmp = w0 * sqrt((1.0 - ((D / 2.0) * ((M_m / d) * (((M_m / d) * (D * h)) / (2.0 * l))))));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(w0, m_m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((((m_m * d) / (2.0d0 * d_1)) ** 2.0d0) <= 5d+102) then
        tmp = w0 * sqrt(((l - ((((h * m_m) * (((m_m / d_1) * d) * 0.25d0)) / d_1) * d)) / l))
    else
        tmp = w0 * sqrt((1.0d0 - ((d / 2.0d0) * ((m_m / d_1) * (((m_m / d_1) * (d * h)) / (2.0d0 * l))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (Math.pow(((M_m * D) / (2.0 * d)), 2.0) <= 5e+102) {
		tmp = w0 * Math.sqrt(((l - ((((h * M_m) * (((M_m / d) * D) * 0.25)) / d) * D)) / l));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((D / 2.0) * ((M_m / d) * (((M_m / d) * (D * h)) / (2.0 * l))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if math.pow(((M_m * D) / (2.0 * d)), 2.0) <= 5e+102:
		tmp = w0 * math.sqrt(((l - ((((h * M_m) * (((M_m / d) * D) * 0.25)) / d) * D)) / l))
	else:
		tmp = w0 * math.sqrt((1.0 - ((D / 2.0) * ((M_m / d) * (((M_m / d) * (D * h)) / (2.0 * l))))))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if ((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) <= 5e+102)
		tmp = Float64(w0 * sqrt(Float64(Float64(l - Float64(Float64(Float64(Float64(h * M_m) * Float64(Float64(Float64(M_m / d) * D) * 0.25)) / d) * D)) / l)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D / 2.0) * Float64(Float64(M_m / d) * Float64(Float64(Float64(M_m / d) * Float64(D * h)) / Float64(2.0 * l)))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if ((((M_m * D) / (2.0 * d)) ^ 2.0) <= 5e+102)
		tmp = w0 * sqrt(((l - ((((h * M_m) * (((M_m / d) * D) * 0.25)) / d) * D)) / l));
	else
		tmp = w0 * sqrt((1.0 - ((D / 2.0) * ((M_m / d) * (((M_m / d) * (D * h)) / (2.0 * l))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 5e+102], N[(w0 * N[Sqrt[N[(N[(l - N[(N[(N[(N[(h * M$95$m), $MachinePrecision] * N[(N[(N[(M$95$m / d), $MachinePrecision] * D), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D / 2.0), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(D * h), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 5e102

   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 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 72.5

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

   5. Applied rewrites 72.5%

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

   7. Applied rewrites 87.0%

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

   9. Applied rewrites 92.1%

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

   if 5e102 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 58.1%

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

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-/l* N/A

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

      20. associate-*r* N/A

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

      21. lower-*.f64 N/A

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

   4. Applied rewrites 65.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 67.0

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

   6. Applied rewrites 67.0%

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

Alternative 4: 82.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -500000000000:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(M\_m \cdot D\right) \cdot \left(M\_m \cdot D\right)}{\left(\ell \cdot d\right) \cdot d}, -0.25 \cdot h, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -500000000000.0)
   (*
    w0
    (sqrt (fma (/ (* (* M_m D) (* M_m D)) (* (* l d) d)) (* -0.25 h) 1.0)))
   (* w0 1.0)))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -500000000000.0) {
		tmp = w0 * sqrt(fma((((M_m * D) * (M_m * D)) / ((l * d) * d)), (-0.25 * h), 1.0));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -500000000000.0)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(M_m * D) * Float64(M_m * D)) / Float64(Float64(l * d) * d)), Float64(-0.25 * h), 1.0)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -500000000000.0], N[(w0 * N[Sqrt[N[(N[(N[(N[(M$95$m * D), $MachinePrecision] * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] * d), $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)) < -5e11

   1. Initial program 62.3%

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

   4. Applied rewrites 62.0%

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

   5. 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 53.7%

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

   9. Applied rewrites 58.4%

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

   if -5e11 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 85.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.5%

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

Alternative 5: 81.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -500000000000:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(M\_m \cdot D\right) \cdot \left(M\_m \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}, -0.25 \cdot h, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -500000000000.0)
   (*
    w0
    (sqrt (fma (/ (* (* M_m D) (* M_m D)) (* (* d d) l)) (* -0.25 h) 1.0)))
   (* w0 1.0)))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -500000000000.0) {
		tmp = w0 * sqrt(fma((((M_m * D) * (M_m * D)) / ((d * d) * l)), (-0.25 * h), 1.0));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -500000000000.0)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(M_m * D) * Float64(M_m * D)) / Float64(Float64(d * d) * l)), Float64(-0.25 * h), 1.0)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -500000000000.0], N[(w0 * N[Sqrt[N[(N[(N[(N[(M$95$m * D), $MachinePrecision] * N[(M$95$m * D), $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)) < -5e11

   1. Initial program 62.3%

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

   4. Applied rewrites 62.0%

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

   5. 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 53.7%

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

   if -5e11 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 85.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.5%

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

Alternative 6: 81.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -500000000000:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, M\_m \cdot \left(\left(M\_m \cdot D\right) \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -500000000000.0)
   (*
    w0
    (sqrt (fma (* h -0.25) (* M_m (* (* M_m D) (/ D (* (* d d) l)))) 1.0)))
   (* w0 1.0)))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -500000000000.0) {
		tmp = w0 * sqrt(fma((h * -0.25), (M_m * ((M_m * D) * (D / ((d * d) * l)))), 1.0));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -500000000000.0)
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(M_m * Float64(Float64(M_m * D) * Float64(D / Float64(Float64(d * d) * l)))), 1.0)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -500000000000.0], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(M$95$m * N[(N[(M$95$m * D), $MachinePrecision] * N[(D / 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)) < -5e11

   1. Initial program 62.3%

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

   3. Taylor expanded in 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 44.3%

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

   7. Applied rewrites 55.4%

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

   if -5e11 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 85.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.5%

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

Alternative 7: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} t_0 := \left(\frac{M\_m}{d} \cdot D\right) \cdot 0.25\\ \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+102}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\ell - \frac{\left(h \cdot M\_m\right) \cdot t\_0}{d} \cdot D}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(t\_0 \cdot \left(\frac{h}{d} \cdot M\_m\right)\right) \cdot \frac{D}{\ell}} \cdot w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (let* ((t_0 (* (* (/ M_m d) D) 0.25)))
   (if (<= (pow (/ (* M_m D) (* 2.0 d)) 2.0) 5e+102)
     (* w0 (sqrt (/ (- l (* (/ (* (* h M_m) t_0) d) D)) l)))
     (* (sqrt (- 1.0 (* (* t_0 (* (/ h d) M_m)) (/ D l)))) w0))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = ((M_m / d) * D) * 0.25;
	double tmp;
	if (pow(((M_m * D) / (2.0 * d)), 2.0) <= 5e+102) {
		tmp = w0 * sqrt(((l - ((((h * M_m) * t_0) / d) * D)) / l));
	} else {
		tmp = sqrt((1.0 - ((t_0 * ((h / d) * M_m)) * (D / l)))) * w0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(w0, m_m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((m_m / d_1) * d) * 0.25d0
    if ((((m_m * d) / (2.0d0 * d_1)) ** 2.0d0) <= 5d+102) then
        tmp = w0 * sqrt(((l - ((((h * m_m) * t_0) / d_1) * d)) / l))
    else
        tmp = sqrt((1.0d0 - ((t_0 * ((h / d_1) * m_m)) * (d / l)))) * w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = ((M_m / d) * D) * 0.25;
	double tmp;
	if (Math.pow(((M_m * D) / (2.0 * d)), 2.0) <= 5e+102) {
		tmp = w0 * Math.sqrt(((l - ((((h * M_m) * t_0) / d) * D)) / l));
	} else {
		tmp = Math.sqrt((1.0 - ((t_0 * ((h / d) * M_m)) * (D / l)))) * w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	t_0 = ((M_m / d) * D) * 0.25
	tmp = 0
	if math.pow(((M_m * D) / (2.0 * d)), 2.0) <= 5e+102:
		tmp = w0 * math.sqrt(((l - ((((h * M_m) * t_0) / d) * D)) / l))
	else:
		tmp = math.sqrt((1.0 - ((t_0 * ((h / d) * M_m)) * (D / l)))) * w0
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(Float64(Float64(M_m / d) * D) * 0.25)
	tmp = 0.0
	if ((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) <= 5e+102)
		tmp = Float64(w0 * sqrt(Float64(Float64(l - Float64(Float64(Float64(Float64(h * M_m) * t_0) / d) * D)) / l)));
	else
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(t_0 * Float64(Float64(h / d) * M_m)) * Float64(D / l)))) * w0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	t_0 = ((M_m / d) * D) * 0.25;
	tmp = 0.0;
	if ((((M_m * D) / (2.0 * d)) ^ 2.0) <= 5e+102)
		tmp = w0 * sqrt(((l - ((((h * M_m) * t_0) / d) * D)) / l));
	else
		tmp = sqrt((1.0 - ((t_0 * ((h / d) * M_m)) * (D / l)))) * w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(N[(M$95$m / d), $MachinePrecision] * D), $MachinePrecision] * 0.25), $MachinePrecision]}, If[LessEqual[N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 5e+102], N[(w0 * N[Sqrt[N[(N[(l - N[(N[(N[(N[(h * M$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] / d), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 - N[(N[(t$95$0 * N[(N[(h / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 5e102

   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 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 72.5

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

   5. Applied rewrites 72.5%

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

   7. Applied rewrites 87.0%

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

   9. Applied rewrites 92.1%

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

   if 5e102 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 58.1%

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

   3. Taylor expanded in 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 42.2

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

   5. Applied rewrites 42.2%

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

   7. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 62.3

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

   9. Applied rewrites 66.6%

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

Alternative 8: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+212}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, h \cdot \left(\frac{M\_m}{d} \cdot \frac{M\_m}{\ell \cdot d}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -4e+212)
   (* w0 (fma (* (* D D) -0.125) (* h (* (/ M_m d) (/ M_m (* l d)))) 1.0))
   (* w0 1.0)))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -4e+212) {
		tmp = w0 * fma(((D * D) * -0.125), (h * ((M_m / d) * (M_m / (l * d)))), 1.0);
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -4e+212)
		tmp = Float64(w0 * fma(Float64(Float64(D * D) * -0.125), Float64(h * Float64(Float64(M_m / d) * Float64(M_m / Float64(l * d)))), 1.0));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -4e+212], N[(w0 * N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] * N[(h * N[(N[(M$95$m / d), $MachinePrecision] * N[(M$95$m / 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)) < -3.9999999999999996e212

   1. Initial program 54.7%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      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 34.0%

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

   7. Applied rewrites 39.9%

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

   9. Applied rewrites 47.2%

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

   if -3.9999999999999996e212 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.0%

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

Alternative 9: 78.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(w0 * N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] * N[(h * N[(M$95$m * N[(M$95$m / N[(N[(d * d), $MachinePrecision] * l), $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)) < -inf.0

   1. Initial program 52.7%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      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 35.5%

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

   7. Applied rewrites 41.6%

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

   9. Applied rewrites 46.2%

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

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

   1. Initial program 86.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 87.7%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, h \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(w0 * N[(N[(N[(M$95$m * N[(N[(h * M$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * N[(-0.125 * D), $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)) < -inf.0

   1. Initial program 52.7%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      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 35.5%

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

   7. Applied rewrites 42.7%

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

   9. Applied rewrites 47.1%

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

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

   1. Initial program 86.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 87.7%

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

Alternative 11: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+248}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, h \cdot \frac{M\_m \cdot M\_m}{\left(\ell \cdot d\right) \cdot d}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)) -5e+248)
   (* w0 (fma (* (* D D) -0.125) (* h (/ (* M_m M_m) (* (* l d) d))) 1.0))
   (* w0 1.0)))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+248) {
		tmp = w0 * fma(((D * D) * -0.125), (h * ((M_m * M_m) / ((l * d) * d))), 1.0);
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -5e+248)
		tmp = Float64(w0 * fma(Float64(Float64(D * D) * -0.125), Float64(h * Float64(Float64(M_m * M_m) / Float64(Float64(l * d) * d))), 1.0));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+248], N[(w0 * N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] * N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] * 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)) < -4.9999999999999996e248

   1. Initial program 54.1%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   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 34.5%

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

   7. Applied rewrites 40.5%

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

   9. Applied rewrites 40.6%

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

   if -4.9999999999999996e248 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 88.6%

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

Alternative 12: 77.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(w0 * N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] * N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $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)) < -inf.0

   1. Initial program 52.7%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      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 35.5%

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

   7. Applied rewrites 41.6%

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

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

   1. Initial program 86.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 87.7%

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

Alternative 13: 76.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D \leq 4 \cdot 10^{-276}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{elif}\;M\_m \cdot D \leq 5 \cdot 10^{-131}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\mathsf{fma}\left(\left(\frac{h}{d} \cdot M\_m\right) \cdot \frac{M\_m}{d}, \left(D \cdot D\right) \cdot -0.25, \ell\right)}{\ell}}\\ \mathbf{elif}\;M\_m \cdot D \leq 4 \cdot 10^{+140}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(M\_m \cdot D\right) \cdot \left(M\_m \cdot D\right)}{\left(\ell \cdot d\right) \cdot d}, -0.25 \cdot h, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{D}{\ell} \cdot \left(\frac{D}{d} \cdot \frac{M\_m \cdot M\_m}{d}\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* M_m D) 4e-276)
   (* w0 1.0)
   (if (<= (* M_m D) 5e-131)
     (*
      w0
      (sqrt (/ (fma (* (* (/ h d) M_m) (/ M_m d)) (* (* D D) -0.25) l) l)))
     (if (<= (* M_m D) 4e+140)
       (*
        w0
        (sqrt (fma (/ (* (* M_m D) (* M_m D)) (* (* l d) d)) (* -0.25 h) 1.0)))
       (*
        w0
        (sqrt
         (fma (* h -0.25) (* (/ D l) (* (/ D d) (/ (* M_m M_m) d))) 1.0)))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((M_m * D) <= 4e-276) {
		tmp = w0 * 1.0;
	} else if ((M_m * D) <= 5e-131) {
		tmp = w0 * sqrt((fma((((h / d) * M_m) * (M_m / d)), ((D * D) * -0.25), l) / l));
	} else if ((M_m * D) <= 4e+140) {
		tmp = w0 * sqrt(fma((((M_m * D) * (M_m * D)) / ((l * d) * d)), (-0.25 * h), 1.0));
	} else {
		tmp = w0 * sqrt(fma((h * -0.25), ((D / l) * ((D / d) * ((M_m * M_m) / d))), 1.0));
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D) <= 4e-276)
		tmp = Float64(w0 * 1.0);
	elseif (Float64(M_m * D) <= 5e-131)
		tmp = Float64(w0 * sqrt(Float64(fma(Float64(Float64(Float64(h / d) * M_m) * Float64(M_m / d)), Float64(Float64(D * D) * -0.25), l) / l)));
	elseif (Float64(M_m * D) <= 4e+140)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(M_m * D) * Float64(M_m * D)) / Float64(Float64(l * d) * d)), Float64(-0.25 * h), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(D / l) * Float64(Float64(D / d) * Float64(Float64(M_m * M_m) / d))), 1.0)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D), $MachinePrecision], 4e-276], N[(w0 * 1.0), $MachinePrecision], If[LessEqual[N[(M$95$m * D), $MachinePrecision], 5e-131], N[(w0 * N[Sqrt[N[(N[(N[(N[(N[(h / d), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * -0.25), $MachinePrecision] + l), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M$95$m * D), $MachinePrecision], 4e+140], N[(w0 * N[Sqrt[N[(N[(N[(N[(M$95$m * D), $MachinePrecision] * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * h), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(D / l), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 M D) < 4e-276

   1. Initial program 79.8%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 70.1%

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

   if 4e-276 < (*.f64 M D) < 5.0000000000000004e-131

   1. Initial program 73.9%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 89.8%

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

   if 5.0000000000000004e-131 < (*.f64 M D) < 4.00000000000000024e140

   1. Initial program 76.3%

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

   4. Applied rewrites 81.6%

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

   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 78.3%

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

   9. Applied rewrites 81.8%

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

   if 4.00000000000000024e140 < (*.f64 M D)

   1. Initial program 63.6%

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

   3. Taylor expanded in 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 45.6%

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

   7. Applied rewrites 67.1%

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

Alternative 14: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D \leq 5 \cdot 10^{-131}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{elif}\;M\_m \cdot D \leq 10^{+172}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(M\_m \cdot D\right) \cdot \left(M\_m \cdot D\right)}{\left(\ell \cdot d\right) \cdot d}, -0.25 \cdot h, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\frac{\frac{h}{d} \cdot \left(\frac{M\_m}{\ell} \cdot M\_m\right)}{d} \cdot D, -0.125 \cdot D, 1\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* M_m D) 5e-131)
   (* w0 1.0)
   (if (<= (* M_m D) 1e+172)
     (*
      w0
      (sqrt (fma (/ (* (* M_m D) (* M_m D)) (* (* l d) d)) (* -0.25 h) 1.0)))
     (* w0 (fma (* (/ (* (/ h d) (* (/ M_m l) M_m)) d) D) (* -0.125 D) 1.0)))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((M_m * D) <= 5e-131) {
		tmp = w0 * 1.0;
	} else if ((M_m * D) <= 1e+172) {
		tmp = w0 * sqrt(fma((((M_m * D) * (M_m * D)) / ((l * d) * d)), (-0.25 * h), 1.0));
	} else {
		tmp = w0 * fma(((((h / d) * ((M_m / l) * M_m)) / d) * D), (-0.125 * D), 1.0);
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D) <= 5e-131)
		tmp = Float64(w0 * 1.0);
	elseif (Float64(M_m * D) <= 1e+172)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(M_m * D) * Float64(M_m * D)) / Float64(Float64(l * d) * d)), Float64(-0.25 * h), 1.0)));
	else
		tmp = Float64(w0 * fma(Float64(Float64(Float64(Float64(h / d) * Float64(Float64(M_m / l) * M_m)) / d) * D), Float64(-0.125 * D), 1.0));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D), $MachinePrecision], 5e-131], N[(w0 * 1.0), $MachinePrecision], If[LessEqual[N[(M$95$m * D), $MachinePrecision], 1e+172], N[(w0 * N[Sqrt[N[(N[(N[(N[(M$95$m * D), $MachinePrecision] * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * h), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[(N[(N[(N[(h / d), $MachinePrecision] * N[(N[(M$95$m / l), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * D), $MachinePrecision] * N[(-0.125 * D), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 5.0000000000000004e-131

   1. Initial program 79.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 72.2%

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

   if 5.0000000000000004e-131 < (*.f64 M D) < 1.0000000000000001e172

   1. Initial program 75.8%

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

   4. Applied rewrites 79.4%

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

   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 76.1%

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

   9. Applied rewrites 79.4%

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

   if 1.0000000000000001e172 < (*.f64 M D)

   1. Initial program 62.7%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      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 48.1%

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

   7. Applied rewrites 56.7%

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

   9. Applied rewrites 63.4%

      \[\leadsto w0 \cdot \mathsf{fma}\left(\frac{\frac{h}{d} \cdot \left(\frac{M}{\ell} \cdot M\right)}{d} \cdot D, -0.125 \cdot D, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 82.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.5 \cdot 10^{-208}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\left(h \cdot \left(M\_m \cdot D\right)\right) \cdot \left(M\_m \cdot D\right)}{-2 \cdot d}}{\left(-2 \cdot d\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\ell - \frac{\left(h \cdot M\_m\right) \cdot \left(\left(\frac{M\_m}{d} \cdot D\right) \cdot 0.25\right)}{d} \cdot D}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= d 1.5e-208)
   (*
    w0
    (sqrt
     (-
      1.0
      (/ (/ (* (* h (* M_m D)) (* M_m D)) (* -2.0 d)) (* (* -2.0 d) l)))))
   (*
    w0
    (sqrt (/ (- l (* (/ (* (* h M_m) (* (* (/ M_m d) D) 0.25)) d) D)) l)))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.5e-208) {
		tmp = w0 * sqrt((1.0 - ((((h * (M_m * D)) * (M_m * D)) / (-2.0 * d)) / ((-2.0 * d) * l))));
	} else {
		tmp = w0 * sqrt(((l - ((((h * M_m) * (((M_m / d) * D) * 0.25)) / d) * D)) / l));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(w0, m_m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 1.5d-208) then
        tmp = w0 * sqrt((1.0d0 - ((((h * (m_m * d)) * (m_m * d)) / ((-2.0d0) * d_1)) / (((-2.0d0) * d_1) * l))))
    else
        tmp = w0 * sqrt(((l - ((((h * m_m) * (((m_m / d_1) * d) * 0.25d0)) / d_1) * d)) / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.5e-208) {
		tmp = w0 * Math.sqrt((1.0 - ((((h * (M_m * D)) * (M_m * D)) / (-2.0 * d)) / ((-2.0 * d) * l))));
	} else {
		tmp = w0 * Math.sqrt(((l - ((((h * M_m) * (((M_m / d) * D) * 0.25)) / d) * D)) / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if d <= 1.5e-208:
		tmp = w0 * math.sqrt((1.0 - ((((h * (M_m * D)) * (M_m * D)) / (-2.0 * d)) / ((-2.0 * d) * l))))
	else:
		tmp = w0 * math.sqrt(((l - ((((h * M_m) * (((M_m / d) * D) * 0.25)) / d) * D)) / l))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (d <= 1.5e-208)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(h * Float64(M_m * D)) * Float64(M_m * D)) / Float64(-2.0 * d)) / Float64(Float64(-2.0 * d) * l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(Float64(l - Float64(Float64(Float64(Float64(h * M_m) * Float64(Float64(Float64(M_m / d) * D) * 0.25)) / d) * D)) / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (d <= 1.5e-208)
		tmp = w0 * sqrt((1.0 - ((((h * (M_m * D)) * (M_m * D)) / (-2.0 * d)) / ((-2.0 * d) * l))));
	else
		tmp = w0 * sqrt(((l - ((((h * M_m) * (((M_m / d) * D) * 0.25)) / d) * D)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[d, 1.5e-208], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(h * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * D), $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[(l - N[(N[(N[(N[(h * M$95$m), $MachinePrecision] * N[(N[(N[(M$95$m / d), $MachinePrecision] * D), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.49999999999999993e-208

   1. Initial program 80.2%

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

   4. Applied rewrites 84.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{h \cdot {\left(D \cdot M\right)}^{2}}{-2 \cdot d}}{\left(-2 \cdot d\right) \cdot \ell}}} \]
   5. 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 84.9

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

      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 84.9

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

      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 84.9

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

   6. Applied rewrites 84.9%

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

   if 1.49999999999999993e-208 < d

   1. Initial program 74.2%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 67.9

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

   5. Applied rewrites 67.9%

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

   7. Applied rewrites 81.9%

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

   9. Applied rewrites 82.8%

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

Alternative 16: 73.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D \leq 2 \cdot 10^{-111}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(\left(\frac{M\_m}{d} \cdot M\_m\right) \cdot \frac{h}{\ell \cdot d}\right) \cdot D, -0.125 \cdot D, 1\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* M_m D) 2e-111)
   (* w0 1.0)
   (* w0 (fma (* (* (* (/ M_m d) M_m) (/ h (* l d))) D) (* -0.125 D) 1.0))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((M_m * D) <= 2e-111) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * fma(((((M_m / d) * M_m) * (h / (l * d))) * D), (-0.125 * D), 1.0);
	}
	return tmp;
}

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D) <= 2e-111)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * fma(Float64(Float64(Float64(Float64(M_m / d) * M_m) * Float64(h / Float64(l * d))) * D), Float64(-0.125 * D), 1.0));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D), $MachinePrecision], 2e-111], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[(N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(h / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * N[(-0.125 * D), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.00000000000000018e-111

   1. Initial program 79.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 72.5%

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

   if 2.00000000000000018e-111 < (*.f64 M D)

   1. Initial program 72.7%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      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 29.1%

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

   7. Applied rewrites 51.4%

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

   9. Applied rewrites 49.9%

      \[\leadsto w0 \cdot \mathsf{fma}\left(\left(\left(\frac{M}{d} \cdot M\right) \cdot \frac{h}{\ell \cdot d}\right) \cdot D, -0.125 \cdot D, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 79.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{\left(M\_m \cdot D\right) \cdot M\_m}{d} \cdot \frac{D}{\ell \cdot d}, 1\right)} \end{array} \]

FPCore

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

C

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

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	return Float64(w0 * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(Float64(M_m * D) * M_m) / d) * Float64(D / Float64(l * d))), 1.0)))
end

Wolfram

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

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in 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 64.2%

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

7. Applied rewrites 73.9%

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

9. Applied rewrites 81.0%

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

Alternative 18: 67.3% accurate, 26.2× speedup

Math

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

FPCore

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

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * 1.0;
}

Fortran

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

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

real(8) function code(w0, m_m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * 1.0d0
end function

Java

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

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	return w0 * 1.0

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	return Float64(w0 * 1.0)
end

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := N[(w0 * 1.0), $MachinePrecision]

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 66.3%

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

Reproduce

herbie shell --seed 2024354 
(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))))))