Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.5% → 87.0%

Time: 10.8s

Alternatives: 14

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: 81.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;w0\_m \cdot \sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 10^{+187}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - {\left(\frac{\frac{M\_m}{d} \cdot D\_m}{2}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{D\_m}{2} \cdot \left(\frac{M\_m}{d} \cdot \frac{\left(h \cdot D\_m\right) \cdot M\_m}{\ell \cdot \left(d \cdot 2\right)}\right)}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (if (<=
       (* w0_m (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)))))
       1e+187)
    (* w0_m (sqrt (- 1.0 (* (pow (/ (* (/ M_m d) D_m) 2.0) 2.0) (/ h l)))))
    (*
     w0_m
     (sqrt
      (-
       1.0
       (*
        (/ D_m 2.0)
        (* (/ M_m d) (/ (* (* h D_m) M_m) (* l (* d 2.0)))))))))))

C

D_m = fabs(D);
M_m = fabs(M);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((w0_m * sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 1e+187) {
		tmp = w0_m * sqrt((1.0 - (pow((((M_m / d) * D_m) / 2.0), 2.0) * (h / l))));
	} else {
		tmp = w0_m * sqrt((1.0 - ((D_m / 2.0) * ((M_m / d) * (((h * D_m) * M_m) / (l * (d * 2.0)))))));
	}
	return w0_s * tmp;
}

Fortran

D_m =     private
M_m =     private
w0\_m =     private
w0\_s =     private
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((w0_m * sqrt((1.0d0 - ((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l))))) <= 1d+187) then
        tmp = w0_m * sqrt((1.0d0 - (((((m_m / d) * d_m) / 2.0d0) ** 2.0d0) * (h / l))))
    else
        tmp = w0_m * sqrt((1.0d0 - ((d_m / 2.0d0) * ((m_m / d) * (((h * d_m) * m_m) / (l * (d * 2.0d0)))))))
    end if
    code = w0_s * tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((w0_m * Math.sqrt((1.0 - (Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 1e+187) {
		tmp = w0_m * Math.sqrt((1.0 - (Math.pow((((M_m / d) * D_m) / 2.0), 2.0) * (h / l))));
	} else {
		tmp = w0_m * Math.sqrt((1.0 - ((D_m / 2.0) * ((M_m / d) * (((h * D_m) * M_m) / (l * (d * 2.0)))))));
	}
	return w0_s * tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d] = sort([w0_m, M_m, D_m, h, l, d])
def code(w0_s, w0_m, M_m, D_m, h, l, d):
	tmp = 0
	if (w0_m * math.sqrt((1.0 - (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 1e+187:
		tmp = w0_m * math.sqrt((1.0 - (math.pow((((M_m / d) * D_m) / 2.0), 2.0) * (h / l))))
	else:
		tmp = w0_m * math.sqrt((1.0 - ((D_m / 2.0) * ((M_m / d) * (((h * D_m) * M_m) / (l * (d * 2.0)))))))
	return w0_s * tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(w0_m * sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))) <= 1e+187)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64((Float64(Float64(Float64(M_m / d) * D_m) / 2.0) ^ 2.0) * Float64(h / l)))));
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(D_m / 2.0) * Float64(Float64(M_m / d) * Float64(Float64(Float64(h * D_m) * M_m) / Float64(l * Float64(d * 2.0))))))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d = num2cell(sort([w0_m, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((w0_m * sqrt((1.0 - ((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l))))) <= 1e+187)
		tmp = w0_m * sqrt((1.0 - (((((M_m / d) * D_m) / 2.0) ^ 2.0) * (h / l))));
	else
		tmp = w0_m * sqrt((1.0 - ((D_m / 2.0) * ((M_m / d) * (((h * D_m) * M_m) / (l * (d * 2.0)))))));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+187], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision] / 2.0), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(N[(h * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))) < 9.99999999999999907e186

   1. Initial program 93.0%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 92.4

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

   4. Applied rewrites 92.4%

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

   if 9.99999999999999907e186 < (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))))

   1. Initial program 61.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. 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 74.4%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{D}{2} \cdot \left(\frac{M}{d} \cdot \left(\left(\frac{h}{\ell} \cdot D\right) \cdot \frac{\frac{M}{d}}{2}\right)\right)}} \]
   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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. *-commutative N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 83.6

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

   6. Applied rewrites 83.6%

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

Alternative 2: 87.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := w0\_m \cdot \sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+187}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{D\_m}{2} \cdot \left(\frac{M\_m}{d} \cdot \frac{\left(h \cdot D\_m\right) \cdot M\_m}{\ell \cdot \left(d \cdot 2\right)}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (let* ((t_0
         (*
          w0_m
          (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)))))))
   (*
    w0_s
    (if (<= t_0 1e+187)
      t_0
      (*
       w0_m
       (sqrt
        (-
         1.0
         (*
          (/ D_m 2.0)
          (* (/ M_m d) (/ (* (* h D_m) M_m) (* l (* d 2.0))))))))))))

C

D_m = fabs(D);
M_m = fabs(M);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double t_0 = w0_m * sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 1e+187) {
		tmp = t_0;
	} else {
		tmp = w0_m * sqrt((1.0 - ((D_m / 2.0) * ((M_m / d) * (((h * D_m) * M_m) / (l * (d * 2.0)))))));
	}
	return w0_s * tmp;
}

Fortran

D_m =     private
M_m =     private
w0\_m =     private
w0\_s =     private
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = w0_m * sqrt((1.0d0 - ((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l))))
    if (t_0 <= 1d+187) then
        tmp = t_0
    else
        tmp = w0_m * sqrt((1.0d0 - ((d_m / 2.0d0) * ((m_m / d) * (((h * d_m) * m_m) / (l * (d * 2.0d0)))))))
    end if
    code = w0_s * tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double t_0 = w0_m * Math.sqrt((1.0 - (Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 1e+187) {
		tmp = t_0;
	} else {
		tmp = w0_m * Math.sqrt((1.0 - ((D_m / 2.0) * ((M_m / d) * (((h * D_m) * M_m) / (l * (d * 2.0)))))));
	}
	return w0_s * tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d] = sort([w0_m, M_m, D_m, h, l, d])
def code(w0_s, w0_m, M_m, D_m, h, l, d):
	t_0 = w0_m * math.sqrt((1.0 - (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))
	tmp = 0
	if t_0 <= 1e+187:
		tmp = t_0
	else:
		tmp = w0_m * math.sqrt((1.0 - ((D_m / 2.0) * ((M_m / d) * (((h * D_m) * M_m) / (l * (d * 2.0)))))))
	return w0_s * tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	t_0 = Float64(w0_m * sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
	tmp = 0.0
	if (t_0 <= 1e+187)
		tmp = t_0;
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(D_m / 2.0) * Float64(Float64(M_m / d) * Float64(Float64(Float64(h * D_m) * M_m) / Float64(l * Float64(d * 2.0))))))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d = num2cell(sort([w0_m, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d)
	t_0 = w0_m * sqrt((1.0 - ((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l))));
	tmp = 0.0;
	if (t_0 <= 1e+187)
		tmp = t_0;
	else
		tmp = w0_m * sqrt((1.0 - ((D_m / 2.0) * ((M_m / d) * (((h * D_m) * M_m) / (l * (d * 2.0)))))));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[t$95$0, 1e+187], t$95$0, N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(N[(h * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))) < 9.99999999999999907e186

   1. Initial program 93.0%

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

   if 9.99999999999999907e186 < (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))))

   1. Initial program 61.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. 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 74.4%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{D}{2} \cdot \left(\frac{M}{d} \cdot \left(\left(\frac{h}{\ell} \cdot D\right) \cdot \frac{\frac{M}{d}}{2}\right)\right)}} \]
   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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. *-commutative N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 83.6

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

   6. Applied rewrites 83.6%

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

Alternative 3: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;\sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 1:\\ \;\;\;\;w0\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{\mathsf{fma}\left(\left(-0.25 \cdot h\right) \cdot \left(\left(\frac{M\_m}{d} \cdot D\_m\right) \cdot \frac{M\_m}{d}\right), \frac{D\_m}{\ell}, 1\right)}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)))) 1.0)
    (* w0_m 1.0)
    (*
     w0_m
     (sqrt
      (fma (* (* -0.25 h) (* (* (/ M_m d) D_m) (/ M_m d))) (/ D_m l) 1.0))))))

C

D_m = fabs(D);
M_m = fabs(M);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)))) <= 1.0) {
		tmp = w0_m * 1.0;
	} else {
		tmp = w0_m * sqrt(fma(((-0.25 * h) * (((M_m / d) * D_m) * (M_m / d))), (D_m / l), 1.0));
	}
	return w0_s * tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0
	if (sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))) <= 1.0)
		tmp = Float64(w0_m * 1.0);
	else
		tmp = Float64(w0_m * sqrt(fma(Float64(Float64(-0.25 * h) * Float64(Float64(Float64(M_m / d) * D_m) * Float64(M_m / d))), Float64(D_m / l), 1.0)));
	end
	return Float64(w0_s * tmp)
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0], N[(w0$95$m * 1.0), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 98.6%

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

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

   1. Initial program 59.8%

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

   3. Taylor expanded in 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 45.3%

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

   7. Applied rewrites 49.9%

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

   9. Applied rewrites 69.1%

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

   11. Applied rewrites 69.0%

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

Alternative 4: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;w0\_m \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{D\_m \cdot M\_m}{\ell \cdot d} \cdot \left(\frac{M\_m}{d} \cdot D\_m\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -1e-9)
    (*
     w0_m
     (sqrt
      (fma (* h -0.25) (* (/ (* D_m M_m) (* l d)) (* (/ M_m d) D_m)) 1.0)))
    (* w0_m 1.0))))

C

D_m = fabs(D);
M_m = fabs(M);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e-9) {
		tmp = w0_m * sqrt(fma((h * -0.25), (((D_m * M_m) / (l * d)) * ((M_m / d) * D_m)), 1.0));
	} else {
		tmp = w0_m * 1.0;
	}
	return w0_s * tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e-9)
		tmp = Float64(w0_m * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(D_m * M_m) / Float64(l * d)) * Float64(Float64(M_m / d) * D_m)), 1.0)));
	else
		tmp = Float64(w0_m * 1.0);
	end
	return Float64(w0_s * tmp)
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e-9], N[(w0$95$m * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.00000000000000006e-9

   1. Initial program 70.8%

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

   3. Taylor expanded in 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 44.4%

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

   7. Applied rewrites 48.5%

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

   9. Applied rewrites 67.3%

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

   11. Applied rewrites 66.2%

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

   if -1.00000000000000006e-9 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 97.1%

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

Alternative 5: 84.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;w0\_m \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \left(\left(\frac{M\_m}{d} \cdot M\_m\right) \cdot D\_m\right) \cdot \frac{D\_m}{\ell \cdot d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -1e-9)
    (*
     w0_m
     (sqrt
      (fma (* h -0.25) (* (* (* (/ M_m d) M_m) D_m) (/ D_m (* l d))) 1.0)))
    (* w0_m 1.0))))

C

D_m = fabs(D);
M_m = fabs(M);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e-9) {
		tmp = w0_m * sqrt(fma((h * -0.25), ((((M_m / d) * M_m) * D_m) * (D_m / (l * d))), 1.0));
	} else {
		tmp = w0_m * 1.0;
	}
	return w0_s * tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e-9)
		tmp = Float64(w0_m * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(Float64(M_m / d) * M_m) * D_m) * Float64(D_m / Float64(l * d))), 1.0)));
	else
		tmp = Float64(w0_m * 1.0);
	end
	return Float64(w0_s * tmp)
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e-9], N[(w0$95$m * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.00000000000000006e-9

   1. Initial program 70.8%

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

   3. Taylor expanded in 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 44.4%

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

   7. Applied rewrites 54.8%

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

   if -1.00000000000000006e-9 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 97.1%

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

Alternative 6: 83.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -500:\\ \;\;\;\;w0\_m \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, M\_m \cdot \left(\left(D\_m \cdot M\_m\right) \cdot \frac{D\_m}{\left(d \cdot \ell\right) \cdot d}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -500.0)
    (*
     w0_m
     (sqrt
      (fma (* h -0.25) (* M_m (* (* D_m M_m) (/ D_m (* (* d l) d)))) 1.0)))
    (* w0_m 1.0))))

C

D_m = fabs(D);
M_m = fabs(M);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -500.0) {
		tmp = w0_m * sqrt(fma((h * -0.25), (M_m * ((D_m * M_m) * (D_m / ((d * l) * d)))), 1.0));
	} else {
		tmp = w0_m * 1.0;
	}
	return w0_s * tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -500.0)
		tmp = Float64(w0_m * sqrt(fma(Float64(h * -0.25), Float64(M_m * Float64(Float64(D_m * M_m) * Float64(D_m / Float64(Float64(d * l) * d)))), 1.0)));
	else
		tmp = Float64(w0_m * 1.0);
	end
	return Float64(w0_s * tmp)
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -500.0], N[(w0$95$m * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(M$95$m * N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m / N[(N[(d * l), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -500

   1. Initial program 70.1%

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

   3. Taylor expanded in h around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. rgt-mult-inverse N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 45.5%

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

   7. Applied rewrites 48.5%

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

   9. Applied rewrites 54.9%

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

   if -500 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.7%

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

Alternative 7: 82.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -500:\\ \;\;\;\;w0\_m \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)}{\left(d \cdot d\right) \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -500.0)
    (*
     w0_m
     (sqrt
      (fma (* h -0.25) (/ (* (* M_m D_m) (* M_m D_m)) (* (* d d) l)) 1.0)))
    (* w0_m 1.0))))

C

D_m = fabs(D);
M_m = fabs(M);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -500.0) {
		tmp = w0_m * sqrt(fma((h * -0.25), (((M_m * D_m) * (M_m * D_m)) / ((d * d) * l)), 1.0));
	} else {
		tmp = w0_m * 1.0;
	}
	return w0_s * tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -500.0)
		tmp = Float64(w0_m * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) / Float64(Float64(d * d) * l)), 1.0)));
	else
		tmp = Float64(w0_m * 1.0);
	end
	return Float64(w0_s * tmp)
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -500.0], N[(w0$95$m * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -500

   1. Initial program 70.1%

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

   3. Taylor expanded in h around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. rgt-mult-inverse N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 45.5%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 49.4%

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

   if -500 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.7%

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

Alternative 8: 80.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot -0.125, \frac{M\_m}{\ell \cdot d} \cdot \frac{\left(h \cdot M\_m\right) \cdot w0\_m}{d}, w0\_m\right)\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -4e+125)
    (fma
     (* (* D_m D_m) -0.125)
     (* (/ M_m (* l d)) (/ (* (* h M_m) w0_m) d))
     w0_m)
    (* w0_m 1.0))))

C

D_m = fabs(D);
M_m = fabs(M);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -4e+125) {
		tmp = fma(((D_m * D_m) * -0.125), ((M_m / (l * d)) * (((h * M_m) * w0_m) / d)), w0_m);
	} else {
		tmp = w0_m * 1.0;
	}
	return w0_s * tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -4e+125)
		tmp = fma(Float64(Float64(D_m * D_m) * -0.125), Float64(Float64(M_m / Float64(l * d)) * Float64(Float64(Float64(h * M_m) * w0_m) / d)), w0_m);
	else
		tmp = Float64(w0_m * 1.0);
	end
	return Float64(w0_s * tmp)
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -4e+125], N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(M$95$m / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * M$95$m), $MachinePrecision] * w0$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + w0$95$m), $MachinePrecision], N[(w0$95$m * 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -3.9999999999999997e125

   1. Initial program 66.0%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 38.4%

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

   7. Applied rewrites 41.4%

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

   9. Applied rewrites 45.0%

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

   if -3.9999999999999997e125 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 92.6%

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

Alternative 9: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot -0.125, \left(\left(M\_m \cdot \frac{h}{d}\right) \cdot M\_m\right) \cdot \frac{w0\_m}{d \cdot \ell}, w0\_m\right)\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -4e+153)
    (fma
     (* (* D_m D_m) -0.125)
     (* (* (* M_m (/ h d)) M_m) (/ w0_m (* d l)))
     w0_m)
    (* w0_m 1.0))))

C

D_m = fabs(D);
M_m = fabs(M);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -4e+153) {
		tmp = fma(((D_m * D_m) * -0.125), (((M_m * (h / d)) * M_m) * (w0_m / (d * l))), w0_m);
	} else {
		tmp = w0_m * 1.0;
	}
	return w0_s * tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -4e+153)
		tmp = fma(Float64(Float64(D_m * D_m) * -0.125), Float64(Float64(Float64(M_m * Float64(h / d)) * M_m) * Float64(w0_m / Float64(d * l))), w0_m);
	else
		tmp = Float64(w0_m * 1.0);
	end
	return Float64(w0_s * tmp)
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -4e+153], N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(M$95$m * N[(h / d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(w0$95$m / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0$95$m), $MachinePrecision], N[(w0$95$m * 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4e153

   1. Initial program 65.5%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 39.0%

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

   7. Applied rewrites 45.5%

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

   if -4e153 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 92.2%

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

Alternative 10: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot -0.125, \left(\frac{w0\_m}{\left(\ell \cdot d\right) \cdot d} \cdot M\_m\right) \cdot \left(h \cdot M\_m\right), w0\_m\right)\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -4e+153)
    (fma
     (* (* D_m D_m) -0.125)
     (* (* (/ w0_m (* (* l d) d)) M_m) (* h M_m))
     w0_m)
    (* w0_m 1.0))))

C

D_m = fabs(D);
M_m = fabs(M);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -4e+153) {
		tmp = fma(((D_m * D_m) * -0.125), (((w0_m / ((l * d) * d)) * M_m) * (h * M_m)), w0_m);
	} else {
		tmp = w0_m * 1.0;
	}
	return w0_s * tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -4e+153)
		tmp = fma(Float64(Float64(D_m * D_m) * -0.125), Float64(Float64(Float64(w0_m / Float64(Float64(l * d) * d)) * M_m) * Float64(h * M_m)), w0_m);
	else
		tmp = Float64(w0_m * 1.0);
	end
	return Float64(w0_s * tmp)
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -4e+153], N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(w0$95$m / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] + w0$95$m), $MachinePrecision], N[(w0$95$m * 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4e153

   1. Initial program 65.5%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 39.0%

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

   7. Applied rewrites 42.0%

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

   9. Applied rewrites 43.8%

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

   if -4e153 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 92.2%

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

Alternative 11: 80.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot -0.125, M\_m \cdot \left(M\_m \cdot \left(\frac{w0\_m}{\left(\ell \cdot d\right) \cdot d} \cdot h\right)\right), w0\_m\right)\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -4e-8)
    (fma
     (* (* D_m D_m) -0.125)
     (* M_m (* M_m (* (/ w0_m (* (* l d) d)) h)))
     w0_m)
    (* w0_m 1.0))))

C

D_m = fabs(D);
M_m = fabs(M);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -4e-8) {
		tmp = fma(((D_m * D_m) * -0.125), (M_m * (M_m * ((w0_m / ((l * d) * d)) * h))), w0_m);
	} else {
		tmp = w0_m * 1.0;
	}
	return w0_s * tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -4e-8)
		tmp = fma(Float64(Float64(D_m * D_m) * -0.125), Float64(M_m * Float64(M_m * Float64(Float64(w0_m / Float64(Float64(l * d) * d)) * h))), w0_m);
	else
		tmp = Float64(w0_m * 1.0);
	end
	return Float64(w0_s * tmp)
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -4e-8], N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(M$95$m * N[(M$95$m * N[(N[(w0$95$m / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0$95$m), $MachinePrecision], N[(w0$95$m * 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.0000000000000001e-8

   1. Initial program 70.4%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 33.7%

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

   7. Applied rewrites 37.6%

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

   9. Applied rewrites 39.7%

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

   if -4.0000000000000001e-8 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.9%

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

Alternative 12: 87.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 5 \cdot 10^{-204} \lor \neg \left(M\_m \cdot D\_m \leq 5 \cdot 10^{+225}\right):\\ \;\;\;\;w0\_m \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \left(\frac{D\_m}{d} \cdot \frac{M\_m}{d}\right) \cdot \left(M\_m \cdot \frac{D\_m}{\ell}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{D\_m \cdot M\_m}{\ell \cdot d} \cdot \left(\frac{M\_m}{d} \cdot D\_m\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (if (or (<= (* M_m D_m) 5e-204) (not (<= (* M_m D_m) 5e+225)))
    (*
     w0_m
     (sqrt
      (fma (* h -0.25) (* (* (/ D_m d) (/ M_m d)) (* M_m (/ D_m l))) 1.0)))
    (*
     w0_m
     (sqrt
      (fma (* h -0.25) (* (/ (* D_m M_m) (* l d)) (* (/ M_m d) D_m)) 1.0))))))

C

D_m = fabs(D);
M_m = fabs(M);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((M_m * D_m) <= 5e-204) || !((M_m * D_m) <= 5e+225)) {
		tmp = w0_m * sqrt(fma((h * -0.25), (((D_m / d) * (M_m / d)) * (M_m * (D_m / l))), 1.0));
	} else {
		tmp = w0_m * sqrt(fma((h * -0.25), (((D_m * M_m) / (l * d)) * ((M_m / d) * D_m)), 1.0));
	}
	return w0_s * tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0
	if ((Float64(M_m * D_m) <= 5e-204) || !(Float64(M_m * D_m) <= 5e+225))
		tmp = Float64(w0_m * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(D_m / d) * Float64(M_m / d)) * Float64(M_m * Float64(D_m / l))), 1.0)));
	else
		tmp = Float64(w0_m * sqrt(fma(Float64(h * -0.25), Float64(Float64(Float64(D_m * M_m) / Float64(l * d)) * Float64(Float64(M_m / d) * D_m)), 1.0)));
	end
	return Float64(w0_s * tmp)
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * If[Or[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e-204], N[Not[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e+225]], $MachinePrecision]], N[(w0$95$m * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] * N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 5.0000000000000002e-204 or 4.99999999999999981e225 < (*.f64 M D)

   1. Initial program 82.8%

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

   3. Taylor expanded in h around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. rgt-mult-inverse N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 66.6%

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

   7. Applied rewrites 68.9%

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

   9. Applied rewrites 85.7%

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

   if 5.0000000000000002e-204 < (*.f64 M D) < 4.99999999999999981e225

   1. Initial program 90.8%

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

   3. Taylor expanded in 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 63.1%

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

   7. Applied rewrites 65.9%

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

   9. Applied rewrites 84.4%

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

   11. Applied rewrites 95.1%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 5 \cdot 10^{-204} \lor \neg \left(M \cdot D \leq 5 \cdot 10^{+225}\right):\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \left(\frac{D}{d} \cdot \frac{M}{d}\right) \cdot \left(M \cdot \frac{D}{\ell}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(h \cdot -0.25, \frac{D \cdot M}{\ell \cdot d} \cdot \left(\frac{M}{d} \cdot D\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;d \leq 3.6 \cdot 10^{+92}:\\ \;\;\;\;w0\_m \cdot \sqrt{\mathsf{fma}\left(\left(-0.25 \cdot h\right) \cdot \left(\left(\frac{M\_m}{d} \cdot D\_m\right) \cdot \frac{M\_m}{d}\right), \frac{D\_m}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{\frac{\ell - \left(\left(\left(\frac{h}{d} \cdot M\_m\right) \cdot \frac{M\_m}{d}\right) \cdot \left(0.25 \cdot D\_m\right)\right) \cdot D\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (if (<= d 3.6e+92)
    (*
     w0_m
     (sqrt
      (fma (* (* -0.25 h) (* (* (/ M_m d) D_m) (/ M_m d))) (/ D_m l) 1.0)))
    (*
     w0_m
     (sqrt
      (/ (- l (* (* (* (* (/ h d) M_m) (/ M_m d)) (* 0.25 D_m)) D_m)) l))))))

C

D_m = fabs(D);
M_m = fabs(M);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 3.6e+92) {
		tmp = w0_m * sqrt(fma(((-0.25 * h) * (((M_m / d) * D_m) * (M_m / d))), (D_m / l), 1.0));
	} else {
		tmp = w0_m * sqrt(((l - (((((h / d) * M_m) * (M_m / d)) * (0.25 * D_m)) * D_m)) / l));
	}
	return w0_s * tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0
	if (d <= 3.6e+92)
		tmp = Float64(w0_m * sqrt(fma(Float64(Float64(-0.25 * h) * Float64(Float64(Float64(M_m / d) * D_m) * Float64(M_m / d))), Float64(D_m / l), 1.0)));
	else
		tmp = Float64(w0_m * sqrt(Float64(Float64(l - Float64(Float64(Float64(Float64(Float64(h / d) * M_m) * Float64(M_m / d)) * Float64(0.25 * D_m)) * D_m)) / l)));
	end
	return Float64(w0_s * tmp)
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[d, 3.6e+92], N[(w0$95$m * N[Sqrt[N[(N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(N[(M$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / l), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(N[(l - N[(N[(N[(N[(N[(h / d), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if d < 3.6e92

   1. Initial program 84.2%

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

   3. Taylor expanded in 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 63.5%

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

   7. Applied rewrites 66.2%

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

   9. Applied rewrites 82.2%

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

   11. Applied rewrites 82.7%

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

   if 3.6e92 < d

   1. Initial program 88.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 71.4

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

   5. Applied rewrites 71.4%

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

   7. Applied rewrites 92.5%

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

Alternative 14: 67.6% accurate, 26.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \left(w0\_m \cdot 1\right) \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d) :precision binary64 (* w0_s (* w0_m 1.0)))

C

D_m = fabs(D);
M_m = fabs(M);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	return w0_s * (w0_m * 1.0);
}

Fortran

D_m =     private
M_m =     private
w0\_m =     private
w0\_s =     private
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0_s * (w0_m * 1.0d0)
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	return w0_s * (w0_m * 1.0);
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d] = sort([w0_m, M_m, D_m, h, l, d])
def code(w0_s, w0_m, M_m, D_m, h, l, d):
	return w0_s * (w0_m * 1.0)

Julia

D_m = abs(D)
M_m = abs(M)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	return Float64(w0_s * Float64(w0_m * 1.0))
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d = num2cell(sort([w0_m, M_m, D_m, h, l, d])){:}
function tmp = code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = w0_s * (w0_m * 1.0);
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * N[(w0$95$m * 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.2%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 70.5%

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

Reproduce

herbie shell --seed 2024356 
(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))))))