Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.5% → 89.5%

Time: 10.3s

Alternatives: 15

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.5% 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: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := w0\_m \cdot \sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}}\\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{D\_m \cdot \left(w0\_m \cdot \sqrt{-0.25 \cdot \frac{{M\_m}^{2} \cdot h}{d\_m \cdot \ell}}\right)}{\sqrt{d\_m}}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \frac{\sqrt{d\_m - \left(M\_m \cdot D\_m\right) \cdot \frac{\frac{\left(M\_m \cdot D\_m\right) \cdot h}{d\_m \cdot \ell}}{4}}}{\sqrt{d\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
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_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0
         (*
          w0_m
          (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)))))))
   (*
    w0_s
    (if (<= t_0 5e+307)
      t_0
      (if (<= t_0 INFINITY)
        (/
         (* D_m (* w0_m (sqrt (- (* 0.25 (/ (* (pow M_m 2.0) h) (* d_m l)))))))
         (sqrt d_m))
        (*
         w0_m
         (/
          (sqrt
           (- d_m (* (* M_m D_m) (/ (/ (* (* M_m D_m) h) (* d_m l)) 4.0))))
          (sqrt d_m))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
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_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = w0_m * sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 5e+307) {
		tmp = t_0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (D_m * (w0_m * sqrt(-(0.25 * ((pow(M_m, 2.0) * h) / (d_m * l)))))) / sqrt(d_m);
	} else {
		tmp = w0_m * (sqrt((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0)))) / sqrt(d_m));
	}
	return w0_s * tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
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_m;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = w0_m * Math.sqrt((1.0 - (Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 5e+307) {
		tmp = t_0;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (D_m * (w0_m * Math.sqrt(-(0.25 * ((Math.pow(M_m, 2.0) * h) / (d_m * l)))))) / Math.sqrt(d_m);
	} else {
		tmp = w0_m * (Math.sqrt((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0)))) / Math.sqrt(d_m));
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d_m] = sort([w0_m, M_m, D_m, h, l, d_m])
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	t_0 = w0_m * math.sqrt((1.0 - (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l))))
	tmp = 0
	if t_0 <= 5e+307:
		tmp = t_0
	elif t_0 <= math.inf:
		tmp = (D_m * (w0_m * math.sqrt(-(0.25 * ((math.pow(M_m, 2.0) * h) / (d_m * l)))))) / math.sqrt(d_m)
	else:
		tmp = w0_m * (math.sqrt((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0)))) / math.sqrt(d_m))
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	t_0 = Float64(w0_m * sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)))))
	tmp = 0.0
	if (t_0 <= 5e+307)
		tmp = t_0;
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(D_m * Float64(w0_m * sqrt(Float64(-Float64(0.25 * Float64(Float64((M_m ^ 2.0) * h) / Float64(d_m * l))))))) / sqrt(d_m));
	else
		tmp = Float64(w0_m * Float64(sqrt(Float64(d_m - Float64(Float64(M_m * D_m) * Float64(Float64(Float64(Float64(M_m * D_m) * h) / Float64(d_m * l)) / 4.0)))) / sqrt(d_m)));
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d_m = num2cell(sort([w0_m, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	t_0 = w0_m * sqrt((1.0 - ((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l))));
	tmp = 0.0;
	if (t_0 <= 5e+307)
		tmp = t_0;
	elseif (t_0 <= Inf)
		tmp = (D_m * (w0_m * sqrt(-(0.25 * (((M_m ^ 2.0) * h) / (d_m * l)))))) / sqrt(d_m);
	else
		tmp = w0_m * (sqrt((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0)))) / sqrt(d_m));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $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_m 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$95$m_] := 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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[t$95$0, 5e+307], t$95$0, If[LessEqual[t$95$0, Infinity], N[(N[(D$95$m * N[(w0$95$m * N[Sqrt[(-N[(0.25 * N[(N[(N[Power[M$95$m, 2.0], $MachinePrecision] * h), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[d$95$m], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[(N[Sqrt[N[(d$95$m - N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[d$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 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))))) < 5e307

   1. Initial program 80.5%

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

   if 5e307 < (*.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))))) < +inf.0

   1. Initial program 80.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 84.5%

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in D around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 21.8%

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

   if +inf.0 < (*.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 80.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 84.5%

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

   5. Applied rewrites 85.4%

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

Alternative 2: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := w0\_m \cdot \sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}}\\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \frac{\sqrt{d\_m - \left(M\_m \cdot D\_m\right) \cdot \frac{\frac{\left(M\_m \cdot D\_m\right) \cdot h}{d\_m \cdot \ell}}{4}}}{\sqrt{d\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
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_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0
         (*
          w0_m
          (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)))))))
   (*
    w0_s
    (if (<= t_0 5e+304)
      t_0
      (*
       w0_m
       (/
        (sqrt (- d_m (* (* M_m D_m) (/ (/ (* (* M_m D_m) h) (* d_m l)) 4.0))))
        (sqrt d_m)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
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_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = w0_m * sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 5e+304) {
		tmp = t_0;
	} else {
		tmp = w0_m * (sqrt((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0)))) / sqrt(d_m));
	}
	return w0_s * tmp;
}

Fortran

M_m =     private
D_m =     private
d_m =     private
w0\_m =     private
w0\_s =     private
NOTE: w0_m, M_m, D_m, h, l, and d_m 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_m_1)
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_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = w0_m * sqrt((1.0d0 - ((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l))))
    if (t_0 <= 5d+304) then
        tmp = t_0
    else
        tmp = w0_m * (sqrt((d_m_1 - ((m_m * d_m) * ((((m_m * d_m) * h) / (d_m_1 * l)) / 4.0d0)))) / sqrt(d_m_1))
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
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_m;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = w0_m * Math.sqrt((1.0 - (Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 5e+304) {
		tmp = t_0;
	} else {
		tmp = w0_m * (Math.sqrt((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0)))) / Math.sqrt(d_m));
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d_m] = sort([w0_m, M_m, D_m, h, l, d_m])
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	t_0 = w0_m * math.sqrt((1.0 - (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l))))
	tmp = 0
	if t_0 <= 5e+304:
		tmp = t_0
	else:
		tmp = w0_m * (math.sqrt((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0)))) / math.sqrt(d_m))
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	t_0 = Float64(w0_m * sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)))))
	tmp = 0.0
	if (t_0 <= 5e+304)
		tmp = t_0;
	else
		tmp = Float64(w0_m * Float64(sqrt(Float64(d_m - Float64(Float64(M_m * D_m) * Float64(Float64(Float64(Float64(M_m * D_m) * h) / Float64(d_m * l)) / 4.0)))) / sqrt(d_m)));
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d_m = num2cell(sort([w0_m, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	t_0 = w0_m * sqrt((1.0 - ((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l))));
	tmp = 0.0;
	if (t_0 <= 5e+304)
		tmp = t_0;
	else
		tmp = w0_m * (sqrt((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0)))) / sqrt(d_m));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $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_m 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$95$m_] := 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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[t$95$0, 5e+304], t$95$0, N[(w0$95$m * N[(N[Sqrt[N[(d$95$m - N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[d$95$m], $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))))) < 4.9999999999999997e304

   1. Initial program 80.5%

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

   if 4.9999999999999997e304 < (*.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 80.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 84.5%

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

   5. Applied rewrites 85.4%

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

Alternative 3: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{d\_m + d\_m} \cdot M\_m\\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;d\_m \leq 10^{-25}:\\ \;\;\;\;w0\_m \cdot \frac{\sqrt{d\_m - \left(M\_m \cdot D\_m\right) \cdot \frac{\frac{\left(M\_m \cdot D\_m\right) \cdot h}{d\_m \cdot \ell}}{4}}}{\sqrt{d\_m}}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - t\_0 \cdot \frac{t\_0 \cdot h}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
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_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* (/ D_m (+ d_m d_m)) M_m)))
   (*
    w0_s
    (if (<= d_m 1e-25)
      (*
       w0_m
       (/
        (sqrt (- d_m (* (* M_m D_m) (/ (/ (* (* M_m D_m) h) (* d_m l)) 4.0))))
        (sqrt d_m)))
      (* w0_m (sqrt (- 1.0 (* t_0 (/ (* t_0 h) l)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
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_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (D_m / (d_m + d_m)) * M_m;
	double tmp;
	if (d_m <= 1e-25) {
		tmp = w0_m * (sqrt((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0)))) / sqrt(d_m));
	} else {
		tmp = w0_m * sqrt((1.0 - (t_0 * ((t_0 * h) / l))));
	}
	return w0_s * tmp;
}

Fortran

M_m =     private
D_m =     private
d_m =     private
w0\_m =     private
w0\_s =     private
NOTE: w0_m, M_m, D_m, h, l, and d_m 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_m_1)
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_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_m / (d_m_1 + d_m_1)) * m_m
    if (d_m_1 <= 1d-25) then
        tmp = w0_m * (sqrt((d_m_1 - ((m_m * d_m) * ((((m_m * d_m) * h) / (d_m_1 * l)) / 4.0d0)))) / sqrt(d_m_1))
    else
        tmp = w0_m * sqrt((1.0d0 - (t_0 * ((t_0 * h) / l))))
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
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_m;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (D_m / (d_m + d_m)) * M_m;
	double tmp;
	if (d_m <= 1e-25) {
		tmp = w0_m * (Math.sqrt((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0)))) / Math.sqrt(d_m));
	} else {
		tmp = w0_m * Math.sqrt((1.0 - (t_0 * ((t_0 * h) / l))));
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d_m] = sort([w0_m, M_m, D_m, h, l, d_m])
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	t_0 = (D_m / (d_m + d_m)) * M_m
	tmp = 0
	if d_m <= 1e-25:
		tmp = w0_m * (math.sqrt((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0)))) / math.sqrt(d_m))
	else:
		tmp = w0_m * math.sqrt((1.0 - (t_0 * ((t_0 * h) / l))))
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(D_m / Float64(d_m + d_m)) * M_m)
	tmp = 0.0
	if (d_m <= 1e-25)
		tmp = Float64(w0_m * Float64(sqrt(Float64(d_m - Float64(Float64(M_m * D_m) * Float64(Float64(Float64(Float64(M_m * D_m) * h) / Float64(d_m * l)) / 4.0)))) / sqrt(d_m)));
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(t_0 * h) / l)))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d_m = num2cell(sort([w0_m, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	t_0 = (D_m / (d_m + d_m)) * M_m;
	tmp = 0.0;
	if (d_m <= 1e-25)
		tmp = w0_m * (sqrt((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0)))) / sqrt(d_m));
	else
		tmp = w0_m * sqrt((1.0 - (t_0 * ((t_0 * h) / l))));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $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_m 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$95$m_] := Block[{t$95$0 = N[(N[(D$95$m / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[d$95$m, 1e-25], N[(w0$95$m * N[(N[Sqrt[N[(d$95$m - N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[d$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(t$95$0 * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.00000000000000004e-25

   1. Initial program 80.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 84.5%

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

   5. Applied rewrites 85.4%

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

   if 1.00000000000000004e-25 < d

   1. Initial program 80.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

   5. Applied rewrites 87.9%

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

Alternative 4: 88.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{d\_m} \cdot M\_m\\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;\sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}} \leq 10^{+140}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{t\_0 \cdot t\_0}{4} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \frac{\sqrt{d\_m - \left(M\_m \cdot D\_m\right) \cdot \frac{\frac{\left(M\_m \cdot D\_m\right) \cdot h}{d\_m \cdot \ell}}{4}}}{\sqrt{d\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
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_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* (/ D_m d_m) M_m)))
   (*
    w0_s
    (if (<=
         (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l))))
         1e+140)
      (* w0_m (sqrt (- 1.0 (* (/ (* t_0 t_0) 4.0) (/ h l)))))
      (*
       w0_m
       (/
        (sqrt (- d_m (* (* M_m D_m) (/ (/ (* (* M_m D_m) h) (* d_m l)) 4.0))))
        (sqrt d_m)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
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_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (D_m / d_m) * M_m;
	double tmp;
	if (sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)))) <= 1e+140) {
		tmp = w0_m * sqrt((1.0 - (((t_0 * t_0) / 4.0) * (h / l))));
	} else {
		tmp = w0_m * (sqrt((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0)))) / sqrt(d_m));
	}
	return w0_s * tmp;
}

Fortran

M_m =     private
D_m =     private
d_m =     private
w0\_m =     private
w0\_s =     private
NOTE: w0_m, M_m, D_m, h, l, and d_m 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_m_1)
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_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_m / d_m_1) * m_m
    if (sqrt((1.0d0 - ((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)))) <= 1d+140) then
        tmp = w0_m * sqrt((1.0d0 - (((t_0 * t_0) / 4.0d0) * (h / l))))
    else
        tmp = w0_m * (sqrt((d_m_1 - ((m_m * d_m) * ((((m_m * d_m) * h) / (d_m_1 * l)) / 4.0d0)))) / sqrt(d_m_1))
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
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_m;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (D_m / d_m) * M_m;
	double tmp;
	if (Math.sqrt((1.0 - (Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)))) <= 1e+140) {
		tmp = w0_m * Math.sqrt((1.0 - (((t_0 * t_0) / 4.0) * (h / l))));
	} else {
		tmp = w0_m * (Math.sqrt((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0)))) / Math.sqrt(d_m));
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d_m] = sort([w0_m, M_m, D_m, h, l, d_m])
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	t_0 = (D_m / d_m) * M_m
	tmp = 0
	if math.sqrt((1.0 - (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)))) <= 1e+140:
		tmp = w0_m * math.sqrt((1.0 - (((t_0 * t_0) / 4.0) * (h / l))))
	else:
		tmp = w0_m * (math.sqrt((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0)))) / math.sqrt(d_m))
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(D_m / d_m) * M_m)
	tmp = 0.0
	if (sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)))) <= 1e+140)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(Float64(t_0 * t_0) / 4.0) * Float64(h / l)))));
	else
		tmp = Float64(w0_m * Float64(sqrt(Float64(d_m - Float64(Float64(M_m * D_m) * Float64(Float64(Float64(Float64(M_m * D_m) * h) / Float64(d_m * l)) / 4.0)))) / sqrt(d_m)));
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d_m = num2cell(sort([w0_m, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	t_0 = (D_m / d_m) * M_m;
	tmp = 0.0;
	if (sqrt((1.0 - ((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)))) <= 1e+140)
		tmp = w0_m * sqrt((1.0 - (((t_0 * t_0) / 4.0) * (h / l))));
	else
		tmp = w0_m * (sqrt((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0)))) / sqrt(d_m));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $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_m 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$95$m_] := Block[{t$95$0 = N[(N[(D$95$m / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]}, 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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1e+140], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[(N[Sqrt[N[(d$95$m - N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[d$95$m], $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)))) < 1.00000000000000006e140

   1. Initial program 80.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. cosh-0-rev N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 79.9%

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

   if 1.00000000000000006e140 < (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 80.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 84.5%

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

   5. Applied rewrites 85.4%

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

Alternative 5: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{d\_m} \cdot M\_m\\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;\sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}} \leq 10^{+140}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{t\_0 \cdot t\_0}{4} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{\frac{d\_m - \left(M\_m \cdot D\_m\right) \cdot \frac{\frac{\left(M\_m \cdot D\_m\right) \cdot h}{d\_m \cdot \ell}}{4}}{d\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
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_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* (/ D_m d_m) M_m)))
   (*
    w0_s
    (if (<=
         (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l))))
         1e+140)
      (* w0_m (sqrt (- 1.0 (* (/ (* t_0 t_0) 4.0) (/ h l)))))
      (*
       w0_m
       (sqrt
        (/
         (- d_m (* (* M_m D_m) (/ (/ (* (* M_m D_m) h) (* d_m l)) 4.0)))
         d_m)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
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_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (D_m / d_m) * M_m;
	double tmp;
	if (sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)))) <= 1e+140) {
		tmp = w0_m * sqrt((1.0 - (((t_0 * t_0) / 4.0) * (h / l))));
	} else {
		tmp = w0_m * sqrt(((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0))) / d_m));
	}
	return w0_s * tmp;
}

Fortran

M_m =     private
D_m =     private
d_m =     private
w0\_m =     private
w0\_s =     private
NOTE: w0_m, M_m, D_m, h, l, and d_m 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_m_1)
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_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_m / d_m_1) * m_m
    if (sqrt((1.0d0 - ((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)))) <= 1d+140) then
        tmp = w0_m * sqrt((1.0d0 - (((t_0 * t_0) / 4.0d0) * (h / l))))
    else
        tmp = w0_m * sqrt(((d_m_1 - ((m_m * d_m) * ((((m_m * d_m) * h) / (d_m_1 * l)) / 4.0d0))) / d_m_1))
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
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_m;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (D_m / d_m) * M_m;
	double tmp;
	if (Math.sqrt((1.0 - (Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)))) <= 1e+140) {
		tmp = w0_m * Math.sqrt((1.0 - (((t_0 * t_0) / 4.0) * (h / l))));
	} else {
		tmp = w0_m * Math.sqrt(((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0))) / d_m));
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d_m] = sort([w0_m, M_m, D_m, h, l, d_m])
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	t_0 = (D_m / d_m) * M_m
	tmp = 0
	if math.sqrt((1.0 - (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)))) <= 1e+140:
		tmp = w0_m * math.sqrt((1.0 - (((t_0 * t_0) / 4.0) * (h / l))))
	else:
		tmp = w0_m * math.sqrt(((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0))) / d_m))
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(D_m / d_m) * M_m)
	tmp = 0.0
	if (sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)))) <= 1e+140)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(Float64(t_0 * t_0) / 4.0) * Float64(h / l)))));
	else
		tmp = Float64(w0_m * sqrt(Float64(Float64(d_m - Float64(Float64(M_m * D_m) * Float64(Float64(Float64(Float64(M_m * D_m) * h) / Float64(d_m * l)) / 4.0))) / d_m)));
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d_m = num2cell(sort([w0_m, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	t_0 = (D_m / d_m) * M_m;
	tmp = 0.0;
	if (sqrt((1.0 - ((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)))) <= 1e+140)
		tmp = w0_m * sqrt((1.0 - (((t_0 * t_0) / 4.0) * (h / l))));
	else
		tmp = w0_m * sqrt(((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0))) / d_m));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $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_m 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$95$m_] := Block[{t$95$0 = N[(N[(D$95$m / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]}, 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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1e+140], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(N[(d$95$m - N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $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)))) < 1.00000000000000006e140

   1. Initial program 80.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. cosh-0-rev N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 79.9%

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

   if 1.00000000000000006e140 < (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 80.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 84.5%

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

   5. Applied rewrites 84.1%

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

Alternative 6: 86.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;\sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}} \leq 10^{+140}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \left(\frac{\frac{0.25}{d\_m} \cdot \left(M\_m \cdot D\_m\right)}{d\_m} \cdot \left(D\_m \cdot M\_m\right)\right) \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{\frac{d\_m - \left(M\_m \cdot D\_m\right) \cdot \frac{\frac{\left(M\_m \cdot D\_m\right) \cdot h}{d\_m \cdot \ell}}{4}}{d\_m}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
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_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<=
       (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l))))
       1e+140)
    (*
     w0_m
     (sqrt
      (-
       1.0
       (* (* (/ (* (/ 0.25 d_m) (* M_m D_m)) d_m) (* D_m M_m)) (/ h l)))))
    (*
     w0_m
     (sqrt
      (/
       (- d_m (* (* M_m D_m) (/ (/ (* (* M_m D_m) h) (* d_m l)) 4.0)))
       d_m))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
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_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)))) <= 1e+140) {
		tmp = w0_m * sqrt((1.0 - (((((0.25 / d_m) * (M_m * D_m)) / d_m) * (D_m * M_m)) * (h / l))));
	} else {
		tmp = w0_m * sqrt(((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0))) / d_m));
	}
	return w0_s * tmp;
}

Fortran

M_m =     private
D_m =     private
d_m =     private
w0\_m =     private
w0\_s =     private
NOTE: w0_m, M_m, D_m, h, l, and d_m 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_m_1)
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_m_1
    real(8) :: tmp
    if (sqrt((1.0d0 - ((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)))) <= 1d+140) then
        tmp = w0_m * sqrt((1.0d0 - (((((0.25d0 / d_m_1) * (m_m * d_m)) / d_m_1) * (d_m * m_m)) * (h / l))))
    else
        tmp = w0_m * sqrt(((d_m_1 - ((m_m * d_m) * ((((m_m * d_m) * h) / (d_m_1 * l)) / 4.0d0))) / d_m_1))
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
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_m;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (Math.sqrt((1.0 - (Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)))) <= 1e+140) {
		tmp = w0_m * Math.sqrt((1.0 - (((((0.25 / d_m) * (M_m * D_m)) / d_m) * (D_m * M_m)) * (h / l))));
	} else {
		tmp = w0_m * Math.sqrt(((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0))) / d_m));
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d_m] = sort([w0_m, M_m, D_m, h, l, d_m])
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	tmp = 0
	if math.sqrt((1.0 - (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)))) <= 1e+140:
		tmp = w0_m * math.sqrt((1.0 - (((((0.25 / d_m) * (M_m * D_m)) / d_m) * (D_m * M_m)) * (h / l))))
	else:
		tmp = w0_m * math.sqrt(((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0))) / d_m))
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)))) <= 1e+140)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(0.25 / d_m) * Float64(M_m * D_m)) / d_m) * Float64(D_m * M_m)) * Float64(h / l)))));
	else
		tmp = Float64(w0_m * sqrt(Float64(Float64(d_m - Float64(Float64(M_m * D_m) * Float64(Float64(Float64(Float64(M_m * D_m) * h) / Float64(d_m * l)) / 4.0))) / d_m)));
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d_m = num2cell(sort([w0_m, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (sqrt((1.0 - ((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)))) <= 1e+140)
		tmp = w0_m * sqrt((1.0 - (((((0.25 / d_m) * (M_m * D_m)) / d_m) * (D_m * M_m)) * (h / l))));
	else
		tmp = w0_m * sqrt(((d_m - ((M_m * D_m) * ((((M_m * D_m) * h) / (d_m * l)) / 4.0))) / d_m));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $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_m 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$95$m_] := 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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1e+140], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(N[(N[(N[(0.25 / d$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(N[(d$95$m - N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $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)))) < 1.00000000000000006e140

   1. Initial program 80.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 79.1

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if 1.00000000000000006e140 < (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 80.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 84.5%

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

   5. Applied rewrites 84.1%

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

Alternative 7: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 5 \cdot 10^{-84}:\\ \;\;\;\;w0\_m \cdot \sqrt{\mathsf{fma}\left(\frac{-0.25}{d\_m}, \frac{M\_m \cdot D\_m}{d\_m} \cdot \left(\left(\frac{M\_m}{\ell} \cdot h\right) \cdot D\_m\right), 1\right)}\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 10^{+36}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(M\_m \cdot D\_m\right) \cdot h, \frac{M\_m \cdot D\_m}{\left(-4 \cdot \left(\ell \cdot d\_m\right)\right) \cdot d\_m}, 1\right)} \cdot w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \left(\frac{D\_m}{d\_m} \cdot \left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right)\right)\right) \cdot \frac{D\_m}{d\_m}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
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_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= (* M_m D_m) 5e-84)
    (*
     w0_m
     (sqrt
      (fma (/ -0.25 d_m) (* (/ (* M_m D_m) d_m) (* (* (/ M_m l) h) D_m)) 1.0)))
    (if (<= (* M_m D_m) 1e+36)
      (*
       (sqrt
        (fma (* (* M_m D_m) h) (/ (* M_m D_m) (* (* -4.0 (* l d_m)) d_m)) 1.0))
       w0_m)
      (*
       w0_m
       (sqrt
        (-
         1.0
         (*
          (* (/ h l) (* (/ D_m d_m) (* (* M_m M_m) 0.25)))
          (/ D_m d_m)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
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_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((M_m * D_m) <= 5e-84) {
		tmp = w0_m * sqrt(fma((-0.25 / d_m), (((M_m * D_m) / d_m) * (((M_m / l) * h) * D_m)), 1.0));
	} else if ((M_m * D_m) <= 1e+36) {
		tmp = sqrt(fma(((M_m * D_m) * h), ((M_m * D_m) / ((-4.0 * (l * d_m)) * d_m)), 1.0)) * w0_m;
	} else {
		tmp = w0_m * sqrt((1.0 - (((h / l) * ((D_m / d_m) * ((M_m * M_m) * 0.25))) * (D_m / d_m))));
	}
	return w0_s * tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 5e-84)
		tmp = Float64(w0_m * sqrt(fma(Float64(-0.25 / d_m), Float64(Float64(Float64(M_m * D_m) / d_m) * Float64(Float64(Float64(M_m / l) * h) * D_m)), 1.0)));
	elseif (Float64(M_m * D_m) <= 1e+36)
		tmp = Float64(sqrt(fma(Float64(Float64(M_m * D_m) * h), Float64(Float64(M_m * D_m) / Float64(Float64(-4.0 * Float64(l * d_m)) * d_m)), 1.0)) * w0_m);
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(Float64(h / l) * Float64(Float64(D_m / d_m) * Float64(Float64(M_m * M_m) * 0.25))) * Float64(D_m / d_m)))));
	end
	return Float64(w0_s * tmp)
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $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_m 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$95$m_] := N[(w0$95$s * If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e-84], N[(w0$95$m * N[Sqrt[N[(N[(-0.25 / d$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(N[(N[(M$95$m / l), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e+36], N[(N[Sqrt[N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(N[(-4.0 * N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0$95$m), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(N[(h / l), $MachinePrecision] * N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 5.0000000000000002e-84

   1. Initial program 80.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 79.1

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-out N/A

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

   7. Applied rewrites 83.8%

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

   if 5.0000000000000002e-84 < (*.f64 M D) < 1.00000000000000004e36

   1. Initial program 80.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 84.5%

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

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 78.9%

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

   if 1.00000000000000004e36 < (*.f64 M D)

   1. Initial program 80.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 77.6%

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

Alternative 8: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \left(\frac{\frac{0.25}{d\_m} \cdot \left(M\_m \cdot D\_m\right)}{d\_m} \cdot \left(D\_m \cdot M\_m\right)\right) \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
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_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -1e-7)
    (*
     w0_m
     (sqrt
      (-
       1.0
       (* (* (/ (* (/ 0.25 d_m) (* M_m D_m)) d_m) (* D_m M_m)) (/ h l)))))
    (* w0_m 1.0))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
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_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -1e-7) {
		tmp = w0_m * sqrt((1.0 - (((((0.25 / d_m) * (M_m * D_m)) / d_m) * (D_m * M_m)) * (h / l))));
	} else {
		tmp = w0_m * 1.0;
	}
	return w0_s * tmp;
}

Fortran

M_m =     private
D_m =     private
d_m =     private
w0\_m =     private
w0\_s =     private
NOTE: w0_m, M_m, D_m, h, l, and d_m 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_m_1)
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_m_1
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)) <= (-1d-7)) then
        tmp = w0_m * sqrt((1.0d0 - (((((0.25d0 / d_m_1) * (m_m * d_m)) / d_m_1) * (d_m * m_m)) * (h / l))))
    else
        tmp = w0_m * 1.0d0
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
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_m;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -1e-7) {
		tmp = w0_m * Math.sqrt((1.0 - (((((0.25 / d_m) * (M_m * D_m)) / d_m) * (D_m * M_m)) * (h / l))));
	} else {
		tmp = w0_m * 1.0;
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d_m] = sort([w0_m, M_m, D_m, h, l, d_m])
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -1e-7:
		tmp = w0_m * math.sqrt((1.0 - (((((0.25 / d_m) * (M_m * D_m)) / d_m) * (D_m * M_m)) * (h / l))))
	else:
		tmp = w0_m * 1.0
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -1e-7)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(0.25 / d_m) * Float64(M_m * D_m)) / d_m) * Float64(D_m * M_m)) * Float64(h / l)))));
	else
		tmp = Float64(w0_m * 1.0);
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d_m = num2cell(sort([w0_m, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -1e-7)
		tmp = w0_m * sqrt((1.0 - (((((0.25 / d_m) * (M_m * D_m)) / d_m) * (D_m * M_m)) * (h / l))));
	else
		tmp = w0_m * 1.0;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $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_m 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$95$m_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e-7], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(N[(N[(N[(0.25 / d$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $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)) < -9.9999999999999995e-8

   1. Initial program 80.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 79.1

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if -9.9999999999999995e-8 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.5%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 67.7%

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

Alternative 9: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{-124}:\\ \;\;\;\;w0\_m \cdot \sqrt{\mathsf{fma}\left(\frac{-0.25}{d\_m}, \frac{M\_m \cdot D\_m}{d\_m} \cdot \left(\left(\frac{M\_m}{\ell} \cdot h\right) \cdot D\_m\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot 0.25\right) \cdot D\_m\right) \cdot h}{d\_m \cdot \ell} \cdot \frac{D\_m}{d\_m}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
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_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) 5e-124)
    (*
     w0_m
     (sqrt
      (fma (/ -0.25 d_m) (* (/ (* M_m D_m) d_m) (* (* (/ M_m l) h) D_m)) 1.0)))
    (*
     w0_m
     (sqrt
      (-
       1.0
       (* (/ (* (* (* (* M_m M_m) 0.25) D_m) h) (* d_m l)) (/ D_m d_m))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
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_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= 5e-124) {
		tmp = w0_m * sqrt(fma((-0.25 / d_m), (((M_m * D_m) / d_m) * (((M_m / l) * h) * D_m)), 1.0));
	} else {
		tmp = w0_m * sqrt((1.0 - ((((((M_m * M_m) * 0.25) * D_m) * h) / (d_m * l)) * (D_m / d_m))));
	}
	return w0_s * tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= 5e-124)
		tmp = Float64(w0_m * sqrt(fma(Float64(-0.25 / d_m), Float64(Float64(Float64(M_m * D_m) / d_m) * Float64(Float64(Float64(M_m / l) * h) * D_m)), 1.0)));
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * 0.25) * D_m) * h) / Float64(d_m * l)) * Float64(D_m / d_m)))));
	end
	return Float64(w0_s * tmp)
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $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_m 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$95$m_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 5e-124], N[(w0$95$m * N[Sqrt[N[(N[(-0.25 / d$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(N[(N[(M$95$m / l), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * 0.25), $MachinePrecision] * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 5.0000000000000003e-124

   1. Initial program 80.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 79.1

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-out N/A

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

   7. Applied rewrites 83.8%

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

   if 5.0000000000000003e-124 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 78.3

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

   5. Applied rewrites 78.3%

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

Alternative 10: 84.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;M\_m \leq 2 \cdot 10^{-122}:\\ \;\;\;\;w0\_m \cdot \sqrt{\mathsf{fma}\left(\frac{-0.25}{d\_m}, \frac{M\_m \cdot D\_m}{d\_m} \cdot \left(\left(\frac{M\_m}{\ell} \cdot h\right) \cdot D\_m\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(h \cdot \frac{D\_m}{\ell \cdot d\_m}\right) \cdot \left(M\_m \cdot M\_m\right)\right) \cdot -0.25, \frac{D\_m}{d\_m}, 1\right)} \cdot w0\_m\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
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_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= M_m 2e-122)
    (*
     w0_m
     (sqrt
      (fma (/ -0.25 d_m) (* (/ (* M_m D_m) d_m) (* (* (/ M_m l) h) D_m)) 1.0)))
    (*
     (sqrt
      (fma (* (* (* h (/ D_m (* l d_m))) (* M_m M_m)) -0.25) (/ D_m d_m) 1.0))
     w0_m))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
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_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 2e-122) {
		tmp = w0_m * sqrt(fma((-0.25 / d_m), (((M_m * D_m) / d_m) * (((M_m / l) * h) * D_m)), 1.0));
	} else {
		tmp = sqrt(fma((((h * (D_m / (l * d_m))) * (M_m * M_m)) * -0.25), (D_m / d_m), 1.0)) * w0_m;
	}
	return w0_s * tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 2e-122)
		tmp = Float64(w0_m * sqrt(fma(Float64(-0.25 / d_m), Float64(Float64(Float64(M_m * D_m) / d_m) * Float64(Float64(Float64(M_m / l) * h) * D_m)), 1.0)));
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(h * Float64(D_m / Float64(l * d_m))) * Float64(M_m * M_m)) * -0.25), Float64(D_m / d_m), 1.0)) * w0_m);
	end
	return Float64(w0_s * tmp)
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $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_m 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$95$m_] := N[(w0$95$s * If[LessEqual[M$95$m, 2e-122], N[(w0$95$m * N[Sqrt[N[(N[(-0.25 / d$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(N[(N[(M$95$m / l), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(h * N[(D$95$m / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if M < 2.00000000000000012e-122

   1. Initial program 80.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 79.1

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-out N/A

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

   7. Applied rewrites 83.8%

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

   if 2.00000000000000012e-122 < M

   1. Initial program 80.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 84.5%

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

   5. Applied rewrites 76.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 76.8

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 76.8

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

   7. Applied rewrites 76.8%

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

Alternative 11: 82.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;D\_m \leq 1.65 \cdot 10^{-5}:\\ \;\;\;\;w0\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(M\_m \cdot D\_m\right) \cdot h, \frac{M\_m \cdot D\_m}{\left(-4 \cdot \left(\ell \cdot d\_m\right)\right) \cdot d\_m}, 1\right)} \cdot w0\_m\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
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_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= D_m 1.65e-5)
    (* w0_m 1.0)
    (*
     (sqrt
      (fma (* (* M_m D_m) h) (/ (* M_m D_m) (* (* -4.0 (* l d_m)) d_m)) 1.0))
     w0_m))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
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_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 1.65e-5) {
		tmp = w0_m * 1.0;
	} else {
		tmp = sqrt(fma(((M_m * D_m) * h), ((M_m * D_m) / ((-4.0 * (l * d_m)) * d_m)), 1.0)) * w0_m;
	}
	return w0_s * tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (D_m <= 1.65e-5)
		tmp = Float64(w0_m * 1.0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(M_m * D_m) * h), Float64(Float64(M_m * D_m) / Float64(Float64(-4.0 * Float64(l * d_m)) * d_m)), 1.0)) * w0_m);
	end
	return Float64(w0_s * tmp)
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $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_m 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$95$m_] := N[(w0$95$s * If[LessEqual[D$95$m, 1.65e-5], N[(w0$95$m * 1.0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(N[(-4.0 * N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if D < 1.6500000000000001e-5

   1. Initial program 80.5%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 67.7%

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

   if 1.6500000000000001e-5 < D

   1. Initial program 80.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 84.5%

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

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 78.9%

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

Alternative 12: 81.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.04:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(M\_m, \frac{\left(\left(D\_m \cdot D\_m\right) \cdot M\_m\right) \cdot h}{\left(-4 \cdot \left(\ell \cdot d\_m\right)\right) \cdot d\_m}, 1\right)} \cdot w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
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_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -0.04)
    (*
     (sqrt
      (fma M_m (/ (* (* (* D_m D_m) M_m) h) (* (* -4.0 (* l d_m)) d_m)) 1.0))
     w0_m)
    (* w0_m 1.0))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
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_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -0.04) {
		tmp = sqrt(fma(M_m, ((((D_m * D_m) * M_m) * h) / ((-4.0 * (l * d_m)) * d_m)), 1.0)) * w0_m;
	} else {
		tmp = w0_m * 1.0;
	}
	return w0_s * tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -0.04)
		tmp = Float64(sqrt(fma(M_m, Float64(Float64(Float64(Float64(D_m * D_m) * M_m) * h) / Float64(Float64(-4.0 * Float64(l * d_m)) * d_m)), 1.0)) * w0_m);
	else
		tmp = Float64(w0_m * 1.0);
	end
	return Float64(w0_s * tmp)
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $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_m 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$95$m_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.04], N[(N[Sqrt[N[(M$95$m * N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(N[(-4.0 * N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $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)) < -0.0400000000000000008

   1. Initial program 80.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 84.5%

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

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 61.6%

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

   if -0.0400000000000000008 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.5%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 67.7%

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

Alternative 13: 79.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.04:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{D\_m \cdot \left(\left(M\_m \cdot M\_m\right) \cdot \left(h \cdot D\_m\right)\right)}{\left(d\_m \cdot d\_m\right) \cdot \ell}, -0.25, 1\right)} \cdot w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
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_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -0.04)
    (*
     (sqrt
      (fma (/ (* D_m (* (* M_m M_m) (* h D_m))) (* (* d_m d_m) l)) -0.25 1.0))
     w0_m)
    (* w0_m 1.0))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
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_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -0.04) {
		tmp = sqrt(fma(((D_m * ((M_m * M_m) * (h * D_m))) / ((d_m * d_m) * l)), -0.25, 1.0)) * w0_m;
	} else {
		tmp = w0_m * 1.0;
	}
	return w0_s * tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -0.04)
		tmp = Float64(sqrt(fma(Float64(Float64(D_m * Float64(Float64(M_m * M_m) * Float64(h * D_m))) / Float64(Float64(d_m * d_m) * l)), -0.25, 1.0)) * w0_m);
	else
		tmp = Float64(w0_m * 1.0);
	end
	return Float64(w0_s * tmp)
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $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_m 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$95$m_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -0.04], N[(N[Sqrt[N[(N[(N[(D$95$m * N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(h * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.25 + 1.0), $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)) < -0.0400000000000000008

   1. Initial program 80.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 84.5%

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

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 65.2%

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

   if -0.0400000000000000008 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.5%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 67.7%

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

Alternative 14: 73.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d_m])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+277}:\\ \;\;\;\;w0\_m \cdot \frac{d\_m \cdot \sqrt{\sqrt{\frac{1}{d\_m} \cdot \frac{1}{d\_m}}}}{\sqrt{d\_m}}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
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_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -1e+277)
    (* w0_m (/ (* d_m (sqrt (sqrt (* (/ 1.0 d_m) (/ 1.0 d_m))))) (sqrt d_m)))
    (* w0_m 1.0))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
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_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -1e+277) {
		tmp = w0_m * ((d_m * sqrt(sqrt(((1.0 / d_m) * (1.0 / d_m))))) / sqrt(d_m));
	} else {
		tmp = w0_m * 1.0;
	}
	return w0_s * tmp;
}

Fortran

M_m =     private
D_m =     private
d_m =     private
w0\_m =     private
w0\_s =     private
NOTE: w0_m, M_m, D_m, h, l, and d_m 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_m_1)
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_m_1
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)) <= (-1d+277)) then
        tmp = w0_m * ((d_m_1 * sqrt(sqrt(((1.0d0 / d_m_1) * (1.0d0 / d_m_1))))) / sqrt(d_m_1))
    else
        tmp = w0_m * 1.0d0
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
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_m;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -1e+277) {
		tmp = w0_m * ((d_m * Math.sqrt(Math.sqrt(((1.0 / d_m) * (1.0 / d_m))))) / Math.sqrt(d_m));
	} else {
		tmp = w0_m * 1.0;
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d_m] = sort([w0_m, M_m, D_m, h, l, d_m])
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -1e+277:
		tmp = w0_m * ((d_m * math.sqrt(math.sqrt(((1.0 / d_m) * (1.0 / d_m))))) / math.sqrt(d_m))
	else:
		tmp = w0_m * 1.0
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d_m = sort([w0_m, M_m, D_m, h, l, d_m])
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -1e+277)
		tmp = Float64(w0_m * Float64(Float64(d_m * sqrt(sqrt(Float64(Float64(1.0 / d_m) * Float64(1.0 / d_m))))) / sqrt(d_m)));
	else
		tmp = Float64(w0_m * 1.0);
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d_m = num2cell(sort([w0_m, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -1e+277)
		tmp = w0_m * ((d_m * sqrt(sqrt(((1.0 / d_m) * (1.0 / d_m))))) / sqrt(d_m));
	else
		tmp = w0_m * 1.0;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $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_m 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$95$m_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+277], N[(w0$95$m * N[(N[(d$95$m * N[Sqrt[N[Sqrt[N[(N[(1.0 / d$95$m), $MachinePrecision] * N[(1.0 / d$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[d$95$m], $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)) < -1e277

   1. Initial program 80.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 84.5%

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in d around inf

      \[\leadsto w0 \cdot \frac{\color{blue}{d \cdot \sqrt{\frac{1}{d}}}}{\sqrt{d}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto w0 \cdot \frac{d \cdot \color{blue}{\sqrt{\frac{1}{d}}}}{\sqrt{d}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto w0 \cdot \frac{d \cdot \sqrt{\frac{1}{d}}}{\sqrt{d}} \]

      3. lower-/.f64 67.3

         \[\leadsto w0 \cdot \frac{d \cdot \sqrt{\frac{1}{d}}}{\sqrt{d}} \]

   8. Applied rewrites 67.3%

      \[\leadsto w0 \cdot \frac{\color{blue}{d \cdot \sqrt{\frac{1}{d}}}}{\sqrt{d}} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto w0 \cdot \frac{d \cdot \sqrt{\sqrt{\frac{1}{d}} \cdot \sqrt{\frac{1}{d}}}}{\sqrt{d}} \]

      2. sqrt-unprod N/A

         \[\leadsto w0 \cdot \frac{d \cdot \sqrt{\sqrt{\frac{1}{d} \cdot \frac{1}{d}}}}{\sqrt{d}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto w0 \cdot \frac{d \cdot \sqrt{\sqrt{\frac{1}{d} \cdot \frac{1}{d}}}}{\sqrt{d}} \]

      4. lower-*.f64 43.1

         \[\leadsto w0 \cdot \frac{d \cdot \sqrt{\sqrt{\frac{1}{d} \cdot \frac{1}{d}}}}{\sqrt{d}} \]

   10. Applied rewrites 43.1%

       \[\leadsto w0 \cdot \frac{d \cdot \sqrt{\sqrt{\frac{1}{d} \cdot \frac{1}{d}}}}{\sqrt{d}} \]

   if -1e277 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 80.5%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 67.7%

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

Alternative 15: 67.7% accurate, 10.1× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
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_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d_m) :precision binary64 (* w0_s (* w0_m 1.0)))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
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_m);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	return w0_s * (w0_m * 1.0);
}

Fortran

M_m =     private
D_m =     private
d_m =     private
w0\_m =     private
w0\_s =     private
NOTE: w0_m, M_m, D_m, h, l, and d_m 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_m_1)
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_m_1
    code = w0_s * (w0_m * 1.0d0)
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
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_m;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	return w0_s * (w0_m * 1.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $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_m 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$95$m_] := N[(w0$95$s * N[(w0$95$m * 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.5%

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

2. Taylor expanded in M around 0

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

4. Applied rewrites 67.7%

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

Reproduce

herbie shell --seed 2025152 
(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))))))