Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.5% → 90.3%

Time: 6.0s

Alternatives: 16

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: 81.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d] = \mathsf{sort}([w0_m, M, D, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D}{d + d} \cdot M\\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;w0\_m \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 2 \cdot 10^{+298}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - {\left(\frac{M \cdot D}{d + d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - t\_0 \cdot \frac{t\_0 \cdot h}{\ell}} \cdot w0\_m\\ \end{array} \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (let* ((t_0 (* (/ D (+ d d)) M)))
   (*
    w0_s
    (if (<=
         (* w0_m (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
         2e+298)
      (* w0_m (sqrt (- 1.0 (* (pow (/ (* M D) (+ d d)) 2.0) (/ h l)))))
      (* (sqrt (- 1.0 (* t_0 (/ (* t_0 h) l)))) w0_m)))))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double t_0 = (D / (d + d)) * M;
	double tmp;
	if ((w0_m * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))) <= 2e+298) {
		tmp = w0_m * sqrt((1.0 - (pow(((M * D) / (d + d)), 2.0) * (h / l))));
	} else {
		tmp = sqrt((1.0 - (t_0 * ((t_0 * h) / l)))) * w0_m;
	}
	return w0_s * tmp;
}

Fortran

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

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d / (d_1 + d_1)) * m
    if ((w0_m * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))) <= 2d+298) then
        tmp = w0_m * sqrt((1.0d0 - ((((m * d) / (d_1 + d_1)) ** 2.0d0) * (h / l))))
    else
        tmp = sqrt((1.0d0 - (t_0 * ((t_0 * h) / l)))) * w0_m
    end if
    code = w0_s * tmp
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M && M < D && D < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double t_0 = (D / (d + d)) * M;
	double tmp;
	if ((w0_m * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))) <= 2e+298) {
		tmp = w0_m * Math.sqrt((1.0 - (Math.pow(((M * D) / (d + d)), 2.0) * (h / l))));
	} else {
		tmp = Math.sqrt((1.0 - (t_0 * ((t_0 * h) / l)))) * w0_m;
	}
	return w0_s * tmp;
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d] = sort([w0_m, M, D, h, l, d])
def code(w0_s, w0_m, M, D, h, l, d):
	t_0 = (D / (d + d)) * M
	tmp = 0
	if (w0_m * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))) <= 2e+298:
		tmp = w0_m * math.sqrt((1.0 - (math.pow(((M * D) / (d + d)), 2.0) * (h / l))))
	else:
		tmp = math.sqrt((1.0 - (t_0 * ((t_0 * h) / l)))) * w0_m
	return w0_s * tmp

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d = sort([w0_m, M, D, h, l, d])
function code(w0_s, w0_m, M, D, h, l, d)
	t_0 = Float64(Float64(D / Float64(d + d)) * M)
	tmp = 0.0
	if (Float64(w0_m * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))) <= 2e+298)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(d + d)) ^ 2.0) * Float64(h / l)))));
	else
		tmp = Float64(sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(t_0 * h) / l)))) * w0_m);
	end
	return Float64(w0_s * tmp)
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d = num2cell(sort([w0_m, M, D, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d)
	t_0 = (D / (d + d)) * M;
	tmp = 0.0;
	if ((w0_m * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))))) <= 2e+298)
		tmp = w0_m * sqrt((1.0 - ((((M * D) / (d + d)) ^ 2.0) * (h / l))));
	else
		tmp = sqrt((1.0 - (t_0 * ((t_0 * h) / l)))) * w0_m;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[N[(w0$95$m * 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], 2e+298], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(t$95$0 * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0$95$m), $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))))) < 1.9999999999999999e298

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 99.8

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

   3. Applied rewrites 99.8%

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

   if 1.9999999999999999e298 < (*.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 45.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 47.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift-+.f64 N/A

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

      21. lift-/.f64 53.6

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

   5. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 71.4

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

   7. Applied rewrites 71.4%

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

Alternative 2: 89.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d] = \mathsf{sort}([w0_m, M, D, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D}{d + d} \cdot M\\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{1 - t\_0 \cdot \left(t\_0 \cdot \frac{h}{\ell}\right)} \cdot w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m\\ \end{array} \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (let* ((t_0 (* (/ D (+ d d)) M)))
   (*
    w0_s
    (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -4e-16)
      (* (sqrt (- 1.0 (* t_0 (* t_0 (/ h l))))) w0_m)
      w0_m))))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double t_0 = (D / (d + d)) * M;
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -4e-16) {
		tmp = sqrt((1.0 - (t_0 * (t_0 * (h / l))))) * w0_m;
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Fortran

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

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d / (d_1 + d_1)) * m
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-4d-16)) then
        tmp = sqrt((1.0d0 - (t_0 * (t_0 * (h / l))))) * w0_m
    else
        tmp = w0_m
    end if
    code = w0_s * tmp
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M && M < D && D < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double t_0 = (D / (d + d)) * M;
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -4e-16) {
		tmp = Math.sqrt((1.0 - (t_0 * (t_0 * (h / l))))) * w0_m;
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d] = sort([w0_m, M, D, h, l, d])
def code(w0_s, w0_m, M, D, h, l, d):
	t_0 = (D / (d + d)) * M
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -4e-16:
		tmp = math.sqrt((1.0 - (t_0 * (t_0 * (h / l))))) * w0_m
	else:
		tmp = w0_m
	return w0_s * tmp

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d = sort([w0_m, M, D, h, l, d])
function code(w0_s, w0_m, M, D, h, l, d)
	t_0 = Float64(Float64(D / Float64(d + d)) * M)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -4e-16)
		tmp = Float64(sqrt(Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(h / l))))) * w0_m);
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d = num2cell(sort([w0_m, M, D, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d)
	t_0 = (D / (d + d)) * M;
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -4e-16)
		tmp = sqrt((1.0 - (t_0 * (t_0 * (h / l))))) * w0_m;
	else
		tmp = w0_m;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -4e-16], N[(N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0$95$m), $MachinePrecision], w0$95$m]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -3.9999999999999999e-16

   1. Initial program 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift-+.f64 N/A

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

      21. lift-/.f64 69.6

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

   5. Applied rewrites 69.6%

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

   if -3.9999999999999999e-16 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.5%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 96.4%

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

Alternative 3: 88.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d] = \mathsf{sort}([w0_m, M, D, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D}{d + d} \cdot M\\ w0\_s \cdot \left(\sqrt{1 - t\_0 \cdot \frac{t\_0 \cdot h}{\ell}} \cdot w0\_m\right) \end{array} \end{array} \]

FPCore

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

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double t_0 = (D / (d + d)) * M;
	return w0_s * (sqrt((1.0 - (t_0 * ((t_0 * h) / l)))) * w0_m);
}

Fortran

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

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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
    real(8) :: t_0
    t_0 = (d / (d_1 + d_1)) * m
    code = w0_s * (sqrt((1.0d0 - (t_0 * ((t_0 * h) / l)))) * w0_m)
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M && M < D && D < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double t_0 = (D / (d + d)) * M;
	return w0_s * (Math.sqrt((1.0 - (t_0 * ((t_0 * h) / l)))) * w0_m);
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d] = sort([w0_m, M, D, h, l, d])
def code(w0_s, w0_m, M, D, h, l, d):
	t_0 = (D / (d + d)) * M
	return w0_s * (math.sqrt((1.0 - (t_0 * ((t_0 * h) / l)))) * w0_m)

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d = sort([w0_m, M, D, h, l, d])
function code(w0_s, w0_m, M, D, h, l, d)
	t_0 = Float64(Float64(D / Float64(d + d)) * M)
	return Float64(w0_s * Float64(sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(t_0 * h) / l)))) * w0_m))
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d = num2cell(sort([w0_m, M, D, h, l, d])){:}
function tmp = code(w0_s, w0_m, M, D, h, l, d)
	t_0 = (D / (d + d)) * M;
	tmp = w0_s * (sqrt((1.0 - (t_0 * ((t_0 * h) / l)))) * w0_m);
end

Wolfram

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, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]}, N[(w0$95$s * N[(N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(t$95$0 * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.5%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

3. Applied rewrites 81.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-+.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lift-/.f64 N/A

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

   20. lift-+.f64 N/A

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

   21. lift-/.f64 83.6

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

5. Applied rewrites 83.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 89.3

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

7. Applied rewrites 89.3%

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

Alternative 4: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d] = \mathsf{sort}([w0_m, M, D, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D}{d + d}\\ t_1 := M \cdot t\_0\\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;w0\_m \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\sqrt{1 - \left(t\_1 \cdot t\_1\right) \cdot \frac{h}{\ell}} \cdot w0\_m\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(t\_0 \cdot M\right) \cdot \frac{0.5 \cdot \left(\left(h \cdot M\right) \cdot D\right)}{\ell \cdot d}} \cdot w0\_m\\ \end{array} \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (let* ((t_0 (/ D (+ d d))) (t_1 (* M t_0)))
   (*
    w0_s
    (if (<=
         (* w0_m (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
         2e+298)
      (* (sqrt (- 1.0 (* (* t_1 t_1) (/ h l)))) w0_m)
      (*
       (sqrt (- 1.0 (* (* t_0 M) (/ (* 0.5 (* (* h M) D)) (* l d)))))
       w0_m)))))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double t_0 = D / (d + d);
	double t_1 = M * t_0;
	double tmp;
	if ((w0_m * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))) <= 2e+298) {
		tmp = sqrt((1.0 - ((t_1 * t_1) * (h / l)))) * w0_m;
	} else {
		tmp = sqrt((1.0 - ((t_0 * M) * ((0.5 * ((h * M) * D)) / (l * d))))) * w0_m;
	}
	return w0_s * tmp;
}

Fortran

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

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = d / (d_1 + d_1)
    t_1 = m * t_0
    if ((w0_m * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))) <= 2d+298) then
        tmp = sqrt((1.0d0 - ((t_1 * t_1) * (h / l)))) * w0_m
    else
        tmp = sqrt((1.0d0 - ((t_0 * m) * ((0.5d0 * ((h * m) * d)) / (l * d_1))))) * w0_m
    end if
    code = w0_s * tmp
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M && M < D && D < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double t_0 = D / (d + d);
	double t_1 = M * t_0;
	double tmp;
	if ((w0_m * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))) <= 2e+298) {
		tmp = Math.sqrt((1.0 - ((t_1 * t_1) * (h / l)))) * w0_m;
	} else {
		tmp = Math.sqrt((1.0 - ((t_0 * M) * ((0.5 * ((h * M) * D)) / (l * d))))) * w0_m;
	}
	return w0_s * tmp;
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d] = sort([w0_m, M, D, h, l, d])
def code(w0_s, w0_m, M, D, h, l, d):
	t_0 = D / (d + d)
	t_1 = M * t_0
	tmp = 0
	if (w0_m * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))) <= 2e+298:
		tmp = math.sqrt((1.0 - ((t_1 * t_1) * (h / l)))) * w0_m
	else:
		tmp = math.sqrt((1.0 - ((t_0 * M) * ((0.5 * ((h * M) * D)) / (l * d))))) * w0_m
	return w0_s * tmp

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d = sort([w0_m, M, D, h, l, d])
function code(w0_s, w0_m, M, D, h, l, d)
	t_0 = Float64(D / Float64(d + d))
	t_1 = Float64(M * t_0)
	tmp = 0.0
	if (Float64(w0_m * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))) <= 2e+298)
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(t_1 * t_1) * Float64(h / l)))) * w0_m);
	else
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(t_0 * M) * Float64(Float64(0.5 * Float64(Float64(h * M) * D)) / Float64(l * d))))) * w0_m);
	end
	return Float64(w0_s * tmp)
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d = num2cell(sort([w0_m, M, D, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d)
	t_0 = D / (d + d);
	t_1 = M * t_0;
	tmp = 0.0;
	if ((w0_m * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))))) <= 2e+298)
		tmp = sqrt((1.0 - ((t_1 * t_1) * (h / l)))) * w0_m;
	else
		tmp = sqrt((1.0 - ((t_0 * M) * ((0.5 * ((h * M) * D)) / (l * d))))) * w0_m;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(M * t$95$0), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[N[(w0$95$m * 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], 2e+298], N[(N[Sqrt[N[(1.0 - N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0$95$m), $MachinePrecision], N[(N[Sqrt[N[(1.0 - N[(N[(t$95$0 * M), $MachinePrecision] * N[(N[(0.5 * N[(N[(h * M), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0$95$m), $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))))) < 1.9999999999999999e298

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 98.7%

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

   if 1.9999999999999999e298 < (*.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 45.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 47.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift-+.f64 N/A

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

      21. lift-/.f64 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 64.8

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

   8. Applied rewrites 64.8%

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

Alternative 5: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d] = \mathsf{sort}([w0_m, M, D, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 1:\\ \;\;\;\;w0\_m\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{0.5 \cdot \left(\left(h \cdot M\right) \cdot D\right)}{\ell \cdot d}} \cdot w0\_m\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))) 1.0)
    w0_m
    (*
     (sqrt (- 1.0 (* (* (/ D (+ d d)) M) (/ (* 0.5 (* (* h M) D)) (* l d)))))
     w0_m))))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if (sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)))) <= 1.0) {
		tmp = w0_m;
	} else {
		tmp = sqrt((1.0 - (((D / (d + d)) * M) * ((0.5 * ((h * M) * D)) / (l * d))))) * w0_m;
	}
	return w0_s * tmp;
}

Fortran

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

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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
    real(8) :: tmp
    if (sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)))) <= 1.0d0) then
        tmp = w0_m
    else
        tmp = sqrt((1.0d0 - (((d / (d_1 + d_1)) * m) * ((0.5d0 * ((h * m) * d)) / (l * d_1))))) * w0_m
    end if
    code = w0_s * tmp
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M && M < D && D < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if (Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)))) <= 1.0) {
		tmp = w0_m;
	} else {
		tmp = Math.sqrt((1.0 - (((D / (d + d)) * M) * ((0.5 * ((h * M) * D)) / (l * d))))) * w0_m;
	}
	return w0_s * tmp;
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d] = sort([w0_m, M, D, h, l, d])
def code(w0_s, w0_m, M, D, h, l, d):
	tmp = 0
	if math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)))) <= 1.0:
		tmp = w0_m
	else:
		tmp = math.sqrt((1.0 - (((D / (d + d)) * M) * ((0.5 * ((h * M) * D)) / (l * d))))) * w0_m
	return w0_s * tmp

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d = sort([w0_m, M, D, h, l, d])
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))) <= 1.0)
		tmp = w0_m;
	else
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(Float64(D / Float64(d + d)) * M) * Float64(Float64(0.5 * Float64(Float64(h * M) * D)) / Float64(l * d))))) * w0_m);
	end
	return Float64(w0_s * tmp)
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d = num2cell(sort([w0_m, M, D, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0;
	if (sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)))) <= 1.0)
		tmp = w0_m;
	else
		tmp = sqrt((1.0 - (((D / (d + d)) * M) * ((0.5 * ((h * M) * D)) / (l * d))))) * w0_m;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[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], 1.0], w0$95$m, N[(N[Sqrt[N[(1.0 - N[(N[(N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * N[(N[(0.5 * N[(N[(h * M), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 99.6%

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

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

   1. Initial program 52.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 53.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift-+.f64 N/A

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

      21. lift-/.f64 58.5

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

   5. Applied rewrites 58.5%

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

   6. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 63.6

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

   8. Applied rewrites 63.6%

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

Alternative 6: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d] = \mathsf{sort}([w0_m, M, D, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{\left(\frac{\left(\left(M \cdot D\right) \cdot M\right) \cdot h}{d} \cdot \frac{D}{\ell \cdot d}\right) \cdot -0.25} \cdot w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -1e+41)
    (* (sqrt (* (* (/ (* (* (* M D) M) h) d) (/ D (* l d))) -0.25)) w0_m)
    w0_m)))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+41) {
		tmp = sqrt(((((((M * D) * M) * h) / d) * (D / (l * d))) * -0.25)) * w0_m;
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Fortran

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

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-1d+41)) then
        tmp = sqrt(((((((m * d) * m) * h) / d_1) * (d / (l * d_1))) * (-0.25d0))) * w0_m
    else
        tmp = w0_m
    end if
    code = w0_s * tmp
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M && M < D && D < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+41) {
		tmp = Math.sqrt(((((((M * D) * M) * h) / d) * (D / (l * d))) * -0.25)) * w0_m;
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d] = sort([w0_m, M, D, h, l, d])
def code(w0_s, w0_m, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+41:
		tmp = math.sqrt(((((((M * D) * M) * h) / d) * (D / (l * d))) * -0.25)) * w0_m
	else:
		tmp = w0_m
	return w0_s * tmp

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d = sort([w0_m, M, D, h, l, d])
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+41)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(M * D) * M) * h) / d) * Float64(D / Float64(l * d))) * -0.25)) * w0_m);
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d = num2cell(sort([w0_m, M, D, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -1e+41)
		tmp = sqrt(((((((M * D) * M) * h) / d) * (D / (l * d))) * -0.25)) * w0_m;
	else
		tmp = w0_m;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+41], N[(N[Sqrt[N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] * N[(D / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * w0$95$m), $MachinePrecision], w0$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.00000000000000001e41

   1. Initial program 63.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 64.3%

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

   4. Taylor expanded in M around inf

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

   6. Applied rewrites 45.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. times-frac N/A

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

      19. lower-*.f64 N/A

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

   8. Applied rewrites 54.9%

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

   if -1.00000000000000001e41 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.8%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 94.9%

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

Alternative 7: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d] = \mathsf{sort}([w0_m, M, D, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\left(\left(\left(\left(h \cdot M\right) \cdot M\right) \cdot D\right) \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot -0.25} \cdot w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+19)
    (* (sqrt (* (* (* (* (* h M) M) D) (/ D (* (* d d) l))) -0.25)) w0_m)
    w0_m)))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+19) {
		tmp = sqrt((((((h * M) * M) * D) * (D / ((d * d) * l))) * -0.25)) * w0_m;
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Fortran

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

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d+19)) then
        tmp = sqrt((((((h * m) * m) * d) * (d / ((d_1 * d_1) * l))) * (-0.25d0))) * w0_m
    else
        tmp = w0_m
    end if
    code = w0_s * tmp
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M && M < D && D < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+19) {
		tmp = Math.sqrt((((((h * M) * M) * D) * (D / ((d * d) * l))) * -0.25)) * w0_m;
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d] = sort([w0_m, M, D, h, l, d])
def code(w0_s, w0_m, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+19:
		tmp = math.sqrt((((((h * M) * M) * D) * (D / ((d * d) * l))) * -0.25)) * w0_m
	else:
		tmp = w0_m
	return w0_s * tmp

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d = sort([w0_m, M, D, h, l, d])
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+19)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(h * M) * M) * D) * Float64(D / Float64(Float64(d * d) * l))) * -0.25)) * w0_m);
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d = num2cell(sort([w0_m, M, D, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+19)
		tmp = sqrt((((((h * M) * M) * D) * (D / ((d * d) * l))) * -0.25)) * w0_m;
	else
		tmp = w0_m;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+19], N[(N[Sqrt[N[(N[(N[(N[(N[(h * M), $MachinePrecision] * M), $MachinePrecision] * D), $MachinePrecision] * N[(D / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * w0$95$m), $MachinePrecision], w0$95$m]), $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)) < -2e19

   1. Initial program 64.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 64.8%

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

   4. Taylor expanded in M around inf

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

   6. Applied rewrites 45.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 44.0

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 46.8

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

   8. Applied rewrites 46.8%

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

   if -2e19 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.7%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 95.6%

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

Alternative 8: 81.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d] = \mathsf{sort}([w0_m, M, D, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{\left(\left(\left(D \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot \frac{D}{\left(\ell \cdot d\right) \cdot d}\right) \cdot -0.25} \cdot w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -1e+41)
    (* (sqrt (* (* (* (* D (* M M)) h) (/ D (* (* l d) d))) -0.25)) w0_m)
    w0_m)))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+41) {
		tmp = sqrt(((((D * (M * M)) * h) * (D / ((l * d) * d))) * -0.25)) * w0_m;
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Fortran

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

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-1d+41)) then
        tmp = sqrt(((((d * (m * m)) * h) * (d / ((l * d_1) * d_1))) * (-0.25d0))) * w0_m
    else
        tmp = w0_m
    end if
    code = w0_s * tmp
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M && M < D && D < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+41) {
		tmp = Math.sqrt(((((D * (M * M)) * h) * (D / ((l * d) * d))) * -0.25)) * w0_m;
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d] = sort([w0_m, M, D, h, l, d])
def code(w0_s, w0_m, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+41:
		tmp = math.sqrt(((((D * (M * M)) * h) * (D / ((l * d) * d))) * -0.25)) * w0_m
	else:
		tmp = w0_m
	return w0_s * tmp

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d = sort([w0_m, M, D, h, l, d])
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+41)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(D * Float64(M * M)) * h) * Float64(D / Float64(Float64(l * d) * d))) * -0.25)) * w0_m);
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d = num2cell(sort([w0_m, M, D, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -1e+41)
		tmp = sqrt(((((D * (M * M)) * h) * (D / ((l * d) * d))) * -0.25)) * w0_m;
	else
		tmp = w0_m;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+41], N[(N[Sqrt[N[(N[(N[(N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(D / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * w0$95$m), $MachinePrecision], w0$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.00000000000000001e41

   1. Initial program 63.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 64.3%

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

   4. Taylor expanded in M around inf

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

   6. Applied rewrites 45.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 47.9

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

   8. Applied rewrites 47.9%

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

   if -1.00000000000000001e41 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.8%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 94.9%

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

Alternative 9: 80.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d] = \mathsf{sort}([w0_m, M, D, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{\left(\left(\left(D \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot -0.25} \cdot w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -1e+41)
    (* (sqrt (* (* (* (* D (* M M)) h) (/ D (* (* d d) l))) -0.25)) w0_m)
    w0_m)))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+41) {
		tmp = sqrt(((((D * (M * M)) * h) * (D / ((d * d) * l))) * -0.25)) * w0_m;
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Fortran

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

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-1d+41)) then
        tmp = sqrt(((((d * (m * m)) * h) * (d / ((d_1 * d_1) * l))) * (-0.25d0))) * w0_m
    else
        tmp = w0_m
    end if
    code = w0_s * tmp
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M && M < D && D < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+41) {
		tmp = Math.sqrt(((((D * (M * M)) * h) * (D / ((d * d) * l))) * -0.25)) * w0_m;
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d] = sort([w0_m, M, D, h, l, d])
def code(w0_s, w0_m, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+41:
		tmp = math.sqrt(((((D * (M * M)) * h) * (D / ((d * d) * l))) * -0.25)) * w0_m
	else:
		tmp = w0_m
	return w0_s * tmp

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d = sort([w0_m, M, D, h, l, d])
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+41)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(D * Float64(M * M)) * h) * Float64(D / Float64(Float64(d * d) * l))) * -0.25)) * w0_m);
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d = num2cell(sort([w0_m, M, D, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -1e+41)
		tmp = sqrt(((((D * (M * M)) * h) * (D / ((d * d) * l))) * -0.25)) * w0_m;
	else
		tmp = w0_m;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+41], N[(N[Sqrt[N[(N[(N[(N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(D / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * w0$95$m), $MachinePrecision], w0$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.00000000000000001e41

   1. Initial program 63.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 64.3%

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

   4. Taylor expanded in M around inf

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

   6. Applied rewrites 45.3%

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

   if -1.00000000000000001e41 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.8%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 94.9%

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

Alternative 10: 80.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d] = \mathsf{sort}([w0_m, M, D, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+136}:\\ \;\;\;\;w0\_m \cdot \left(\frac{\left(D \cdot \frac{D}{d \cdot d}\right) \cdot \left(\left(h \cdot M\right) \cdot M\right)}{\ell} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\_m\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+136)
    (* w0_m (* (/ (* (* D (/ D (* d d))) (* (* h M) M)) l) -0.125))
    w0_m)))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+136) {
		tmp = w0_m * ((((D * (D / (d * d))) * ((h * M) * M)) / l) * -0.125);
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Fortran

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

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d+136)) then
        tmp = w0_m * ((((d * (d / (d_1 * d_1))) * ((h * m) * m)) / l) * (-0.125d0))
    else
        tmp = w0_m
    end if
    code = w0_s * tmp
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M && M < D && D < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+136) {
		tmp = w0_m * ((((D * (D / (d * d))) * ((h * M) * M)) / l) * -0.125);
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d] = sort([w0_m, M, D, h, l, d])
def code(w0_s, w0_m, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+136:
		tmp = w0_m * ((((D * (D / (d * d))) * ((h * M) * M)) / l) * -0.125)
	else:
		tmp = w0_m
	return w0_s * tmp

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d = sort([w0_m, M, D, h, l, d])
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+136)
		tmp = Float64(w0_m * Float64(Float64(Float64(Float64(D * Float64(D / Float64(d * d))) * Float64(Float64(h * M) * M)) / l) * -0.125));
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d = num2cell(sort([w0_m, M, D, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+136)
		tmp = w0_m * ((((D * (D / (d * d))) * ((h * M) * M)) / l) * -0.125);
	else
		tmp = w0_m;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+136], N[(w0$95$m * N[(N[(N[(N[(D * N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision], w0$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.00000000000000012e136

   1. Initial program 60.9%

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

   2. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 39.3

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

   4. Applied rewrites 39.3%

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

   5. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 39.2

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

   7. Applied rewrites 41.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. frac-times N/A

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

      13. pow2 N/A

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

      14. pow2 N/A

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

      15. associate-*r/ N/A

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

      16. lower-/.f64 N/A

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

   9. Applied rewrites 44.4%

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

   if -2.00000000000000012e136 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.1%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 92.1%

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

Alternative 11: 79.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d] = \mathsf{sort}([w0_m, M, D, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+136}:\\ \;\;\;\;\left(\left(\frac{\left(\left(M \cdot D\right) \cdot M\right) \cdot h}{d} \cdot \frac{D}{\ell \cdot d}\right) \cdot -0.125\right) \cdot w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+136)
    (* (* (* (/ (* (* (* M D) M) h) d) (/ D (* l d))) -0.125) w0_m)
    w0_m)))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+136) {
		tmp = ((((((M * D) * M) * h) / d) * (D / (l * d))) * -0.125) * w0_m;
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Fortran

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

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d+136)) then
        tmp = ((((((m * d) * m) * h) / d_1) * (d / (l * d_1))) * (-0.125d0)) * w0_m
    else
        tmp = w0_m
    end if
    code = w0_s * tmp
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M && M < D && D < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+136) {
		tmp = ((((((M * D) * M) * h) / d) * (D / (l * d))) * -0.125) * w0_m;
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d] = sort([w0_m, M, D, h, l, d])
def code(w0_s, w0_m, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+136:
		tmp = ((((((M * D) * M) * h) / d) * (D / (l * d))) * -0.125) * w0_m
	else:
		tmp = w0_m
	return w0_s * tmp

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d = sort([w0_m, M, D, h, l, d])
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+136)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(M * D) * M) * h) / d) * Float64(D / Float64(l * d))) * -0.125) * w0_m);
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d = num2cell(sort([w0_m, M, D, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+136)
		tmp = ((((((M * D) * M) * h) / d) * (D / (l * d))) * -0.125) * w0_m;
	else
		tmp = w0_m;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+136], N[(N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] * N[(D / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * w0$95$m), $MachinePrecision], w0$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.00000000000000012e136

   1. Initial program 60.9%

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

   2. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 39.3

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

   4. Applied rewrites 39.3%

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

   5. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 39.2

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

   7. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.9

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

   9. Applied rewrites 42.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. pow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. pow2 N/A

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

       9. lift-/.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. lift-*.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. associate-/l* N/A

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

       16. lift-*.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. times-frac N/A

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

       19. lower-*.f64 N/A

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

   11. Applied rewrites 48.6%

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

   if -2.00000000000000012e136 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.1%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 92.1%

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

Alternative 12: 79.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d] = \mathsf{sort}([w0_m, M, D, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+136}:\\ \;\;\;\;w0\_m \cdot \left(\frac{M \cdot \left(\left(h \cdot M\right) \cdot \left(D \cdot D\right)\right)}{d \cdot \left(d \cdot \ell\right)} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\_m\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+136)
    (* w0_m (* (/ (* M (* (* h M) (* D D))) (* d (* d l))) -0.125))
    w0_m)))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+136) {
		tmp = w0_m * (((M * ((h * M) * (D * D))) / (d * (d * l))) * -0.125);
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Fortran

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

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d+136)) then
        tmp = w0_m * (((m * ((h * m) * (d * d))) / (d_1 * (d_1 * l))) * (-0.125d0))
    else
        tmp = w0_m
    end if
    code = w0_s * tmp
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M && M < D && D < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+136) {
		tmp = w0_m * (((M * ((h * M) * (D * D))) / (d * (d * l))) * -0.125);
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d] = sort([w0_m, M, D, h, l, d])
def code(w0_s, w0_m, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+136:
		tmp = w0_m * (((M * ((h * M) * (D * D))) / (d * (d * l))) * -0.125)
	else:
		tmp = w0_m
	return w0_s * tmp

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d = sort([w0_m, M, D, h, l, d])
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+136)
		tmp = Float64(w0_m * Float64(Float64(Float64(M * Float64(Float64(h * M) * Float64(D * D))) / Float64(d * Float64(d * l))) * -0.125));
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d = num2cell(sort([w0_m, M, D, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+136)
		tmp = w0_m * (((M * ((h * M) * (D * D))) / (d * (d * l))) * -0.125);
	else
		tmp = w0_m;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+136], N[(w0$95$m * N[(N[(N[(M * N[(N[(h * M), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision], w0$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.00000000000000012e136

   1. Initial program 60.9%

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

   2. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 39.3

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

   4. Applied rewrites 39.3%

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

   5. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 39.2

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

   7. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 43.4

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

   9. Applied rewrites 43.4%

      \[\leadsto w0 \cdot \left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot \left(d \cdot \ell\right)} \cdot -0.125\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. pow2 N/A

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

       18. lift-*.f64 43.4

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

   11. Applied rewrites 43.4%

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

   if -2.00000000000000012e136 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.1%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 92.1%

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

Alternative 13: 79.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d] = \mathsf{sort}([w0_m, M, D, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+136}:\\ \;\;\;\;\left(\left(\left(\left(\left(h \cdot M\right) \cdot M\right) \cdot D\right) \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot -0.125\right) \cdot w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+136)
    (* (* (* (* (* (* h M) M) D) (/ D (* (* d d) l))) -0.125) w0_m)
    w0_m)))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+136) {
		tmp = (((((h * M) * M) * D) * (D / ((d * d) * l))) * -0.125) * w0_m;
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Fortran

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

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d+136)) then
        tmp = (((((h * m) * m) * d) * (d / ((d_1 * d_1) * l))) * (-0.125d0)) * w0_m
    else
        tmp = w0_m
    end if
    code = w0_s * tmp
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M && M < D && D < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+136) {
		tmp = (((((h * M) * M) * D) * (D / ((d * d) * l))) * -0.125) * w0_m;
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d] = sort([w0_m, M, D, h, l, d])
def code(w0_s, w0_m, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+136:
		tmp = (((((h * M) * M) * D) * (D / ((d * d) * l))) * -0.125) * w0_m
	else:
		tmp = w0_m
	return w0_s * tmp

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d = sort([w0_m, M, D, h, l, d])
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+136)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(h * M) * M) * D) * Float64(D / Float64(Float64(d * d) * l))) * -0.125) * w0_m);
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d = num2cell(sort([w0_m, M, D, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+136)
		tmp = (((((h * M) * M) * D) * (D / ((d * d) * l))) * -0.125) * w0_m;
	else
		tmp = w0_m;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+136], N[(N[(N[(N[(N[(N[(h * M), $MachinePrecision] * M), $MachinePrecision] * D), $MachinePrecision] * N[(D / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * w0$95$m), $MachinePrecision], w0$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.00000000000000012e136

   1. Initial program 60.9%

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

   2. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 39.3

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

   4. Applied rewrites 39.3%

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

   5. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 39.2

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

   7. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.9

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

   9. Applied rewrites 42.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 42.1

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 43.7

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

   11. Applied rewrites 43.7%

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

   if -2.00000000000000012e136 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.1%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 92.1%

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

Alternative 14: 79.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d] = \mathsf{sort}([w0_m, M, D, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+136}:\\ \;\;\;\;\left(\left(\left(\left(D \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot \frac{D}{\left(\ell \cdot d\right) \cdot d}\right) \cdot -0.125\right) \cdot w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+136)
    (* (* (* (* (* D (* M M)) h) (/ D (* (* l d) d))) -0.125) w0_m)
    w0_m)))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+136) {
		tmp = ((((D * (M * M)) * h) * (D / ((l * d) * d))) * -0.125) * w0_m;
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Fortran

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

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d+136)) then
        tmp = ((((d * (m * m)) * h) * (d / ((l * d_1) * d_1))) * (-0.125d0)) * w0_m
    else
        tmp = w0_m
    end if
    code = w0_s * tmp
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M && M < D && D < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+136) {
		tmp = ((((D * (M * M)) * h) * (D / ((l * d) * d))) * -0.125) * w0_m;
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d] = sort([w0_m, M, D, h, l, d])
def code(w0_s, w0_m, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+136:
		tmp = ((((D * (M * M)) * h) * (D / ((l * d) * d))) * -0.125) * w0_m
	else:
		tmp = w0_m
	return w0_s * tmp

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d = sort([w0_m, M, D, h, l, d])
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+136)
		tmp = Float64(Float64(Float64(Float64(Float64(D * Float64(M * M)) * h) * Float64(D / Float64(Float64(l * d) * d))) * -0.125) * w0_m);
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d = num2cell(sort([w0_m, M, D, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+136)
		tmp = ((((D * (M * M)) * h) * (D / ((l * d) * d))) * -0.125) * w0_m;
	else
		tmp = w0_m;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+136], N[(N[(N[(N[(N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(D / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * w0$95$m), $MachinePrecision], w0$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.00000000000000012e136

   1. Initial program 60.9%

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

   2. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 39.3

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

   4. Applied rewrites 39.3%

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

   5. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 39.2

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

   7. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.9

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

   9. Applied rewrites 42.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 44.3

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

   11. Applied rewrites 44.3%

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

   if -2.00000000000000012e136 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.1%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 92.1%

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

Alternative 15: 78.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d] = \mathsf{sort}([w0_m, M, D, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+136}:\\ \;\;\;\;\left(\left(\left(\left(D \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot \frac{D}{\left(d \cdot d\right) \cdot \ell}\right) \cdot -0.125\right) \cdot w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m\\ \end{array} \end{array} \]

FPCore

w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+136)
    (* (* (* (* (* D (* M M)) h) (/ D (* (* d d) l))) -0.125) w0_m)
    w0_m)))

C

w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+136) {
		tmp = ((((D * (M * M)) * h) * (D / ((d * d) * l))) * -0.125) * w0_m;
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Fortran

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

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d+136)) then
        tmp = ((((d * (m * m)) * h) * (d / ((d_1 * d_1) * l))) * (-0.125d0)) * w0_m
    else
        tmp = w0_m
    end if
    code = w0_s * tmp
end function

Java

w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M && M < D && D < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+136) {
		tmp = ((((D * (M * M)) * h) * (D / ((d * d) * l))) * -0.125) * w0_m;
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Python

w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d] = sort([w0_m, M, D, h, l, d])
def code(w0_s, w0_m, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+136:
		tmp = ((((D * (M * M)) * h) * (D / ((d * d) * l))) * -0.125) * w0_m
	else:
		tmp = w0_m
	return w0_s * tmp

Julia

w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d = sort([w0_m, M, D, h, l, d])
function code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+136)
		tmp = Float64(Float64(Float64(Float64(Float64(D * Float64(M * M)) * h) * Float64(D / Float64(Float64(d * d) * l))) * -0.125) * w0_m);
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

MATLAB

w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d = num2cell(sort([w0_m, M, D, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+136)
		tmp = ((((D * (M * M)) * h) * (D / ((d * d) * l))) * -0.125) * w0_m;
	else
		tmp = w0_m;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+136], N[(N[(N[(N[(N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(D / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * w0$95$m), $MachinePrecision], w0$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.00000000000000012e136

   1. Initial program 60.9%

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

   2. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 39.3

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

   4. Applied rewrites 39.3%

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

   5. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 39.2

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

   7. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.9

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

   9. Applied rewrites 42.8%

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

   if -2.00000000000000012e136 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 89.1%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 92.1%

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

Alternative 16: 68.6% accurate, 39.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(w0_s, w0_m, m, d, h, l, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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_s * w0_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d_] := N[(w0$95$s * w0$95$m), $MachinePrecision]

Derivation

1. Initial program 81.5%

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

2. Taylor expanded in M around 0

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

4. Applied rewrites 68.6%

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

Reproduce

herbie shell --seed 2025114 
(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))))))