Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 89.8%

Time: 5.6s

Alternatives: 6

Speedup: 0.6×

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: 81.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := w0\_m \cdot \sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+290}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\sqrt{\left(\frac{M\_m}{d} \cdot \frac{h \cdot M\_m}{\ell \cdot d}\right) \cdot -0.25} \cdot D\_m\right) \cdot w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{D\_m}{d} \cdot \left(\frac{D\_m}{d} \cdot \frac{\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot 0.25}{\ell}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (let* ((t_0
         (*
          w0_m
          (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)))))))
   (*
    w0_s
    (if (<= t_0 5e+290)
      t_0
      (if (<= t_0 INFINITY)
        (* (* (sqrt (* (* (/ M_m d) (/ (* h M_m) (* l d))) -0.25)) D_m) w0_m)
        (*
         w0_m
         (sqrt
          (-
           1.0
           (* (/ D_m d) (* (/ D_m d) (/ (* (* (* M_m M_m) h) 0.25) l)))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double t_0 = w0_m * sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 5e+290) {
		tmp = t_0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((((M_m / d) * ((h * M_m) / (l * d))) * -0.25)) * D_m) * w0_m;
	} else {
		tmp = w0_m * sqrt((1.0 - ((D_m / d) * ((D_m / d) * ((((M_m * M_m) * h) * 0.25) / l)))));
	}
	return w0_s * tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double t_0 = w0_m * Math.sqrt((1.0 - (Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 5e+290) {
		tmp = t_0;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((((M_m / d) * ((h * M_m) / (l * d))) * -0.25)) * D_m) * w0_m;
	} else {
		tmp = w0_m * Math.sqrt((1.0 - ((D_m / d) * ((D_m / d) * ((((M_m * M_m) * h) * 0.25) / l)))));
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d] = sort([w0_m, M_m, D_m, h, l, d])
def code(w0_s, w0_m, M_m, D_m, h, l, d):
	t_0 = w0_m * math.sqrt((1.0 - (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))))
	tmp = 0
	if t_0 <= 5e+290:
		tmp = t_0
	elif t_0 <= math.inf:
		tmp = (math.sqrt((((M_m / d) * ((h * M_m) / (l * d))) * -0.25)) * D_m) * w0_m
	else:
		tmp = w0_m * math.sqrt((1.0 - ((D_m / d) * ((D_m / d) * ((((M_m * M_m) * h) * 0.25) / l)))))
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	t_0 = Float64(w0_m * sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
	tmp = 0.0
	if (t_0 <= 5e+290)
		tmp = t_0;
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(M_m / d) * Float64(Float64(h * M_m) / Float64(l * d))) * -0.25)) * D_m) * w0_m);
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(D_m / d) * Float64(Float64(D_m / d) * Float64(Float64(Float64(Float64(M_m * M_m) * h) * 0.25) / l))))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d = num2cell(sort([w0_m, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d)
	t_0 = w0_m * sqrt((1.0 - ((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l))));
	tmp = 0.0;
	if (t_0 <= 5e+290)
		tmp = t_0;
	elseif (t_0 <= Inf)
		tmp = (sqrt((((M_m / d) * ((h * M_m) / (l * d))) * -0.25)) * D_m) * w0_m;
	else
		tmp = w0_m * sqrt((1.0 - ((D_m / d) * ((D_m / d) * ((((M_m * M_m) * h) * 0.25) / l)))));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[t$95$0, 5e+290], t$95$0, If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(h * M$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * D$95$m), $MachinePrecision] * w0$95$m), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * 0.25), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))) < 4.9999999999999998e290

   1. Initial program 81.4%

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

   if 4.9999999999999998e290 < (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))))) < +inf.0

   1. Initial program 81.4%

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

   2. Taylor expanded in D around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 17.8

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

   4. Applied rewrites 17.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 17.8

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

   6. Applied rewrites 17.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 25.5

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

   8. Applied rewrites 25.5%

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

   if +inf.0 < (*.f64 w0 (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))))

   1. Initial program 81.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow-prod-down N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. mult-flip N/A

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

      18. metadata-eval N/A

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

      19. swap-sqr N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 53.7

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

   3. Applied rewrites 53.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. lift-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-pow.f64 N/A

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

      21. lift-*.f64 N/A

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

   5. Applied rewrites 82.3%

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

Alternative 2: 89.7% accurate, 0.6× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -2e+124)
    (* (* (sqrt (* (* (/ M_m d) (/ (* h M_m) (* l d))) -0.25)) D_m) w0_m)
    (*
     w0_m
     (sqrt
      (fma
       (* (/ M_m (+ d d)) D_m)
       (/ (* (/ D_m (+ d d)) (* M_m h)) (- l))
       1.0))))))

C

M_m = fabs(M);
D_m = fabs(D);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+124) {
		tmp = (sqrt((((M_m / d) * ((h * M_m) / (l * d))) * -0.25)) * D_m) * w0_m;
	} else {
		tmp = w0_m * sqrt(fma(((M_m / (d + d)) * D_m), (((D_m / (d + d)) * (M_m * h)) / -l), 1.0));
	}
	return w0_s * tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+124)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(M_m / d) * Float64(Float64(h * M_m) / Float64(l * d))) * -0.25)) * D_m) * w0_m);
	else
		tmp = Float64(w0_m * sqrt(fma(Float64(Float64(M_m / Float64(d + d)) * D_m), Float64(Float64(Float64(D_m / Float64(d + d)) * Float64(M_m * h)) / Float64(-l)), 1.0)));
	end
	return Float64(w0_s * tmp)
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+124], N[(N[(N[Sqrt[N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(h * M$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * D$95$m), $MachinePrecision] * w0$95$m), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(N[(N[(M$95$m / N[(d + d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(N[(D$95$m / N[(d + d), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.9999999999999999e124

   1. Initial program 81.4%

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

   2. Taylor expanded in D around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 17.8

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

   4. Applied rewrites 17.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 17.8

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

   6. Applied rewrites 17.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 25.5

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

   8. Applied rewrites 25.5%

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

   if -1.9999999999999999e124 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 81.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 87.0%

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

Alternative 3: 88.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+21}:\\ \;\;\;\;\left(\sqrt{\left(\frac{M\_m}{d} \cdot \frac{h \cdot M\_m}{\ell \cdot d}\right) \cdot -0.25} \cdot D\_m\right) \cdot w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{D\_m}{d} \cdot \left(\frac{D\_m}{d} \cdot \frac{\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot 0.25}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -2e+21)
    (* (* (sqrt (* (* (/ M_m d) (/ (* h M_m) (* l d))) -0.25)) D_m) w0_m)
    (*
     w0_m
     (sqrt
      (- 1.0 (* (/ D_m d) (* (/ D_m d) (/ (* (* (* M_m M_m) h) 0.25) l)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+21) {
		tmp = (sqrt((((M_m / d) * ((h * M_m) / (l * d))) * -0.25)) * D_m) * w0_m;
	} else {
		tmp = w0_m * sqrt((1.0 - ((D_m / d) * ((D_m / d) * ((((M_m * M_m) * h) * 0.25) / l)))));
	}
	return w0_s * tmp;
}

Fortran

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

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

real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-2d+21)) then
        tmp = (sqrt((((m_m / d) * ((h * m_m) / (l * d))) * (-0.25d0))) * d_m) * w0_m
    else
        tmp = w0_m * sqrt((1.0d0 - ((d_m / d) * ((d_m / d) * ((((m_m * m_m) * h) * 0.25d0) / l)))))
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+21) {
		tmp = (Math.sqrt((((M_m / d) * ((h * M_m) / (l * d))) * -0.25)) * D_m) * w0_m;
	} else {
		tmp = w0_m * Math.sqrt((1.0 - ((D_m / d) * ((D_m / d) * ((((M_m * M_m) * h) * 0.25) / l)))));
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d] = sort([w0_m, M_m, D_m, h, l, d])
def code(w0_s, w0_m, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+21:
		tmp = (math.sqrt((((M_m / d) * ((h * M_m) / (l * d))) * -0.25)) * D_m) * w0_m
	else:
		tmp = w0_m * math.sqrt((1.0 - ((D_m / d) * ((D_m / d) * ((((M_m * M_m) * h) * 0.25) / l)))))
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+21)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(M_m / d) * Float64(Float64(h * M_m) / Float64(l * d))) * -0.25)) * D_m) * w0_m);
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(D_m / d) * Float64(Float64(D_m / d) * Float64(Float64(Float64(Float64(M_m * M_m) * h) * 0.25) / l))))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d = num2cell(sort([w0_m, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+21)
		tmp = (sqrt((((M_m / d) * ((h * M_m) / (l * d))) * -0.25)) * D_m) * w0_m;
	else
		tmp = w0_m * sqrt((1.0 - ((D_m / d) * ((D_m / d) * ((((M_m * M_m) * h) * 0.25) / l)))));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+21], N[(N[(N[Sqrt[N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(h * M$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * D$95$m), $MachinePrecision] * w0$95$m), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * 0.25), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2e21

   1. Initial program 81.4%

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

   2. Taylor expanded in D around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 17.8

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

   4. Applied rewrites 17.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 17.8

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

   6. Applied rewrites 17.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 25.5

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

   8. Applied rewrites 25.5%

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

   if -2e21 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 81.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow-prod-down N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. mult-flip N/A

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

      18. metadata-eval N/A

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

      19. swap-sqr N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 53.7

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

   3. Applied rewrites 53.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. lift-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-pow.f64 N/A

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

      21. lift-*.f64 N/A

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

   5. Applied rewrites 82.3%

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

Alternative 4: 88.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+21}:\\ \;\;\;\;\left(\sqrt{\left(\frac{M\_m}{d} \cdot \frac{h \cdot M\_m}{\ell \cdot d}\right) \cdot -0.25} \cdot D\_m\right) \cdot w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -2e+21)
    (* (* (sqrt (* (* (/ M_m d) (/ (* h M_m) (* l d))) -0.25)) D_m) w0_m)
    (* w0_m 1.0))))

C

M_m = fabs(M);
D_m = fabs(D);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+21) {
		tmp = (sqrt((((M_m / d) * ((h * M_m) / (l * d))) * -0.25)) * D_m) * w0_m;
	} else {
		tmp = w0_m * 1.0;
	}
	return w0_s * tmp;
}

Fortran

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

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

real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-2d+21)) then
        tmp = (sqrt((((m_m / d) * ((h * m_m) / (l * d))) * (-0.25d0))) * d_m) * w0_m
    else
        tmp = w0_m * 1.0d0
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+21) {
		tmp = (Math.sqrt((((M_m / d) * ((h * M_m) / (l * d))) * -0.25)) * D_m) * w0_m;
	} else {
		tmp = w0_m * 1.0;
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d] = sort([w0_m, M_m, D_m, h, l, d])
def code(w0_s, w0_m, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e+21:
		tmp = (math.sqrt((((M_m / d) * ((h * M_m) / (l * d))) * -0.25)) * D_m) * w0_m
	else:
		tmp = w0_m * 1.0
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+21)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(M_m / d) * Float64(Float64(h * M_m) / Float64(l * d))) * -0.25)) * D_m) * w0_m);
	else
		tmp = Float64(w0_m * 1.0);
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d = num2cell(sort([w0_m, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+21)
		tmp = (sqrt((((M_m / d) * ((h * M_m) / (l * d))) * -0.25)) * D_m) * w0_m;
	else
		tmp = w0_m * 1.0;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+21], N[(N[(N[Sqrt[N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(h * M$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * D$95$m), $MachinePrecision] * w0$95$m), $MachinePrecision], N[(w0$95$m * 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2e21

   1. Initial program 81.4%

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

   2. Taylor expanded in D around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 17.8

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

   4. Applied rewrites 17.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 17.8

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

   6. Applied rewrites 17.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 25.5

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

   8. Applied rewrites 25.5%

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

   if -2e21 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 81.4%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 68.5%

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

Alternative 5: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M_m, D_m, h, l, d] = \mathsf{sort}([w0_m, M_m, D_m, h, l, d])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+67}:\\ \;\;\;\;\left(\sqrt{\left(\left(h \cdot \frac{M\_m}{\left(d \cdot d\right) \cdot \ell}\right) \cdot M\_m\right) \cdot -0.25} \cdot D\_m\right) \cdot w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M_m D_m h l d)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -1e+67)
    (* (* (sqrt (* (* (* h (/ M_m (* (* d d) l))) M_m) -0.25)) D_m) w0_m)
    (* w0_m 1.0))))

C

M_m = fabs(M);
D_m = fabs(D);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e+67) {
		tmp = (sqrt((((h * (M_m / ((d * d) * l))) * M_m) * -0.25)) * D_m) * w0_m;
	} else {
		tmp = w0_m * 1.0;
	}
	return w0_s * tmp;
}

Fortran

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

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

real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d)
use fmin_fmax_functions
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-1d+67)) then
        tmp = (sqrt((((h * (m_m / ((d * d) * l))) * m_m) * (-0.25d0))) * d_m) * w0_m
    else
        tmp = w0_m * 1.0d0
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e+67) {
		tmp = (Math.sqrt((((h * (M_m / ((d * d) * l))) * M_m) * -0.25)) * D_m) * w0_m;
	} else {
		tmp = w0_m * 1.0;
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M_m, D_m, h, l, d] = sort([w0_m, M_m, D_m, h, l, d])
def code(w0_s, w0_m, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -1e+67:
		tmp = (math.sqrt((((h * (M_m / ((d * d) * l))) * M_m) * -0.25)) * D_m) * w0_m
	else:
		tmp = w0_m * 1.0
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M_m, D_m, h, l, d = sort([w0_m, M_m, D_m, h, l, d])
function code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+67)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(h * Float64(M_m / Float64(Float64(d * d) * l))) * M_m) * -0.25)) * D_m) * w0_m);
	else
		tmp = Float64(w0_m * 1.0);
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M_m, D_m, h, l, d = num2cell(sort([w0_m, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -1e+67)
		tmp = (sqrt((((h * (M_m / ((d * d) * l))) * M_m) * -0.25)) * D_m) * w0_m;
	else
		tmp = w0_m * 1.0;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: w0_m, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+67], N[(N[(N[Sqrt[N[(N[(N[(h * N[(M$95$m / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * D$95$m), $MachinePrecision] * w0$95$m), $MachinePrecision], N[(w0$95$m * 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -9.99999999999999983e66

   1. Initial program 81.4%

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

   2. Taylor expanded in D around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 17.8

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

   4. Applied rewrites 17.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 17.8

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

   6. Applied rewrites 17.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 20.5

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

   8. Applied rewrites 20.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 21.5

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

   10. Applied rewrites 21.5%

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

   if -9.99999999999999983e66 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 81.4%

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

   2. Taylor expanded in M around 0

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

   4. Applied rewrites 68.5%

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

Alternative 6: 68.5% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

2. Taylor expanded in M around 0

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

4. Applied rewrites 68.5%

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

Reproduce

herbie shell --seed 2025156 
(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))))))