Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.7% → 85.7%

Time: 7.5s

Alternatives: 13

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.7% 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: 85.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 10^{+54}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \frac{\left(D\_m \cdot M\_m\right) \cdot \left(D\_m \cdot \frac{M\_m}{d\_m}\right)}{d\_m \cdot 4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\ell - \left(\frac{h}{d\_m} \cdot D\_m\right) \cdot \left(\left(0.25 \cdot D\_m\right) \cdot \left(\frac{M\_m}{d\_m} \cdot M\_m\right)\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ (* M_m D_m) (* 2.0 d_m)) 1e+54)
   (*
    w0
    (sqrt
     (- 1.0 (/ (* h (/ (* (* D_m M_m) (* D_m (/ M_m d_m))) (* d_m 4.0))) l))))
   (*
    w0
    (sqrt
     (/ (- l (* (* (/ h d_m) D_m) (* (* 0.25 D_m) (* (/ M_m d_m) M_m)))) l)))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1e+54) {
		tmp = w0 * sqrt((1.0 - ((h * (((D_m * M_m) * (D_m * (M_m / d_m))) / (d_m * 4.0))) / l)));
	} else {
		tmp = w0 * sqrt(((l - (((h / d_m) * D_m) * ((0.25 * D_m) * ((M_m / d_m) * M_m)))) / l));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(w0, m_m, d_m, h, l, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((m_m * d_m) / (2.0d0 * d_m_1)) <= 1d+54) then
        tmp = w0 * sqrt((1.0d0 - ((h * (((d_m * m_m) * (d_m * (m_m / d_m_1))) / (d_m_1 * 4.0d0))) / l)))
    else
        tmp = w0 * sqrt(((l - (((h / d_m_1) * d_m) * ((0.25d0 * d_m) * ((m_m / d_m_1) * m_m)))) / l))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1e+54) {
		tmp = w0 * Math.sqrt((1.0 - ((h * (((D_m * M_m) * (D_m * (M_m / d_m))) / (d_m * 4.0))) / l)));
	} else {
		tmp = w0 * Math.sqrt(((l - (((h / d_m) * D_m) * ((0.25 * D_m) * ((M_m / d_m) * M_m)))) / l));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if ((M_m * D_m) / (2.0 * d_m)) <= 1e+54:
		tmp = w0 * math.sqrt((1.0 - ((h * (((D_m * M_m) * (D_m * (M_m / d_m))) / (d_m * 4.0))) / l)))
	else:
		tmp = w0 * math.sqrt(((l - (((h / d_m) * D_m) * ((0.25 * D_m) * ((M_m / d_m) * M_m)))) / l))
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) <= 1e+54)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(Float64(Float64(D_m * M_m) * Float64(D_m * Float64(M_m / d_m))) / Float64(d_m * 4.0))) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(Float64(l - Float64(Float64(Float64(h / d_m) * D_m) * Float64(Float64(0.25 * D_m) * Float64(Float64(M_m / d_m) * M_m)))) / l)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1e+54)
		tmp = w0 * sqrt((1.0 - ((h * (((D_m * M_m) * (D_m * (M_m / d_m))) / (d_m * 4.0))) / l)));
	else
		tmp = w0 * sqrt(((l - (((h / d_m) * D_m) * ((0.25 * D_m) * ((M_m / d_m) * M_m)))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 1e+54], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(l - N[(N[(N[(h / d$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(N[(M$95$m / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1.0000000000000001e54

   1. Initial program 86.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.2

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 90.8

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

   4. Applied rewrites 90.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{h \cdot \frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}}}{\ell}} \]

      12. lower-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{h \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}}}{\ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{h \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1}}{\ell}} \]

      14. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{h \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}}{\ell}} \]

      15. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{h \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}}{\ell}} \]

      16. *-commutative N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{h \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1}}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{h \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1}}{\ell}} \]

      18. metadata-eval 90.8

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

   6. Applied rewrites 90.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

   if 1.0000000000000001e54 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 61.5%

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

   3. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 51.7

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

   5. Applied rewrites 51.7%

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

   7. Applied rewrites 62.6%

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

   9. Applied rewrites 68.3%

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

Alternative 2: 83.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}} \leq 5 \cdot 10^{+20}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m, D\_m \cdot \left(\left(-0.125 \cdot \left(\frac{M\_m}{d\_m} \cdot h\right)\right) \cdot \frac{\frac{M\_m}{d\_m}}{\ell}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\ell - \left(\left(\left(0.25 \cdot D\_m\right) \cdot \frac{D\_m \cdot h}{d\_m}\right) \cdot M\_m\right) \cdot \frac{M\_m}{d\_m}}{\ell}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<=
      (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l))))
      5e+20)
   (*
    w0
    (fma D_m (* D_m (* (* -0.125 (* (/ M_m d_m) h)) (/ (/ M_m d_m) l))) 1.0))
   (*
    w0
    (sqrt
     (/ (- l (* (* (* (* 0.25 D_m) (/ (* D_m h) d_m)) M_m) (/ M_m d_m))) l)))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)))) <= 5e+20) {
		tmp = w0 * fma(D_m, (D_m * ((-0.125 * ((M_m / d_m) * h)) * ((M_m / d_m) / l))), 1.0);
	} else {
		tmp = w0 * sqrt(((l - ((((0.25 * D_m) * ((D_m * h) / d_m)) * M_m) * (M_m / d_m))) / l));
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)))) <= 5e+20)
		tmp = Float64(w0 * fma(D_m, Float64(D_m * Float64(Float64(-0.125 * Float64(Float64(M_m / d_m) * h)) * Float64(Float64(M_m / d_m) / l))), 1.0));
	else
		tmp = Float64(w0 * sqrt(Float64(Float64(l - Float64(Float64(Float64(Float64(0.25 * D_m) * Float64(Float64(D_m * h) / d_m)) * M_m) * Float64(M_m / d_m))) / l)));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5e+20], N[(w0 * N[(D$95$m * N[(D$95$m * N[(N[(-0.125 * N[(N[(M$95$m / d$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(l - N[(N[(N[(N[(0.25 * D$95$m), $MachinePrecision] * N[(N[(D$95$m * h), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))) < 5e20

   1. Initial program 100.0%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 61.5%

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

   7. Applied rewrites 62.2%

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

   9. Applied rewrites 66.8%

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

   11. Applied rewrites 94.9%

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

   if 5e20 < (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 55.2%

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

   3. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 44.3

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

   5. Applied rewrites 44.3%

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

   7. Applied rewrites 58.8%

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

   9. Applied rewrites 63.2%

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

Alternative 3: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{M\_m}{d\_m}}{\ell}\\ \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+20}:\\ \;\;\;\;w0 \cdot \sqrt{\left(\left(\left(-0.25 \cdot D\_m\right) \cdot h\right) \cdot \left(\frac{t\_0}{d\_m} \cdot M\_m\right)\right) \cdot D\_m}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m, D\_m \cdot \left(\left(-0.125 \cdot \left(\frac{M\_m}{d\_m} \cdot h\right)\right) \cdot t\_0\right), 1\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ (/ M_m d_m) l)))
   (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -2e+20)
     (* w0 (sqrt (* (* (* (* -0.25 D_m) h) (* (/ t_0 d_m) M_m)) D_m)))
     (* w0 (fma D_m (* D_m (* (* -0.125 (* (/ M_m d_m) h)) t_0)) 1.0)))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m / d_m) / l;
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+20) {
		tmp = w0 * sqrt(((((-0.25 * D_m) * h) * ((t_0 / d_m) * M_m)) * D_m));
	} else {
		tmp = w0 * fma(D_m, (D_m * ((-0.125 * ((M_m / d_m) * h)) * t_0)), 1.0);
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(M_m / d_m) / l)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e+20)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(Float64(-0.25 * D_m) * h) * Float64(Float64(t_0 / d_m) * M_m)) * D_m)));
	else
		tmp = Float64(w0 * fma(D_m, Float64(D_m * Float64(Float64(-0.125 * Float64(Float64(M_m / d_m) * h)) * t_0)), 1.0));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(M$95$m / d$95$m), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+20], N[(w0 * N[Sqrt[N[(N[(N[(N[(-0.25 * D$95$m), $MachinePrecision] * h), $MachinePrecision] * N[(N[(t$95$0 / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(D$95$m * N[(D$95$m * N[(N[(-0.125 * N[(N[(M$95$m / d$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 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)) < -2e20

   1. Initial program 65.7%

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

   3. Taylor expanded in M around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 40.6

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

   5. Applied rewrites 40.6%

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

   7. Applied rewrites 47.7%

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

   9. Applied rewrites 51.0%

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

   11. Applied rewrites 56.3%

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

   if -2e20 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.2%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 60.6%

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

   7. Applied rewrites 61.3%

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

   9. Applied rewrites 65.5%

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

   11. Applied rewrites 92.9%

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

Alternative 4: 82.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+20}:\\ \;\;\;\;w0 \cdot \sqrt{\left(-0.25 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \frac{\left(\frac{M\_m}{d\_m} \cdot M\_m\right) \cdot h}{\ell \cdot d\_m}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m, D\_m \cdot \left(\left(-0.125 \cdot \left(\frac{M\_m}{d\_m} \cdot h\right)\right) \cdot \frac{\frac{M\_m}{d\_m}}{\ell}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -2e+20)
   (*
    w0
    (sqrt (* (* -0.25 (* D_m D_m)) (/ (* (* (/ M_m d_m) M_m) h) (* l d_m)))))
   (*
    w0
    (fma D_m (* D_m (* (* -0.125 (* (/ M_m d_m) h)) (/ (/ M_m d_m) l))) 1.0))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+20) {
		tmp = w0 * sqrt(((-0.25 * (D_m * D_m)) * ((((M_m / d_m) * M_m) * h) / (l * d_m))));
	} else {
		tmp = w0 * fma(D_m, (D_m * ((-0.125 * ((M_m / d_m) * h)) * ((M_m / d_m) / l))), 1.0);
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e+20)
		tmp = Float64(w0 * sqrt(Float64(Float64(-0.25 * Float64(D_m * D_m)) * Float64(Float64(Float64(Float64(M_m / d_m) * M_m) * h) / Float64(l * d_m)))));
	else
		tmp = Float64(w0 * fma(D_m, Float64(D_m * Float64(Float64(-0.125 * Float64(Float64(M_m / d_m) * h)) * Float64(Float64(M_m / d_m) / l))), 1.0));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+20], N[(w0 * N[Sqrt[N[(N[(-0.25 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M$95$m / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(D$95$m * N[(D$95$m * N[(N[(-0.125 * N[(N[(M$95$m / d$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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)) < -2e20

   1. Initial program 65.7%

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

   3. Taylor expanded in M around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 40.6

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

   5. Applied rewrites 40.6%

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

   7. Applied rewrites 49.7%

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

   if -2e20 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.2%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 60.6%

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

   7. Applied rewrites 61.3%

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

   9. Applied rewrites 65.5%

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

   11. Applied rewrites 92.9%

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

Alternative 5: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+83}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot -0.125, h \cdot \left(\frac{M\_m}{d\_m} \cdot \frac{M\_m}{\ell \cdot d\_m}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m \cdot \left(D\_m \cdot \frac{\frac{M\_m \cdot M\_m}{\ell} \cdot h}{d\_m}\right), \frac{-0.125}{d\_m}, 1\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -5e+83)
   (*
    w0
    (fma (* (* D_m D_m) -0.125) (* h (* (/ M_m d_m) (/ M_m (* l d_m)))) 1.0))
   (*
    w0
    (fma (* D_m (* D_m (/ (* (/ (* M_m M_m) l) h) d_m))) (/ -0.125 d_m) 1.0))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -5e+83) {
		tmp = w0 * fma(((D_m * D_m) * -0.125), (h * ((M_m / d_m) * (M_m / (l * d_m)))), 1.0);
	} else {
		tmp = w0 * fma((D_m * (D_m * ((((M_m * M_m) / l) * h) / d_m))), (-0.125 / d_m), 1.0);
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -5e+83)
		tmp = Float64(w0 * fma(Float64(Float64(D_m * D_m) * -0.125), Float64(h * Float64(Float64(M_m / d_m) * Float64(M_m / Float64(l * d_m)))), 1.0));
	else
		tmp = Float64(w0 * fma(Float64(D_m * Float64(D_m * Float64(Float64(Float64(Float64(M_m * M_m) / l) * h) / d_m))), Float64(-0.125 / d_m), 1.0));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+83], N[(w0 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(h * N[(N[(M$95$m / d$95$m), $MachinePrecision] * N[(M$95$m / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[(D$95$m * N[(D$95$m * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / l), $MachinePrecision] * h), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.125 / d$95$m), $MachinePrecision] + 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)) < -5.00000000000000029e83

   1. Initial program 64.2%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 38.1%

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

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 42.4%

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

   if -5.00000000000000029e83 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.5%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in d around 0

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

   8. Applied rewrites 68.2%

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

   10. Applied rewrites 78.7%

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

Alternative 6: 80.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+83}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot -0.125, h \cdot \left(\frac{M\_m}{d\_m} \cdot \frac{M\_m}{\ell \cdot d\_m}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -5e+83)
   (*
    w0
    (fma (* (* D_m D_m) -0.125) (* h (* (/ M_m d_m) (/ M_m (* l d_m)))) 1.0))
   (* w0 1.0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -5e+83) {
		tmp = w0 * fma(((D_m * D_m) * -0.125), (h * ((M_m / d_m) * (M_m / (l * d_m)))), 1.0);
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -5e+83)
		tmp = Float64(w0 * fma(Float64(Float64(D_m * D_m) * -0.125), Float64(h * Float64(Float64(M_m / d_m) * Float64(M_m / Float64(l * d_m)))), 1.0));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+83], N[(w0 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(h * N[(N[(M$95$m / d$95$m), $MachinePrecision] * N[(M$95$m / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.00000000000000029e83

   1. Initial program 64.2%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 38.1%

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

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 42.4%

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

   if -5.00000000000000029e83 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 94.4%

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

Alternative 7: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+107}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\frac{\left(\left(D\_m \cdot D\_m\right) \cdot M\_m\right) \cdot \left(M\_m \cdot h\right)}{d\_m \cdot \ell}, \frac{-0.125}{d\_m}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -1e+107)
   (*
    w0
    (fma (/ (* (* (* D_m D_m) M_m) (* M_m h)) (* d_m l)) (/ -0.125 d_m) 1.0))
   (* w0 1.0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -1e+107) {
		tmp = w0 * fma(((((D_m * D_m) * M_m) * (M_m * h)) / (d_m * l)), (-0.125 / d_m), 1.0);
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -1e+107)
		tmp = Float64(w0 * fma(Float64(Float64(Float64(Float64(D_m * D_m) * M_m) * Float64(M_m * h)) / Float64(d_m * l)), Float64(-0.125 / d_m), 1.0));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+107], N[(w0 * N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision] * N[(-0.125 / d$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -9.9999999999999997e106

   1. Initial program 62.9%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 39.5%

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

   6. Taylor expanded in d around 0

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

   8. Applied rewrites 42.7%

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

   10. Applied rewrites 43.8%

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

   if -9.9999999999999997e106 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.6%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 92.9%

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

Alternative 8: 79.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+83}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot -0.125, h \cdot \left(M\_m \cdot \frac{M\_m}{\left(d\_m \cdot d\_m\right) \cdot \ell}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -5e+83)
   (*
    w0
    (fma (* (* D_m D_m) -0.125) (* h (* M_m (/ M_m (* (* d_m d_m) l)))) 1.0))
   (* w0 1.0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -5e+83) {
		tmp = w0 * fma(((D_m * D_m) * -0.125), (h * (M_m * (M_m / ((d_m * d_m) * l)))), 1.0);
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -5e+83)
		tmp = Float64(w0 * fma(Float64(Float64(D_m * D_m) * -0.125), Float64(h * Float64(M_m * Float64(M_m / Float64(Float64(d_m * d_m) * l)))), 1.0));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+83], N[(w0 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(h * N[(M$95$m * N[(M$95$m / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5.00000000000000029e83

   1. Initial program 64.2%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 38.1%

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

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 39.9%

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

   if -5.00000000000000029e83 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 94.4%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+83}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, h \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+279}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(D\_m \cdot D\_m\right) \cdot -0.125, M\_m \cdot \frac{M\_m \cdot h}{\left(d\_m \cdot \ell\right) \cdot d\_m}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -2e+279)
   (*
    w0
    (fma (* (* D_m D_m) -0.125) (* M_m (/ (* M_m h) (* (* d_m l) d_m))) 1.0))
   (* w0 1.0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+279) {
		tmp = w0 * fma(((D_m * D_m) * -0.125), (M_m * ((M_m * h) / ((d_m * l) * d_m))), 1.0);
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e+279)
		tmp = Float64(w0 * fma(Float64(Float64(D_m * D_m) * -0.125), Float64(M_m * Float64(Float64(M_m * h) / Float64(Float64(d_m * l) * d_m))), 1.0));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+279], N[(w0 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(M$95$m * N[(N[(M$95$m * h), $MachinePrecision] / N[(N[(d$95$m * l), $MachinePrecision] * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.00000000000000012e279

   1. Initial program 55.8%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 45.4%

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

   7. Applied rewrites 46.9%

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

   9. Applied rewrites 48.4%

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

   11. Applied rewrites 48.7%

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

   if -2.00000000000000012e279 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 86.4%

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

Alternative 10: 83.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d\_m} \cdot M\_m\\ \mathbf{if}\;D\_m \leq 10^{+84}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\ell - \frac{\left(t\_0 \cdot h\right) \cdot \left(0.25 \cdot \left(D\_m \cdot D\_m\right)\right)}{d\_m}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\ell - \left(\frac{h}{d\_m} \cdot D\_m\right) \cdot \left(\left(0.25 \cdot D\_m\right) \cdot t\_0\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d_m) M_m)))
   (if (<= D_m 1e+84)
     (* w0 (sqrt (/ (- l (/ (* (* t_0 h) (* 0.25 (* D_m D_m))) d_m)) l)))
     (* w0 (sqrt (/ (- l (* (* (/ h d_m) D_m) (* (* 0.25 D_m) t_0))) l))))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m / d_m) * M_m;
	double tmp;
	if (D_m <= 1e+84) {
		tmp = w0 * sqrt(((l - (((t_0 * h) * (0.25 * (D_m * D_m))) / d_m)) / l));
	} else {
		tmp = w0 * sqrt(((l - (((h / d_m) * D_m) * ((0.25 * D_m) * t_0))) / l));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(w0, m_m, d_m, h, l, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m_m / d_m_1) * m_m
    if (d_m <= 1d+84) then
        tmp = w0 * sqrt(((l - (((t_0 * h) * (0.25d0 * (d_m * d_m))) / d_m_1)) / l))
    else
        tmp = w0 * sqrt(((l - (((h / d_m_1) * d_m) * ((0.25d0 * d_m) * t_0))) / l))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m / d_m) * M_m;
	double tmp;
	if (D_m <= 1e+84) {
		tmp = w0 * Math.sqrt(((l - (((t_0 * h) * (0.25 * (D_m * D_m))) / d_m)) / l));
	} else {
		tmp = w0 * Math.sqrt(((l - (((h / d_m) * D_m) * ((0.25 * D_m) * t_0))) / l));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = (M_m / d_m) * M_m
	tmp = 0
	if D_m <= 1e+84:
		tmp = w0 * math.sqrt(((l - (((t_0 * h) * (0.25 * (D_m * D_m))) / d_m)) / l))
	else:
		tmp = w0 * math.sqrt(((l - (((h / d_m) * D_m) * ((0.25 * D_m) * t_0))) / l))
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(M_m / d_m) * M_m)
	tmp = 0.0
	if (D_m <= 1e+84)
		tmp = Float64(w0 * sqrt(Float64(Float64(l - Float64(Float64(Float64(t_0 * h) * Float64(0.25 * Float64(D_m * D_m))) / d_m)) / l)));
	else
		tmp = Float64(w0 * sqrt(Float64(Float64(l - Float64(Float64(Float64(h / d_m) * D_m) * Float64(Float64(0.25 * D_m) * t_0))) / l)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = (M_m / d_m) * M_m;
	tmp = 0.0;
	if (D_m <= 1e+84)
		tmp = w0 * sqrt(((l - (((t_0 * h) * (0.25 * (D_m * D_m))) / d_m)) / l));
	else
		tmp = w0 * sqrt(((l - (((h / d_m) * D_m) * ((0.25 * D_m) * t_0))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(M$95$m / d$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[D$95$m, 1e+84], N[(w0 * N[Sqrt[N[(N[(l - N[(N[(N[(t$95$0 * h), $MachinePrecision] * N[(0.25 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(l - N[(N[(N[(h / d$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(N[(0.25 * D$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 1.00000000000000006e84

   1. Initial program 83.9%

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

   3. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 73.6%

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

   if 1.00000000000000006e84 < D

   1. Initial program 69.9%

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

   3. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 41.2

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

   5. Applied rewrites 41.2%

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

   7. Applied rewrites 57.5%

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

   9. Applied rewrites 72.8%

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

Alternative 11: 86.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d\_m} \cdot D\_m\\ w0 \cdot \sqrt{1 - \frac{h \cdot \frac{t\_0 \cdot t\_0}{4}}{\ell}} \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m d_m) D_m)))
   (* w0 (sqrt (- 1.0 (/ (* h (/ (* t_0 t_0) 4.0)) l))))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m / d_m) * D_m;
	return w0 * sqrt((1.0 - ((h * ((t_0 * t_0) / 4.0)) / l)));
}

Fortran

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

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

real(8) function code(w0, m_m, d_m, h, l, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    t_0 = (m_m / d_m_1) * d_m
    code = w0 * sqrt((1.0d0 - ((h * ((t_0 * t_0) / 4.0d0)) / l)))
end function

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m / d_m) * D_m;
	return w0 * Math.sqrt((1.0 - ((h * ((t_0 * t_0) / 4.0)) / l)));
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = (M_m / d_m) * D_m
	return w0 * math.sqrt((1.0 - ((h * ((t_0 * t_0) / 4.0)) / l)))

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(M_m / d_m) * D_m)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(Float64(t_0 * t_0) / 4.0)) / l))))
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp = code(w0, M_m, D_m, h, l, d_m)
	t_0 = (M_m / d_m) * D_m;
	tmp = w0 * sqrt((1.0 - ((h * ((t_0 * t_0) / 4.0)) / l)));
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(M$95$m / d$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 85.8

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 86.6

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

4. Applied rewrites 86.6%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*l/ N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

       \[\leadsto w0 \cdot \sqrt{1 - \frac{h \cdot \frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\color{blue}{3 + 1}}}{\ell}} \]

   12. lower-/.f64 N/A

       \[\leadsto w0 \cdot \sqrt{1 - \frac{h \cdot \color{blue}{\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}}}{\ell}} \]

   13. lower-*.f64 N/A

       \[\leadsto w0 \cdot \sqrt{1 - \frac{h \cdot \frac{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}{3 + 1}}{\ell}} \]

   14. *-commutative N/A

       \[\leadsto w0 \cdot \sqrt{1 - \frac{h \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}}{\ell}} \]

   15. lower-*.f64 N/A

       \[\leadsto w0 \cdot \sqrt{1 - \frac{h \cdot \frac{\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \left(D \cdot \frac{M}{d}\right)}{3 + 1}}{\ell}} \]

   16. *-commutative N/A

       \[\leadsto w0 \cdot \sqrt{1 - \frac{h \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1}}{\ell}} \]

   17. lower-*.f64 N/A

       \[\leadsto w0 \cdot \sqrt{1 - \frac{h \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}}{3 + 1}}{\ell}} \]

   18. metadata-eval 86.6

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

6. Applied rewrites 86.6%

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

Alternative 12: 81.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ w0 \cdot \mathsf{fma}\left(D\_m, D\_m \cdot \left(\left(-0.125 \cdot \left(\frac{M\_m}{d\_m} \cdot h\right)\right) \cdot \frac{\frac{M\_m}{d\_m}}{\ell}\right), 1\right) \end{array} \]

FPCore

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

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0 * fma(D_m, (D_m * ((-0.125 * ((M_m / d_m) * h)) * ((M_m / d_m) / l))), 1.0);
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	return Float64(w0 * fma(D_m, Float64(D_m * Float64(Float64(-0.125 * Float64(Float64(M_m / d_m) * h)) * Float64(Float64(M_m / d_m) / l))), 1.0))
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0 * N[(D$95$m * N[(D$95$m * N[(N[(-0.125 * N[(N[(M$95$m / d$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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. Add Preprocessing

3. Taylor expanded in M around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

5. Applied rewrites 52.0%

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

7. Applied rewrites 52.8%

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

9. Applied rewrites 56.0%

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

11. Applied rewrites 76.0%

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

Alternative 13: 68.4% accurate, 26.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(w0, m_m, d_m, h, l, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0 * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 63.6%

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

Reproduce

herbie shell --seed 2025017 
(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))))))