Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.9% → 89.7%

Time: 16.8s

Alternatives: 14

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.9% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{2 \cdot d}\\ \mathbf{if}\;M\_m \cdot D\_m \leq 5 \cdot 10^{+159}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{t\_0}{\ell} \cdot \frac{t\_0}{\frac{-1}{h}}}\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 2 \cdot 10^{+190}:\\ \;\;\;\;w0 \cdot \left(D\_m \cdot \sqrt{-0.25 \cdot \left(M\_m \cdot \left(h \cdot \frac{M\_m}{\ell \cdot \left(d \cdot d\right)}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D\_m}{d} \cdot \left(\frac{M\_m \cdot \left(h \cdot \frac{D\_m}{d}\right)}{\ell} \cdot \left(M\_m \cdot 0.25\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (/ (* M_m D_m) (* 2.0 d))))
   (if (<= (* M_m D_m) 5e+159)
     (* w0 (sqrt (+ 1.0 (* (/ t_0 l) (/ t_0 (/ -1.0 h))))))
     (if (<= (* M_m D_m) 2e+190)
       (* w0 (* D_m (sqrt (* -0.25 (* M_m (* h (/ M_m (* l (* d d)))))))))
       (*
        w0
        (sqrt
         (-
          1.0
          (* (/ D_m d) (* (/ (* M_m (* h (/ D_m d))) l) (* M_m 0.25))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (M_m * D_m) / (2.0 * d);
	double tmp;
	if ((M_m * D_m) <= 5e+159) {
		tmp = w0 * sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))));
	} else if ((M_m * D_m) <= 2e+190) {
		tmp = w0 * (D_m * sqrt((-0.25 * (M_m * (h * (M_m / (l * (d * d))))))));
	} else {
		tmp = w0 * sqrt((1.0 - ((D_m / d) * (((M_m * (h * (D_m / d))) / l) * (M_m * 0.25)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m_m * d_m) / (2.0d0 * d)
    if ((m_m * d_m) <= 5d+159) then
        tmp = w0 * sqrt((1.0d0 + ((t_0 / l) * (t_0 / ((-1.0d0) / h)))))
    else if ((m_m * d_m) <= 2d+190) then
        tmp = w0 * (d_m * sqrt(((-0.25d0) * (m_m * (h * (m_m / (l * (d * d))))))))
    else
        tmp = w0 * sqrt((1.0d0 - ((d_m / d) * (((m_m * (h * (d_m / d))) / l) * (m_m * 0.25d0)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (M_m * D_m) / (2.0 * d);
	double tmp;
	if ((M_m * D_m) <= 5e+159) {
		tmp = w0 * Math.sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))));
	} else if ((M_m * D_m) <= 2e+190) {
		tmp = w0 * (D_m * Math.sqrt((-0.25 * (M_m * (h * (M_m / (l * (d * d))))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((D_m / d) * (((M_m * (h * (D_m / d))) / l) * (M_m * 0.25)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = (M_m * D_m) / (2.0 * d)
	tmp = 0
	if (M_m * D_m) <= 5e+159:
		tmp = w0 * math.sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))))
	elif (M_m * D_m) <= 2e+190:
		tmp = w0 * (D_m * math.sqrt((-0.25 * (M_m * (h * (M_m / (l * (d * d))))))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((D_m / d) * (((M_m * (h * (D_m / d))) / l) * (M_m * 0.25)))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(Float64(M_m * D_m) / Float64(2.0 * d))
	tmp = 0.0
	if (Float64(M_m * D_m) <= 5e+159)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(t_0 / l) * Float64(t_0 / Float64(-1.0 / h))))));
	elseif (Float64(M_m * D_m) <= 2e+190)
		tmp = Float64(w0 * Float64(D_m * sqrt(Float64(-0.25 * Float64(M_m * Float64(h * Float64(M_m / Float64(l * Float64(d * d)))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D_m / d) * Float64(Float64(Float64(M_m * Float64(h * Float64(D_m / d))) / l) * Float64(M_m * 0.25))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = (M_m * D_m) / (2.0 * d);
	tmp = 0.0;
	if ((M_m * D_m) <= 5e+159)
		tmp = w0 * sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))));
	elseif ((M_m * D_m) <= 2e+190)
		tmp = w0 * (D_m * sqrt((-0.25 * (M_m * (h * (M_m / (l * (d * d))))))));
	else
		tmp = w0 * sqrt((1.0 - ((D_m / d) * (((M_m * (h * (D_m / d))) / l) * (M_m * 0.25)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e+159], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(t$95$0 / l), $MachinePrecision] * N[(t$95$0 / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 2e+190], N[(w0 * N[(D$95$m * N[Sqrt[N[(-0.25 * N[(M$95$m * N[(h * N[(M$95$m / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(N[(M$95$m * N[(h * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(M$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 5.00000000000000003e159

   1. Initial program 83.6%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. unpow2 N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 91.7

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

   4. Applied egg-rr 91.7%

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

   if 5.00000000000000003e159 < (*.f64 M D) < 2.0000000000000001e190

   1. Initial program 28.0%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. unpow2 N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 28.0

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

   4. Applied egg-rr 28.0%

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

   5. Taylor expanded in M around inf

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   6. 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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. /-lowering-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 28.0

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

   7. Simplified 28.0%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 26.4%

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

   if 2.0000000000000001e190 < (*.f64 M D)

   1. Initial program 75.5%

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 65.8%

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

      1. associate-*r* N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 71.4

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

   6. Applied egg-rr 71.4%

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

      1. associate-/r* N/A

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

      2. associate-*l/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 76.2

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

   8. Applied egg-rr 76.2%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 5 \cdot 10^{+159}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{-1}{h}}}\\ \mathbf{elif}\;M \cdot D \leq 2 \cdot 10^{+190}:\\ \;\;\;\;w0 \cdot \left(D \cdot \sqrt{-0.25 \cdot \left(M \cdot \left(h \cdot \frac{M}{\ell \cdot \left(d \cdot d\right)}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(\frac{M \cdot \left(h \cdot \frac{D}{d}\right)}{\ell} \cdot \left(M \cdot 0.25\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 4 \cdot 10^{-169}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m \cdot D\_m, \frac{\frac{-0.125 \cdot \frac{M\_m \cdot \left(M\_m \cdot h\right)}{\ell}}{d}}{d}, 1\right)\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 4 \cdot 10^{+150}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)}{d \cdot \ell} \cdot \frac{h}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M\_m \cdot 0.25\right) \cdot \left(\frac{D\_m}{d} \cdot \left(\frac{M\_m}{d \cdot \ell} \cdot \left(D\_m \cdot h\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 4e-169)
   (* w0 (fma (* D_m D_m) (/ (/ (* -0.125 (/ (* M_m (* M_m h)) l)) d) d) 1.0))
   (if (<= (* M_m D_m) 4e+150)
     (*
      w0
      (sqrt
       (fma -0.25 (* (/ (* (* M_m D_m) (* M_m D_m)) (* d l)) (/ h d)) 1.0)))
     (*
      w0
      (sqrt
       (-
        1.0
        (* (* M_m 0.25) (* (/ D_m d) (* (/ M_m (* d l)) (* D_m h))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 4e-169) {
		tmp = w0 * fma((D_m * D_m), (((-0.125 * ((M_m * (M_m * h)) / l)) / d) / d), 1.0);
	} else if ((M_m * D_m) <= 4e+150) {
		tmp = w0 * sqrt(fma(-0.25, ((((M_m * D_m) * (M_m * D_m)) / (d * l)) * (h / d)), 1.0));
	} else {
		tmp = w0 * sqrt((1.0 - ((M_m * 0.25) * ((D_m / d) * ((M_m / (d * l)) * (D_m * h))))));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 4e-169)
		tmp = Float64(w0 * fma(Float64(D_m * D_m), Float64(Float64(Float64(-0.125 * Float64(Float64(M_m * Float64(M_m * h)) / l)) / d) / d), 1.0));
	elseif (Float64(M_m * D_m) <= 4e+150)
		tmp = Float64(w0 * sqrt(fma(-0.25, Float64(Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) / Float64(d * l)) * Float64(h / d)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m * 0.25) * Float64(Float64(D_m / d) * Float64(Float64(M_m / Float64(d * l)) * Float64(D_m * h)))))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 4e-169], N[(w0 * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(-0.125 * N[(N[(M$95$m * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 4e+150], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m * 0.25), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(M$95$m / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 4.00000000000000008e-169

   1. Initial program 81.5%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 58.8

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

   5. Simplified 58.8%

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

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 49.9%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-/l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 63.3

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

   10. Applied egg-rr 63.3%

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

   if 4.00000000000000008e-169 < (*.f64 M D) < 3.99999999999999992e150

   1. Initial program 88.9%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 64.0

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

   5. Simplified 64.0%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 63.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. /-lowering-/.f64 95.4

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

   9. Applied egg-rr 95.4%

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

   if 3.99999999999999992e150 < (*.f64 M D)

   1. Initial program 70.1%

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 62.6%

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

      1. associate-*r* N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 66.9

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

   6. Applied egg-rr 66.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 77.8

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

   8. Applied egg-rr 77.8%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 4 \cdot 10^{-169}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-0.125 \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}}{d}}{d}, 1\right)\\ \mathbf{elif}\;M \cdot D \leq 4 \cdot 10^{+150}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d \cdot \ell} \cdot \frac{h}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot 0.25\right) \cdot \left(\frac{D}{d} \cdot \left(\frac{M}{d \cdot \ell} \cdot \left(D \cdot h\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 4 \cdot 10^{-169}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m \cdot D\_m, \frac{\frac{-0.125 \cdot \frac{M\_m \cdot \left(M\_m \cdot h\right)}{\ell}}{d}}{d}, 1\right)\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 4 \cdot 10^{+150}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)}{d \cdot \ell} \cdot \frac{h}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D\_m}{d} \cdot \left(\left(M\_m \cdot 0.25\right) \cdot \left(M\_m \cdot \frac{D\_m \cdot h}{d \cdot \ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 4e-169)
   (* w0 (fma (* D_m D_m) (/ (/ (* -0.125 (/ (* M_m (* M_m h)) l)) d) d) 1.0))
   (if (<= (* M_m D_m) 4e+150)
     (*
      w0
      (sqrt
       (fma -0.25 (* (/ (* (* M_m D_m) (* M_m D_m)) (* d l)) (/ h d)) 1.0)))
     (*
      w0
      (sqrt
       (-
        1.0
        (* (/ D_m d) (* (* M_m 0.25) (* M_m (/ (* D_m h) (* d l)))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 4e-169) {
		tmp = w0 * fma((D_m * D_m), (((-0.125 * ((M_m * (M_m * h)) / l)) / d) / d), 1.0);
	} else if ((M_m * D_m) <= 4e+150) {
		tmp = w0 * sqrt(fma(-0.25, ((((M_m * D_m) * (M_m * D_m)) / (d * l)) * (h / d)), 1.0));
	} else {
		tmp = w0 * sqrt((1.0 - ((D_m / d) * ((M_m * 0.25) * (M_m * ((D_m * h) / (d * l)))))));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 4e-169)
		tmp = Float64(w0 * fma(Float64(D_m * D_m), Float64(Float64(Float64(-0.125 * Float64(Float64(M_m * Float64(M_m * h)) / l)) / d) / d), 1.0));
	elseif (Float64(M_m * D_m) <= 4e+150)
		tmp = Float64(w0 * sqrt(fma(-0.25, Float64(Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) / Float64(d * l)) * Float64(h / d)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D_m / d) * Float64(Float64(M_m * 0.25) * Float64(M_m * Float64(Float64(D_m * h) / Float64(d * l))))))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 4e-169], N[(w0 * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(-0.125 * N[(N[(M$95$m * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 4e+150], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(M$95$m * 0.25), $MachinePrecision] * N[(M$95$m * N[(N[(D$95$m * h), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 4.00000000000000008e-169

   1. Initial program 81.5%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 58.8

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

   5. Simplified 58.8%

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

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 49.9%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-/l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 63.3

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

   10. Applied egg-rr 63.3%

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

   if 4.00000000000000008e-169 < (*.f64 M D) < 3.99999999999999992e150

   1. Initial program 88.9%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 64.0

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

   5. Simplified 64.0%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 63.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. /-lowering-/.f64 95.4

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

   9. Applied egg-rr 95.4%

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

   if 3.99999999999999992e150 < (*.f64 M D)

   1. Initial program 70.1%

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 62.6%

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

      1. associate-*r* N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 66.9

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

   6. Applied egg-rr 66.9%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 4 \cdot 10^{-169}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-0.125 \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}}{d}}{d}, 1\right)\\ \mathbf{elif}\;M \cdot D \leq 4 \cdot 10^{+150}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d \cdot \ell} \cdot \frac{h}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(\left(M \cdot 0.25\right) \cdot \left(M \cdot \frac{D \cdot h}{d \cdot \ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 4 \cdot 10^{-169}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m \cdot D\_m, \frac{\frac{-0.125 \cdot \frac{M\_m \cdot \left(M\_m \cdot h\right)}{\ell}}{d}}{d}, 1\right)\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 10^{+154}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)}{d \cdot \ell} \cdot \frac{h}{d}, 1\right)}\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 2 \cdot 10^{+190}:\\ \;\;\;\;w0 \cdot \left(D\_m \cdot \sqrt{-0.25 \cdot \left(M\_m \cdot \left(h \cdot \frac{M\_m}{\ell \cdot \left(d \cdot d\right)}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D\_m, w0 \cdot \left(\frac{h}{d} \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot \left(D\_m \cdot -0.125\right)}{d \cdot \ell}\right), w0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 4e-169)
   (* w0 (fma (* D_m D_m) (/ (/ (* -0.125 (/ (* M_m (* M_m h)) l)) d) d) 1.0))
   (if (<= (* M_m D_m) 1e+154)
     (*
      w0
      (sqrt
       (fma -0.25 (* (/ (* (* M_m D_m) (* M_m D_m)) (* d l)) (/ h d)) 1.0)))
     (if (<= (* M_m D_m) 2e+190)
       (* w0 (* D_m (sqrt (* -0.25 (* M_m (* h (/ M_m (* l (* d d)))))))))
       (fma
        D_m
        (* w0 (* (/ h d) (/ (* (* M_m M_m) (* D_m -0.125)) (* d l))))
        w0)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 4e-169) {
		tmp = w0 * fma((D_m * D_m), (((-0.125 * ((M_m * (M_m * h)) / l)) / d) / d), 1.0);
	} else if ((M_m * D_m) <= 1e+154) {
		tmp = w0 * sqrt(fma(-0.25, ((((M_m * D_m) * (M_m * D_m)) / (d * l)) * (h / d)), 1.0));
	} else if ((M_m * D_m) <= 2e+190) {
		tmp = w0 * (D_m * sqrt((-0.25 * (M_m * (h * (M_m / (l * (d * d))))))));
	} else {
		tmp = fma(D_m, (w0 * ((h / d) * (((M_m * M_m) * (D_m * -0.125)) / (d * l)))), w0);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 4e-169)
		tmp = Float64(w0 * fma(Float64(D_m * D_m), Float64(Float64(Float64(-0.125 * Float64(Float64(M_m * Float64(M_m * h)) / l)) / d) / d), 1.0));
	elseif (Float64(M_m * D_m) <= 1e+154)
		tmp = Float64(w0 * sqrt(fma(-0.25, Float64(Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) / Float64(d * l)) * Float64(h / d)), 1.0)));
	elseif (Float64(M_m * D_m) <= 2e+190)
		tmp = Float64(w0 * Float64(D_m * sqrt(Float64(-0.25 * Float64(M_m * Float64(h * Float64(M_m / Float64(l * Float64(d * d)))))))));
	else
		tmp = fma(D_m, Float64(w0 * Float64(Float64(h / d) * Float64(Float64(Float64(M_m * M_m) * Float64(D_m * -0.125)) / Float64(d * l)))), w0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 4e-169], N[(w0 * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(-0.125 * N[(N[(M$95$m * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e+154], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 2e+190], N[(w0 * N[(D$95$m * N[Sqrt[N[(-0.25 * N[(M$95$m * N[(h * N[(M$95$m / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(D$95$m * N[(w0 * N[(N[(h / d), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * -0.125), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 M D) < 4.00000000000000008e-169

   1. Initial program 81.5%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 58.8

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

   5. Simplified 58.8%

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

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 49.9%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-/l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 63.3

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

   10. Applied egg-rr 63.3%

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

   if 4.00000000000000008e-169 < (*.f64 M D) < 1.00000000000000004e154

   1. Initial program 89.2%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 62.3

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

   5. Simplified 62.3%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 63.2%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. /-lowering-/.f64 95.5

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

   9. Applied egg-rr 95.5%

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

   if 1.00000000000000004e154 < (*.f64 M D) < 2.0000000000000001e190

   1. Initial program 28.0%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. unpow2 N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 28.0

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

   4. Applied egg-rr 28.0%

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

   5. Taylor expanded in M around inf

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   6. 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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. /-lowering-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 28.0

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

   7. Simplified 28.0%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 26.4%

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

   if 2.0000000000000001e190 < (*.f64 M D)

   1. Initial program 75.5%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 46.0

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

   5. Simplified 46.0%

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

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 46.0%

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   10. Applied egg-rr 41.0%

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

       1. associate-*r/ N/A

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

       2. associate-*r* N/A

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

       3. times-frac N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-commutative N/A

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

       6. /-lowering-/.f64 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-lowering-*.f64 N/A

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

       12. *-lowering-*.f64 N/A

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

       13. /-lowering-/.f64 65.8

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

   12. Applied egg-rr 65.8%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 4 \cdot 10^{-169}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-0.125 \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}}{d}}{d}, 1\right)\\ \mathbf{elif}\;M \cdot D \leq 10^{+154}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d \cdot \ell} \cdot \frac{h}{d}, 1\right)}\\ \mathbf{elif}\;M \cdot D \leq 2 \cdot 10^{+190}:\\ \;\;\;\;w0 \cdot \left(D \cdot \sqrt{-0.25 \cdot \left(M \cdot \left(h \cdot \frac{M}{\ell \cdot \left(d \cdot d\right)}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D, w0 \cdot \left(\frac{h}{d} \cdot \frac{\left(M \cdot M\right) \cdot \left(D \cdot -0.125\right)}{d \cdot \ell}\right), w0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \ell \cdot \left(d \cdot d\right)\\ \mathbf{if}\;M\_m \cdot D\_m \leq 2 \cdot 10^{-147}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m \cdot D\_m, \frac{\frac{-0.125 \cdot \frac{M\_m \cdot \left(M\_m \cdot h\right)}{\ell}}{d}}{d}, 1\right)\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 10^{+154}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)\right) \cdot \frac{h}{t\_0}, 1\right)}\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 2 \cdot 10^{+190}:\\ \;\;\;\;w0 \cdot \left(D\_m \cdot \sqrt{-0.25 \cdot \left(M\_m \cdot \left(h \cdot \frac{M\_m}{t\_0}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D\_m, w0 \cdot \left(\frac{h}{d} \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot \left(D\_m \cdot -0.125\right)}{d \cdot \ell}\right), w0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* l (* d d))))
   (if (<= (* M_m D_m) 2e-147)
     (*
      w0
      (fma (* D_m D_m) (/ (/ (* -0.125 (/ (* M_m (* M_m h)) l)) d) d) 1.0))
     (if (<= (* M_m D_m) 1e+154)
       (* w0 (sqrt (fma -0.25 (* (* (* M_m D_m) (* M_m D_m)) (/ h t_0)) 1.0)))
       (if (<= (* M_m D_m) 2e+190)
         (* w0 (* D_m (sqrt (* -0.25 (* M_m (* h (/ M_m t_0)))))))
         (fma
          D_m
          (* w0 (* (/ h d) (/ (* (* M_m M_m) (* D_m -0.125)) (* d l))))
          w0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = l * (d * d);
	double tmp;
	if ((M_m * D_m) <= 2e-147) {
		tmp = w0 * fma((D_m * D_m), (((-0.125 * ((M_m * (M_m * h)) / l)) / d) / d), 1.0);
	} else if ((M_m * D_m) <= 1e+154) {
		tmp = w0 * sqrt(fma(-0.25, (((M_m * D_m) * (M_m * D_m)) * (h / t_0)), 1.0));
	} else if ((M_m * D_m) <= 2e+190) {
		tmp = w0 * (D_m * sqrt((-0.25 * (M_m * (h * (M_m / t_0))))));
	} else {
		tmp = fma(D_m, (w0 * ((h / d) * (((M_m * M_m) * (D_m * -0.125)) / (d * l)))), w0);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(l * Float64(d * d))
	tmp = 0.0
	if (Float64(M_m * D_m) <= 2e-147)
		tmp = Float64(w0 * fma(Float64(D_m * D_m), Float64(Float64(Float64(-0.125 * Float64(Float64(M_m * Float64(M_m * h)) / l)) / d) / d), 1.0));
	elseif (Float64(M_m * D_m) <= 1e+154)
		tmp = Float64(w0 * sqrt(fma(-0.25, Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) * Float64(h / t_0)), 1.0)));
	elseif (Float64(M_m * D_m) <= 2e+190)
		tmp = Float64(w0 * Float64(D_m * sqrt(Float64(-0.25 * Float64(M_m * Float64(h * Float64(M_m / t_0)))))));
	else
		tmp = fma(D_m, Float64(w0 * Float64(Float64(h / d) * Float64(Float64(Float64(M_m * M_m) * Float64(D_m * -0.125)) / Float64(d * l)))), w0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 2e-147], N[(w0 * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(-0.125 * N[(N[(M$95$m * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e+154], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(h / t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 2e+190], N[(w0 * N[(D$95$m * N[Sqrt[N[(-0.25 * N[(M$95$m * N[(h * N[(M$95$m / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(D$95$m * N[(w0 * N[(N[(h / d), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * -0.125), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 M D) < 1.9999999999999999e-147

   1. Initial program 81.3%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 59.2

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

   5. Simplified 59.2%

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

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 50.5%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-/l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 63.6

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

   10. Applied egg-rr 63.6%

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

   if 1.9999999999999999e-147 < (*.f64 M D) < 1.00000000000000004e154

   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 w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 61.4

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

   5. Simplified 61.4%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 62.5%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 87.4

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

   9. Applied egg-rr 87.4%

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

   if 1.00000000000000004e154 < (*.f64 M D) < 2.0000000000000001e190

   1. Initial program 28.0%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. unpow2 N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 28.0

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

   4. Applied egg-rr 28.0%

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

   5. Taylor expanded in M around inf

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   6. 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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. /-lowering-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 28.0

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

   7. Simplified 28.0%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 26.4%

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

   if 2.0000000000000001e190 < (*.f64 M D)

   1. Initial program 75.5%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 46.0

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

   5. Simplified 46.0%

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

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 46.0%

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   10. Applied egg-rr 41.0%

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

       1. associate-*r/ N/A

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

       2. associate-*r* N/A

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

       3. times-frac N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-commutative N/A

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

       6. /-lowering-/.f64 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-lowering-*.f64 N/A

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

       12. *-lowering-*.f64 N/A

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

       13. /-lowering-/.f64 65.8

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

   12. Applied egg-rr 65.8%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{-147}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-0.125 \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}}{d}}{d}, 1\right)\\ \mathbf{elif}\;M \cdot D \leq 10^{+154}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{\ell \cdot \left(d \cdot d\right)}, 1\right)}\\ \mathbf{elif}\;M \cdot D \leq 2 \cdot 10^{+190}:\\ \;\;\;\;w0 \cdot \left(D \cdot \sqrt{-0.25 \cdot \left(M \cdot \left(h \cdot \frac{M}{\ell \cdot \left(d \cdot d\right)}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D, w0 \cdot \left(\frac{h}{d} \cdot \frac{\left(M \cdot M\right) \cdot \left(D \cdot -0.125\right)}{d \cdot \ell}\right), w0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \ell \cdot \left(d \cdot d\right)\\ \mathbf{if}\;M\_m \cdot D\_m \leq 2 \cdot 10^{-147}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 10^{+154}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)\right) \cdot \frac{h}{t\_0}, 1\right)}\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 2 \cdot 10^{+190}:\\ \;\;\;\;w0 \cdot \left(D\_m \cdot \sqrt{-0.25 \cdot \left(M\_m \cdot \left(h \cdot \frac{M\_m}{t\_0}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D\_m, w0 \cdot \left(\frac{h}{d} \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot \left(D\_m \cdot -0.125\right)}{d \cdot \ell}\right), w0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* l (* d d))))
   (if (<= (* M_m D_m) 2e-147)
     w0
     (if (<= (* M_m D_m) 1e+154)
       (* w0 (sqrt (fma -0.25 (* (* (* M_m D_m) (* M_m D_m)) (/ h t_0)) 1.0)))
       (if (<= (* M_m D_m) 2e+190)
         (* w0 (* D_m (sqrt (* -0.25 (* M_m (* h (/ M_m t_0)))))))
         (fma
          D_m
          (* w0 (* (/ h d) (/ (* (* M_m M_m) (* D_m -0.125)) (* d l))))
          w0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = l * (d * d);
	double tmp;
	if ((M_m * D_m) <= 2e-147) {
		tmp = w0;
	} else if ((M_m * D_m) <= 1e+154) {
		tmp = w0 * sqrt(fma(-0.25, (((M_m * D_m) * (M_m * D_m)) * (h / t_0)), 1.0));
	} else if ((M_m * D_m) <= 2e+190) {
		tmp = w0 * (D_m * sqrt((-0.25 * (M_m * (h * (M_m / t_0))))));
	} else {
		tmp = fma(D_m, (w0 * ((h / d) * (((M_m * M_m) * (D_m * -0.125)) / (d * l)))), w0);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(l * Float64(d * d))
	tmp = 0.0
	if (Float64(M_m * D_m) <= 2e-147)
		tmp = w0;
	elseif (Float64(M_m * D_m) <= 1e+154)
		tmp = Float64(w0 * sqrt(fma(-0.25, Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) * Float64(h / t_0)), 1.0)));
	elseif (Float64(M_m * D_m) <= 2e+190)
		tmp = Float64(w0 * Float64(D_m * sqrt(Float64(-0.25 * Float64(M_m * Float64(h * Float64(M_m / t_0)))))));
	else
		tmp = fma(D_m, Float64(w0 * Float64(Float64(h / d) * Float64(Float64(Float64(M_m * M_m) * Float64(D_m * -0.125)) / Float64(d * l)))), w0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 2e-147], w0, If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e+154], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(h / t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 2e+190], N[(w0 * N[(D$95$m * N[Sqrt[N[(-0.25 * N[(M$95$m * N[(h * N[(M$95$m / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(D$95$m * N[(w0 * N[(N[(h / d), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * -0.125), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 M D) < 1.9999999999999999e-147

   1. Initial program 81.3%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 81.0%

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

   if 1.9999999999999999e-147 < (*.f64 M D) < 1.00000000000000004e154

   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 w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 61.4

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

   5. Simplified 61.4%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 62.5%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 87.4

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

   9. Applied egg-rr 87.4%

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

   if 1.00000000000000004e154 < (*.f64 M D) < 2.0000000000000001e190

   1. Initial program 28.0%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. unpow2 N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 28.0

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

   4. Applied egg-rr 28.0%

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

   5. Taylor expanded in M around inf

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   6. 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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. /-lowering-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 28.0

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

   7. Simplified 28.0%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 26.4%

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

   if 2.0000000000000001e190 < (*.f64 M D)

   1. Initial program 75.5%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 46.0

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

   5. Simplified 46.0%

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

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 46.0%

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   10. Applied egg-rr 41.0%

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

       1. associate-*r/ N/A

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

       2. associate-*r* N/A

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

       3. times-frac N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-commutative N/A

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

       6. /-lowering-/.f64 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-lowering-*.f64 N/A

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

       12. *-lowering-*.f64 N/A

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

       13. /-lowering-/.f64 65.8

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

   12. Applied egg-rr 65.8%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{-147}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \cdot D \leq 10^{+154}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h}{\ell \cdot \left(d \cdot d\right)}, 1\right)}\\ \mathbf{elif}\;M \cdot D \leq 2 \cdot 10^{+190}:\\ \;\;\;\;w0 \cdot \left(D \cdot \sqrt{-0.25 \cdot \left(M \cdot \left(h \cdot \frac{M}{\ell \cdot \left(d \cdot d\right)}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D, w0 \cdot \left(\frac{h}{d} \cdot \frac{\left(M \cdot M\right) \cdot \left(D \cdot -0.125\right)}{d \cdot \ell}\right), w0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 4 \cdot 10^{-169}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m \cdot D\_m, \frac{\frac{-0.125 \cdot \frac{M\_m \cdot \left(M\_m \cdot h\right)}{\ell}}{d}}{d}, 1\right)\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 4 \cdot 10^{+150}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)}{d \cdot \ell} \cdot \frac{h}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D\_m}{d}, -0.25 \cdot \left(M\_m \cdot \frac{M\_m \cdot \left(D\_m \cdot h\right)}{d \cdot \ell}\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 4e-169)
   (* w0 (fma (* D_m D_m) (/ (/ (* -0.125 (/ (* M_m (* M_m h)) l)) d) d) 1.0))
   (if (<= (* M_m D_m) 4e+150)
     (*
      w0
      (sqrt
       (fma -0.25 (* (/ (* (* M_m D_m) (* M_m D_m)) (* d l)) (/ h d)) 1.0)))
     (*
      w0
      (sqrt
       (fma (/ D_m d) (* -0.25 (* M_m (/ (* M_m (* D_m h)) (* d l)))) 1.0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 4e-169) {
		tmp = w0 * fma((D_m * D_m), (((-0.125 * ((M_m * (M_m * h)) / l)) / d) / d), 1.0);
	} else if ((M_m * D_m) <= 4e+150) {
		tmp = w0 * sqrt(fma(-0.25, ((((M_m * D_m) * (M_m * D_m)) / (d * l)) * (h / d)), 1.0));
	} else {
		tmp = w0 * sqrt(fma((D_m / d), (-0.25 * (M_m * ((M_m * (D_m * h)) / (d * l)))), 1.0));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 4e-169)
		tmp = Float64(w0 * fma(Float64(D_m * D_m), Float64(Float64(Float64(-0.125 * Float64(Float64(M_m * Float64(M_m * h)) / l)) / d) / d), 1.0));
	elseif (Float64(M_m * D_m) <= 4e+150)
		tmp = Float64(w0 * sqrt(fma(-0.25, Float64(Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) / Float64(d * l)) * Float64(h / d)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(D_m / d), Float64(-0.25 * Float64(M_m * Float64(Float64(M_m * Float64(D_m * h)) / Float64(d * l)))), 1.0)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 4e-169], N[(w0 * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(-0.125 * N[(N[(M$95$m * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 4e+150], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(D$95$m / d), $MachinePrecision] * N[(-0.25 * N[(M$95$m * N[(N[(M$95$m * N[(D$95$m * h), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 4.00000000000000008e-169

   1. Initial program 81.5%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 58.8

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

   5. Simplified 58.8%

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

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 49.9%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-/l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 63.3

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

   10. Applied egg-rr 63.3%

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

   if 4.00000000000000008e-169 < (*.f64 M D) < 3.99999999999999992e150

   1. Initial program 88.9%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 64.0

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

   5. Simplified 64.0%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 63.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. /-lowering-/.f64 95.4

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

   9. Applied egg-rr 95.4%

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

   if 3.99999999999999992e150 < (*.f64 M D)

   1. Initial program 70.1%

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 62.6%

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

      1. associate-*r* N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 66.9

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

   6. Applied egg-rr 66.9%

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

      1. associate-/r* N/A

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

      2. associate-*l/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 70.6

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

   8. Applied egg-rr 70.6%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   10. Applied egg-rr 70.5%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 4 \cdot 10^{-169}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-0.125 \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}}{d}}{d}, 1\right)\\ \mathbf{elif}\;M \cdot D \leq 4 \cdot 10^{+150}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d \cdot \ell} \cdot \frac{h}{d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D}{d}, -0.25 \cdot \left(M \cdot \frac{M \cdot \left(D \cdot h\right)}{d \cdot \ell}\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 5 \cdot 10^{+53}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M\_m \cdot D\_m}{\left(2 \cdot d\right) \cdot \ell}, \frac{\left(M\_m \cdot D\_m\right) \cdot h}{0 - 2 \cdot d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D\_m}{d} \cdot \left(\left(M\_m \cdot 0.25\right) \cdot \left(M\_m \cdot \left(\frac{D\_m}{d} \cdot \frac{h}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* M_m D_m) 5e+53)
   (*
    w0
    (sqrt
     (fma
      (/ (* M_m D_m) (* (* 2.0 d) l))
      (/ (* (* M_m D_m) h) (- 0.0 (* 2.0 d)))
      1.0)))
   (*
    w0
    (sqrt
     (- 1.0 (* (/ D_m d) (* (* M_m 0.25) (* M_m (* (/ D_m d) (/ h l))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((M_m * D_m) <= 5e+53) {
		tmp = w0 * sqrt(fma(((M_m * D_m) / ((2.0 * d) * l)), (((M_m * D_m) * h) / (0.0 - (2.0 * d))), 1.0));
	} else {
		tmp = w0 * sqrt((1.0 - ((D_m / d) * ((M_m * 0.25) * (M_m * ((D_m / d) * (h / l)))))));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 5e+53)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M_m * D_m) / Float64(Float64(2.0 * d) * l)), Float64(Float64(Float64(M_m * D_m) * h) / Float64(0.0 - Float64(2.0 * d))), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D_m / d) * Float64(Float64(M_m * 0.25) * Float64(M_m * Float64(Float64(D_m / d) * Float64(h / l))))))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e+53], N[(w0 * N[Sqrt[N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(N[(2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(0.0 - N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(M$95$m * 0.25), $MachinePrecision] * N[(M$95$m * N[(N[(D$95$m / d), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 5.0000000000000004e53

   1. Initial program 82.8%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. unpow2 N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{M \cdot D}}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{\color{blue}{2 \cdot d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}}} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d}}}{\frac{1}{h}}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{\color{blue}{M \cdot D}}{2 \cdot d}}{\frac{1}{h}}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{\color{blue}{2 \cdot d}}}{\frac{1}{h}}} \]

      15. /-lowering-/.f64 91.5

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\color{blue}{\frac{1}{h}}}} \]

   4. Applied egg-rr 91.5%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}}} \]
   5. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right) + 1}} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \left(\mathsf{neg}\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)} + 1} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}, \mathsf{neg}\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right), 1\right)}} \]

   6. Applied egg-rr 87.5%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{\left(2 \cdot d\right) \cdot \ell}, -\frac{\left(M \cdot D\right) \cdot h}{2 \cdot d}, 1\right)}} \]

   if 5.0000000000000004e53 < (*.f64 M D)

   1. Initial program 78.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}} \]

      3. times-frac N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \frac{D}{d}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right)\right) \cdot \frac{D}{d}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right)\right) \cdot \frac{D}{d}}} \]

   4. Applied egg-rr 62.1%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)\right) \cdot \frac{D}{d}}} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{D}{d}\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)} \cdot \frac{D}{d}} \]

      2. associate-*l* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{h}{\ell} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(M \cdot \left(M \cdot \frac{1}{4}\right)\right)}\right) \cdot \frac{D}{d}} \]

      3. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left(\left(\frac{h}{\ell} \cdot \frac{D}{d}\right) \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right)} \cdot \frac{D}{d}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left(\left(\frac{h}{\ell} \cdot \frac{D}{d}\right) \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right)} \cdot \frac{D}{d}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{D}{d}\right) \cdot M\right)} \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      6. frac-times N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\color{blue}{\frac{h \cdot D}{\ell \cdot d}} \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      7. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{h \cdot D}{\color{blue}{d \cdot \ell}} \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\color{blue}{\frac{h \cdot D}{d \cdot \ell}} \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{\color{blue}{h \cdot D}}{d \cdot \ell} \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{h \cdot D}{\color{blue}{d \cdot \ell}} \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      11. *-lowering-*.f64 72.2

          \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{h \cdot D}{d \cdot \ell} \cdot M\right) \cdot \color{blue}{\left(M \cdot 0.25\right)}\right) \cdot \frac{D}{d}} \]

   6. Applied egg-rr 72.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left(\frac{h \cdot D}{d \cdot \ell} \cdot M\right) \cdot \left(M \cdot 0.25\right)\right)} \cdot \frac{D}{d}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{h \cdot D}{\color{blue}{\ell \cdot d}} \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      2. times-frac N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\color{blue}{\left(\frac{h}{\ell} \cdot \frac{D}{d}\right)} \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\color{blue}{\left(\frac{h}{\ell} \cdot \frac{D}{d}\right)} \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\left(\color{blue}{\frac{h}{\ell}} \cdot \frac{D}{d}\right) \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      5. /-lowering-/.f64 72.0

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\left(\frac{h}{\ell} \cdot \color{blue}{\frac{D}{d}}\right) \cdot M\right) \cdot \left(M \cdot 0.25\right)\right) \cdot \frac{D}{d}} \]

   8. Applied egg-rr 72.0%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\color{blue}{\left(\frac{h}{\ell} \cdot \frac{D}{d}\right)} \cdot M\right) \cdot \left(M \cdot 0.25\right)\right) \cdot \frac{D}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 5 \cdot 10^{+53}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{\left(2 \cdot d\right) \cdot \ell}, \frac{\left(M \cdot D\right) \cdot h}{0 - 2 \cdot d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(\left(M \cdot 0.25\right) \cdot \left(M \cdot \left(\frac{D}{d} \cdot \frac{h}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 5 \cdot 10^{-165}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M\_m \cdot D\_m}{\left(2 \cdot d\right) \cdot \ell}, \frac{\left(M\_m \cdot D\_m\right) \cdot h}{0 - 2 \cdot d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D\_m}{d} \cdot \left(\frac{M\_m \cdot \left(h \cdot \frac{D\_m}{d}\right)}{\ell} \cdot \left(M\_m \cdot 0.25\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 5e-165)
   (*
    w0
    (sqrt
     (fma
      (/ (* M_m D_m) (* (* 2.0 d) l))
      (/ (* (* M_m D_m) h) (- 0.0 (* 2.0 d)))
      1.0)))
   (*
    w0
    (sqrt
     (- 1.0 (* (/ D_m d) (* (/ (* M_m (* h (/ D_m d))) l) (* M_m 0.25))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 5e-165) {
		tmp = w0 * sqrt(fma(((M_m * D_m) / ((2.0 * d) * l)), (((M_m * D_m) * h) / (0.0 - (2.0 * d))), 1.0));
	} else {
		tmp = w0 * sqrt((1.0 - ((D_m / d) * (((M_m * (h * (D_m / d))) / l) * (M_m * 0.25)))));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 5e-165)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M_m * D_m) / Float64(Float64(2.0 * d) * l)), Float64(Float64(Float64(M_m * D_m) * h) / Float64(0.0 - Float64(2.0 * d))), 1.0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D_m / d) * Float64(Float64(Float64(M_m * Float64(h * Float64(D_m / d))) / l) * Float64(M_m * 0.25))))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 5e-165], N[(w0 * N[Sqrt[N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(N[(2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(0.0 - N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(N[(M$95$m * N[(h * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(M$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.99999999999999981e-165

   1. Initial program 84.0%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}}} \]

      2. un-div-inv N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}}} \]

      3. unpow2 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}}{\frac{\ell}{h}}} \]

      4. div-inv N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\color{blue}{\ell \cdot \frac{1}{h}}}} \]

      5. times-frac N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{\color{blue}{M \cdot D}}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{\color{blue}{2 \cdot d}}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}}} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d}}}{\frac{1}{h}}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{\color{blue}{M \cdot D}}{2 \cdot d}}{\frac{1}{h}}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{\color{blue}{2 \cdot d}}}{\frac{1}{h}}} \]

      15. /-lowering-/.f64 92.1

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\color{blue}{\frac{1}{h}}}} \]

   4. Applied egg-rr 92.1%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}}} \]
   5. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right) + 1}} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \left(\mathsf{neg}\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)} + 1} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}, \mathsf{neg}\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right), 1\right)}} \]

   6. Applied egg-rr 88.2%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{\left(2 \cdot d\right) \cdot \ell}, -\frac{\left(M \cdot D\right) \cdot h}{2 \cdot d}, 1\right)}} \]

   if 4.99999999999999981e-165 < M

   1. Initial program 78.2%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}} \]

      3. times-frac N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right) \cdot \frac{D}{d}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right)\right) \cdot \frac{D}{d}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M}{2}\right)\right) \cdot \frac{D}{d}}} \]

   4. Applied egg-rr 67.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)\right) \cdot \frac{D}{d}}} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{D}{d}\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)} \cdot \frac{D}{d}} \]

      2. associate-*l* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{h}{\ell} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(M \cdot \left(M \cdot \frac{1}{4}\right)\right)}\right) \cdot \frac{D}{d}} \]

      3. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left(\left(\frac{h}{\ell} \cdot \frac{D}{d}\right) \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right)} \cdot \frac{D}{d}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left(\left(\frac{h}{\ell} \cdot \frac{D}{d}\right) \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right)} \cdot \frac{D}{d}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{D}{d}\right) \cdot M\right)} \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      6. frac-times N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\color{blue}{\frac{h \cdot D}{\ell \cdot d}} \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      7. *-commutative N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{h \cdot D}{\color{blue}{d \cdot \ell}} \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\color{blue}{\frac{h \cdot D}{d \cdot \ell}} \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{\color{blue}{h \cdot D}}{d \cdot \ell} \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{h \cdot D}{\color{blue}{d \cdot \ell}} \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      11. *-lowering-*.f64 85.4

          \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{h \cdot D}{d \cdot \ell} \cdot M\right) \cdot \color{blue}{\left(M \cdot 0.25\right)}\right) \cdot \frac{D}{d}} \]

   6. Applied egg-rr 85.4%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left(\frac{h \cdot D}{d \cdot \ell} \cdot M\right) \cdot \left(M \cdot 0.25\right)\right)} \cdot \frac{D}{d}} \]
   7. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\color{blue}{\frac{\frac{h \cdot D}{d}}{\ell}} \cdot M\right) \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      2. associate-*l/ N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\frac{\frac{h \cdot D}{d} \cdot M}{\ell}} \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\frac{\frac{h \cdot D}{d} \cdot M}{\ell}} \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\color{blue}{\frac{h \cdot D}{d} \cdot M}}{\ell} \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      5. associate-/l* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\color{blue}{\left(h \cdot \frac{D}{d}\right)} \cdot M}{\ell} \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\color{blue}{\left(h \cdot \frac{D}{d}\right)} \cdot M}{\ell} \cdot \left(M \cdot \frac{1}{4}\right)\right) \cdot \frac{D}{d}} \]

      7. /-lowering-/.f64 89.9

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(h \cdot \color{blue}{\frac{D}{d}}\right) \cdot M}{\ell} \cdot \left(M \cdot 0.25\right)\right) \cdot \frac{D}{d}} \]

   8. Applied egg-rr 89.9%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\frac{\left(h \cdot \frac{D}{d}\right) \cdot M}{\ell}} \cdot \left(M \cdot 0.25\right)\right) \cdot \frac{D}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{-165}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{\left(2 \cdot d\right) \cdot \ell}, \frac{\left(M \cdot D\right) \cdot h}{0 - 2 \cdot d}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(\frac{M \cdot \left(h \cdot \frac{D}{d}\right)}{\ell} \cdot \left(M \cdot 0.25\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 2 \cdot 10^{+44}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m \cdot D\_m, \frac{-0.125}{d} \cdot \frac{M\_m \cdot \left(M\_m \cdot h\right)}{d \cdot \ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D\_m, w0 \cdot \left(\left(M\_m \cdot \left(h \cdot \frac{M\_m}{\ell \cdot \left(d \cdot d\right)}\right)\right) \cdot \left(D\_m \cdot -0.125\right)\right), w0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= D_m 2e+44)
   (* w0 (fma (* D_m D_m) (* (/ -0.125 d) (/ (* M_m (* M_m h)) (* d l))) 1.0))
   (fma D_m (* w0 (* (* M_m (* h (/ M_m (* l (* d d))))) (* D_m -0.125))) w0)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (D_m <= 2e+44) {
		tmp = w0 * fma((D_m * D_m), ((-0.125 / d) * ((M_m * (M_m * h)) / (d * l))), 1.0);
	} else {
		tmp = fma(D_m, (w0 * ((M_m * (h * (M_m / (l * (d * d))))) * (D_m * -0.125))), w0);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (D_m <= 2e+44)
		tmp = Float64(w0 * fma(Float64(D_m * D_m), Float64(Float64(-0.125 / d) * Float64(Float64(M_m * Float64(M_m * h)) / Float64(d * l))), 1.0));
	else
		tmp = fma(D_m, Float64(w0 * Float64(Float64(M_m * Float64(h * Float64(M_m / Float64(l * Float64(d * d))))) * Float64(D_m * -0.125))), w0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[D$95$m, 2e+44], N[(w0 * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(-0.125 / d), $MachinePrecision] * N[(N[(M$95$m * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(D$95$m * N[(w0 * N[(N[(M$95$m * N[(h * N[(M$95$m / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.0000000000000002e44

   1. Initial program 83.0%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in w0 around 0

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto w0 \cdot \color{blue}{\sqrt{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]

      3. --lowering--.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]

      4. associate-*r/ N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}} \]

      7. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}} \]

      9. unpow2 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left({D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      10. associate-*r* N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(\left({D}^{2} \cdot M\right) \cdot M\right)} \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(\left({D}^{2} \cdot M\right) \cdot M\right)} \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\color{blue}{\left({D}^{2} \cdot M\right)} \cdot M\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      13. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\color{blue}{\left(D \cdot D\right)} \cdot M\right) \cdot M\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\color{blue}{\left(D \cdot D\right)} \cdot M\right) \cdot M\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}} \]

      16. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

      17. *-lowering-*.f64 62.8

          \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

   5. Simplified 62.8%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell}}} \]

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

         \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right)} \]

   8. Simplified 52.5%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(D \cdot D, \frac{\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot h}{\ell \cdot \left(d \cdot d\right)}, 1\right)} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\color{blue}{\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)}}{\ell \cdot \left(d \cdot d\right)}, 1\right) \]

      2. *-commutative N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}, 1\right) \]

      3. associate-*r* N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}, 1\right) \]

      4. times-frac N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\frac{-1}{8}}{d} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot \ell}}, 1\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\frac{-1}{8}}{d} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot \ell}}, 1\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\frac{-1}{8}}{d}} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot \ell}, 1\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8}}{d} \cdot \color{blue}{\frac{\left(M \cdot M\right) \cdot h}{d \cdot \ell}}, 1\right) \]

      8. associate-*l* N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8}}{d} \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{d \cdot \ell}, 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8}}{d} \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{d \cdot \ell}, 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8}}{d} \cdot \frac{M \cdot \color{blue}{\left(M \cdot h\right)}}{d \cdot \ell}, 1\right) \]

      11. *-lowering-*.f64 66.2

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{-0.125}{d} \cdot \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot \ell}}, 1\right) \]

   10. Applied egg-rr 66.2%

       \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{-0.125}{d} \cdot \frac{M \cdot \left(M \cdot h\right)}{d \cdot \ell}}, 1\right) \]

   if 2.0000000000000002e44 < D

   1. Initial program 78.7%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in w0 around 0

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto w0 \cdot \color{blue}{\sqrt{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]

      3. --lowering--.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]

      4. associate-*r/ N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}} \]

      7. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}} \]

      9. unpow2 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left({D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      10. associate-*r* N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(\left({D}^{2} \cdot M\right) \cdot M\right)} \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(\left({D}^{2} \cdot M\right) \cdot M\right)} \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\color{blue}{\left({D}^{2} \cdot M\right)} \cdot M\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      13. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\color{blue}{\left(D \cdot D\right)} \cdot M\right) \cdot M\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\color{blue}{\left(D \cdot D\right)} \cdot M\right) \cdot M\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}} \]

      16. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

      17. *-lowering-*.f64 39.2

          \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

   5. Simplified 39.2%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell}}} \]

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

         \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right)} \]

   8. Simplified 32.9%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(D \cdot D, \frac{\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot h}{\ell \cdot \left(d \cdot d\right)}, 1\right)} \]
   9. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\ell \cdot \left(d \cdot d\right)}\right) \cdot w0 + 1 \cdot w0} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{\left(D \cdot \left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\ell \cdot \left(d \cdot d\right)}\right)\right)} \cdot w0 + 1 \cdot w0 \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{D \cdot \left(\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\ell \cdot \left(d \cdot d\right)}\right) \cdot w0\right)} + 1 \cdot w0 \]

      4. *-lft-identity N/A

         \[\leadsto D \cdot \left(\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\ell \cdot \left(d \cdot d\right)}\right) \cdot w0\right) + \color{blue}{w0} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(D, \left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\ell \cdot \left(d \cdot d\right)}\right) \cdot w0, w0\right)} \]

   10. Applied egg-rr 57.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(D, \left(\left(D \cdot \left(-0.125 \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{h}{\ell \cdot \left(d \cdot d\right)}\right) \cdot w0, w0\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(D, \left(\color{blue}{\left(\left(D \cdot \frac{-1}{8}\right) \cdot \left(M \cdot M\right)\right)} \cdot \frac{h}{\ell \cdot \left(d \cdot d\right)}\right) \cdot w0, w0\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(D, \color{blue}{\left(\left(D \cdot \frac{-1}{8}\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{\ell \cdot \left(d \cdot d\right)}\right)\right)} \cdot w0, w0\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot \frac{-1}{8}\right) \cdot \color{blue}{\frac{\left(M \cdot M\right) \cdot h}{\ell \cdot \left(d \cdot d\right)}}\right) \cdot w0, w0\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot \frac{-1}{8}\right) \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{\ell \cdot \left(d \cdot d\right)}\right) \cdot w0, w0\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(D, \color{blue}{\left(\left(D \cdot \frac{-1}{8}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell \cdot \left(d \cdot d\right)}\right)} \cdot w0, w0\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(D, \left(\color{blue}{\left(D \cdot \frac{-1}{8}\right)} \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell \cdot \left(d \cdot d\right)}\right) \cdot w0, w0\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(M \cdot \frac{M \cdot h}{\ell \cdot \left(d \cdot d\right)}\right)}\right) \cdot w0, w0\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(M \cdot \frac{M \cdot h}{\ell \cdot \left(d \cdot d\right)}\right)}\right) \cdot w0, w0\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot \frac{-1}{8}\right) \cdot \left(M \cdot \frac{\color{blue}{h \cdot M}}{\ell \cdot \left(d \cdot d\right)}\right)\right) \cdot w0, w0\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot \frac{-1}{8}\right) \cdot \left(M \cdot \color{blue}{\left(h \cdot \frac{M}{\ell \cdot \left(d \cdot d\right)}\right)}\right)\right) \cdot w0, w0\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot \frac{-1}{8}\right) \cdot \left(M \cdot \color{blue}{\left(h \cdot \frac{M}{\ell \cdot \left(d \cdot d\right)}\right)}\right)\right) \cdot w0, w0\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot \frac{-1}{8}\right) \cdot \left(M \cdot \left(h \cdot \color{blue}{\frac{M}{\ell \cdot \left(d \cdot d\right)}}\right)\right)\right) \cdot w0, w0\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot \frac{-1}{8}\right) \cdot \left(M \cdot \left(h \cdot \frac{M}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)\right)\right) \cdot w0, w0\right) \]

       14. *-lowering-*.f64 68.1

           \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot -0.125\right) \cdot \left(M \cdot \left(h \cdot \frac{M}{\ell \cdot \color{blue}{\left(d \cdot d\right)}}\right)\right)\right) \cdot w0, w0\right) \]

   12. Applied egg-rr 68.1%

       \[\leadsto \mathsf{fma}\left(D, \color{blue}{\left(\left(D \cdot -0.125\right) \cdot \left(M \cdot \left(h \cdot \frac{M}{\ell \cdot \left(d \cdot d\right)}\right)\right)\right)} \cdot w0, w0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2 \cdot 10^{+44}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{-0.125}{d} \cdot \frac{M \cdot \left(M \cdot h\right)}{d \cdot \ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D, w0 \cdot \left(\left(M \cdot \left(h \cdot \frac{M}{\ell \cdot \left(d \cdot d\right)}\right)\right) \cdot \left(D \cdot -0.125\right)\right), w0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 5.2 \cdot 10^{+59}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D\_m, w0 \cdot \left(\left(M\_m \cdot \left(h \cdot \frac{M\_m}{\ell \cdot \left(d \cdot d\right)}\right)\right) \cdot \left(D\_m \cdot -0.125\right)\right), w0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= D_m 5.2e+59)
   w0
   (fma D_m (* w0 (* (* M_m (* h (/ M_m (* l (* d d))))) (* D_m -0.125))) w0)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (D_m <= 5.2e+59) {
		tmp = w0;
	} else {
		tmp = fma(D_m, (w0 * ((M_m * (h * (M_m / (l * (d * d))))) * (D_m * -0.125))), w0);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (D_m <= 5.2e+59)
		tmp = w0;
	else
		tmp = fma(D_m, Float64(w0 * Float64(Float64(M_m * Float64(h * Float64(M_m / Float64(l * Float64(d * d))))) * Float64(D_m * -0.125))), w0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[D$95$m, 5.2e+59], w0, N[(D$95$m * N[(w0 * N[(N[(M$95$m * N[(h * N[(M$95$m / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 5.19999999999999999e59

   1. Initial program 83.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   4. Step-by-step derivation

   5. Simplified 75.8%

      \[\leadsto \color{blue}{w0} \]

   if 5.19999999999999999e59 < D

   1. Initial program 77.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in w0 around 0

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto w0 \cdot \color{blue}{\sqrt{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]

      3. --lowering--.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]

      4. associate-*r/ N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}} \]

      7. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}} \]

      9. unpow2 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left({D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      10. associate-*r* N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(\left({D}^{2} \cdot M\right) \cdot M\right)} \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(\left({D}^{2} \cdot M\right) \cdot M\right)} \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\color{blue}{\left({D}^{2} \cdot M\right)} \cdot M\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      13. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\color{blue}{\left(D \cdot D\right)} \cdot M\right) \cdot M\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\color{blue}{\left(D \cdot D\right)} \cdot M\right) \cdot M\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}} \]

      16. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

      17. *-lowering-*.f64 38.8

          \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

   5. Simplified 38.8%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell}}} \]

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

         \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right)} \]

   8. Simplified 32.2%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(D \cdot D, \frac{\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot h}{\ell \cdot \left(d \cdot d\right)}, 1\right)} \]
   9. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(D \cdot D\right) \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\ell \cdot \left(d \cdot d\right)}\right) \cdot w0 + 1 \cdot w0} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{\left(D \cdot \left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\ell \cdot \left(d \cdot d\right)}\right)\right)} \cdot w0 + 1 \cdot w0 \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{D \cdot \left(\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\ell \cdot \left(d \cdot d\right)}\right) \cdot w0\right)} + 1 \cdot w0 \]

      4. *-lft-identity N/A

         \[\leadsto D \cdot \left(\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\ell \cdot \left(d \cdot d\right)}\right) \cdot w0\right) + \color{blue}{w0} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(D, \left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\ell \cdot \left(d \cdot d\right)}\right) \cdot w0, w0\right)} \]

   10. Applied egg-rr 57.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(D, \left(\left(D \cdot \left(-0.125 \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{h}{\ell \cdot \left(d \cdot d\right)}\right) \cdot w0, w0\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(D, \left(\color{blue}{\left(\left(D \cdot \frac{-1}{8}\right) \cdot \left(M \cdot M\right)\right)} \cdot \frac{h}{\ell \cdot \left(d \cdot d\right)}\right) \cdot w0, w0\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(D, \color{blue}{\left(\left(D \cdot \frac{-1}{8}\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{\ell \cdot \left(d \cdot d\right)}\right)\right)} \cdot w0, w0\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot \frac{-1}{8}\right) \cdot \color{blue}{\frac{\left(M \cdot M\right) \cdot h}{\ell \cdot \left(d \cdot d\right)}}\right) \cdot w0, w0\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot \frac{-1}{8}\right) \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{\ell \cdot \left(d \cdot d\right)}\right) \cdot w0, w0\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(D, \color{blue}{\left(\left(D \cdot \frac{-1}{8}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell \cdot \left(d \cdot d\right)}\right)} \cdot w0, w0\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(D, \left(\color{blue}{\left(D \cdot \frac{-1}{8}\right)} \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell \cdot \left(d \cdot d\right)}\right) \cdot w0, w0\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(M \cdot \frac{M \cdot h}{\ell \cdot \left(d \cdot d\right)}\right)}\right) \cdot w0, w0\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(M \cdot \frac{M \cdot h}{\ell \cdot \left(d \cdot d\right)}\right)}\right) \cdot w0, w0\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot \frac{-1}{8}\right) \cdot \left(M \cdot \frac{\color{blue}{h \cdot M}}{\ell \cdot \left(d \cdot d\right)}\right)\right) \cdot w0, w0\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot \frac{-1}{8}\right) \cdot \left(M \cdot \color{blue}{\left(h \cdot \frac{M}{\ell \cdot \left(d \cdot d\right)}\right)}\right)\right) \cdot w0, w0\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot \frac{-1}{8}\right) \cdot \left(M \cdot \color{blue}{\left(h \cdot \frac{M}{\ell \cdot \left(d \cdot d\right)}\right)}\right)\right) \cdot w0, w0\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot \frac{-1}{8}\right) \cdot \left(M \cdot \left(h \cdot \color{blue}{\frac{M}{\ell \cdot \left(d \cdot d\right)}}\right)\right)\right) \cdot w0, w0\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot \frac{-1}{8}\right) \cdot \left(M \cdot \left(h \cdot \frac{M}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)\right)\right) \cdot w0, w0\right) \]

       14. *-lowering-*.f64 68.5

           \[\leadsto \mathsf{fma}\left(D, \left(\left(D \cdot -0.125\right) \cdot \left(M \cdot \left(h \cdot \frac{M}{\ell \cdot \color{blue}{\left(d \cdot d\right)}}\right)\right)\right) \cdot w0, w0\right) \]

   12. Applied egg-rr 68.5%

       \[\leadsto \mathsf{fma}\left(D, \color{blue}{\left(\left(D \cdot -0.125\right) \cdot \left(M \cdot \left(h \cdot \frac{M}{\ell \cdot \left(d \cdot d\right)}\right)\right)\right)} \cdot w0, w0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 5.2 \cdot 10^{+59}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D, w0 \cdot \left(\left(M \cdot \left(h \cdot \frac{M}{\ell \cdot \left(d \cdot d\right)}\right)\right) \cdot \left(D \cdot -0.125\right)\right), w0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 7.3 \cdot 10^{+74}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \frac{\left(D\_m \cdot D\_m\right) \cdot \left(-0.125 \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 7.3e+74)
   w0
   (* w0 (/ (* (* D_m D_m) (* -0.125 (* h (* M_m M_m)))) (* d (* d l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 7.3e+74) {
		tmp = w0;
	} else {
		tmp = w0 * (((D_m * D_m) * (-0.125 * (h * (M_m * M_m)))) / (d * (d * l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    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
    real(8) :: tmp
    if (m_m <= 7.3d+74) then
        tmp = w0
    else
        tmp = w0 * (((d_m * d_m) * ((-0.125d0) * (h * (m_m * m_m)))) / (d * (d * l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 7.3e+74) {
		tmp = w0;
	} else {
		tmp = w0 * (((D_m * D_m) * (-0.125 * (h * (M_m * M_m)))) / (d * (d * l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 7.3e+74:
		tmp = w0
	else:
		tmp = w0 * (((D_m * D_m) * (-0.125 * (h * (M_m * M_m)))) / (d * (d * l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 7.3e+74)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(Float64(Float64(D_m * D_m) * Float64(-0.125 * Float64(h * Float64(M_m * M_m)))) / Float64(d * Float64(d * l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 7.3e+74)
		tmp = w0;
	else
		tmp = w0 * (((D_m * D_m) * (-0.125 * (h * (M_m * M_m)))) / (d * (d * l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 7.3e+74], w0, N[(w0 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(-0.125 * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.3000000000000005e74

   1. Initial program 84.7%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   4. Step-by-step derivation

   5. Simplified 72.9%

      \[\leadsto \color{blue}{w0} \]

   if 7.3000000000000005e74 < M

   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 w0 around 0

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto w0 \cdot \color{blue}{\sqrt{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]

      3. --lowering--.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]

      4. associate-*r/ N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}} \]

      7. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}} \]

      9. unpow2 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left({D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      10. associate-*r* N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(\left({D}^{2} \cdot M\right) \cdot M\right)} \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(\left({D}^{2} \cdot M\right) \cdot M\right)} \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\color{blue}{\left({D}^{2} \cdot M\right)} \cdot M\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      13. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\color{blue}{\left(D \cdot D\right)} \cdot M\right) \cdot M\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\color{blue}{\left(D \cdot D\right)} \cdot M\right) \cdot M\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}} \]

      16. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

      17. *-lowering-*.f64 38.7

          \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

   5. Simplified 38.7%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell}}} \]

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

         \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right)} \]

   8. Simplified 16.8%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(D \cdot D, \frac{\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot h}{\ell \cdot \left(d \cdot d\right)}, 1\right)} \]

   9. Taylor expanded in D around inf

      \[\leadsto w0 \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto w0 \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto w0 \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} \]

       3. *-commutative N/A

          \[\leadsto w0 \cdot \frac{\frac{-1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell} \]

       4. associate-*r* N/A

          \[\leadsto w0 \cdot \frac{\color{blue}{\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot {D}^{2}}}{{d}^{2} \cdot \ell} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \frac{\color{blue}{\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot {D}^{2}}}{{d}^{2} \cdot \ell} \]

       6. *-commutative N/A

          \[\leadsto w0 \cdot \frac{\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot \frac{-1}{8}\right)} \cdot {D}^{2}}{{d}^{2} \cdot \ell} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \frac{\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot \frac{-1}{8}\right)} \cdot {D}^{2}}{{d}^{2} \cdot \ell} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \frac{\left(\color{blue}{\left({M}^{2} \cdot h\right)} \cdot \frac{-1}{8}\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell} \]

       9. unpow2 N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot \frac{-1}{8}\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto w0 \cdot \frac{\left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot \frac{-1}{8}\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell} \]

       11. unpow2 N/A

           \[\leadsto w0 \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2} \cdot \ell} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto w0 \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2} \cdot \ell} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto w0 \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot \left(D \cdot D\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

       14. unpow2 N/A

           \[\leadsto w0 \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell} \]

       15. *-lowering-*.f64 13.7

           \[\leadsto w0 \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot -0.125\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell} \]

   11. Simplified 13.7%

       \[\leadsto w0 \cdot \color{blue}{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot -0.125\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}} \]
   12. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot \left(D \cdot D\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}} \]

       2. *-commutative N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot \ell\right) \cdot d}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot \ell\right) \cdot d}} \]

       4. *-lowering-*.f64 16.8

          \[\leadsto w0 \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot -0.125\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot \ell\right)} \cdot d} \]

   13. Applied egg-rr 16.8%

       \[\leadsto w0 \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot -0.125\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot \ell\right) \cdot d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.3 \cdot 10^{+74}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \frac{\left(D \cdot D\right) \cdot \left(-0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 5 \cdot 10^{+74}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\left(\left(M\_m \cdot M\_m\right) \cdot \left(D\_m \cdot \left(h \cdot -0.125\right)\right)\right) \cdot \frac{D\_m}{\ell \cdot \left(d \cdot d\right)}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 5e+74)
   w0
   (* w0 (* (* (* M_m M_m) (* D_m (* h -0.125))) (/ D_m (* l (* d d)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 5e+74) {
		tmp = w0;
	} else {
		tmp = w0 * (((M_m * M_m) * (D_m * (h * -0.125))) * (D_m / (l * (d * d))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    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
    real(8) :: tmp
    if (m_m <= 5d+74) then
        tmp = w0
    else
        tmp = w0 * (((m_m * m_m) * (d_m * (h * (-0.125d0)))) * (d_m / (l * (d * d))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 5e+74) {
		tmp = w0;
	} else {
		tmp = w0 * (((M_m * M_m) * (D_m * (h * -0.125))) * (D_m / (l * (d * d))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 5e+74:
		tmp = w0
	else:
		tmp = w0 * (((M_m * M_m) * (D_m * (h * -0.125))) * (D_m / (l * (d * d))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 5e+74)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(Float64(Float64(M_m * M_m) * Float64(D_m * Float64(h * -0.125))) * Float64(D_m / Float64(l * Float64(d * d)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 5e+74)
		tmp = w0;
	else
		tmp = w0 * (((M_m * M_m) * (D_m * (h * -0.125))) * (D_m / (l * (d * d))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 5e+74], w0, N[(w0 * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * N[(h * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.99999999999999963e74

   1. Initial program 84.7%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in M around 0

      \[\leadsto \color{blue}{w0} \]
   4. Step-by-step derivation

   5. Simplified 72.9%

      \[\leadsto \color{blue}{w0} \]

   if 4.99999999999999963e74 < M

   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 w0 around 0

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto w0 \cdot \color{blue}{\sqrt{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]

      3. --lowering--.f64 N/A

         \[\leadsto w0 \cdot \sqrt{\color{blue}{1 - \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}} \]

      4. associate-*r/ N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}}{{d}^{2} \cdot \ell}} \]

      7. associate-*r* N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}} \]

      9. unpow2 N/A

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left({D}^{2} \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      10. associate-*r* N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(\left({D}^{2} \cdot M\right) \cdot M\right)} \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\color{blue}{\left(\left({D}^{2} \cdot M\right) \cdot M\right)} \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\color{blue}{\left({D}^{2} \cdot M\right)} \cdot M\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      13. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\color{blue}{\left(D \cdot D\right)} \cdot M\right) \cdot M\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\color{blue}{\left(D \cdot D\right)} \cdot M\right) \cdot M\right) \cdot h\right)}{{d}^{2} \cdot \ell}} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}} \]

      16. unpow2 N/A

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{1}{4} \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

      17. *-lowering-*.f64 38.7

          \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]

   5. Simplified 38.7%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell}}} \]

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

         \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right)} \]

   8. Simplified 16.8%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(D \cdot D, \frac{\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot h}{\ell \cdot \left(d \cdot d\right)}, 1\right)} \]

   9. Taylor expanded in D around inf

      \[\leadsto w0 \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto w0 \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto w0 \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} \]

       3. *-commutative N/A

          \[\leadsto w0 \cdot \frac{\frac{-1}{8} \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell} \]

       4. associate-*r* N/A

          \[\leadsto w0 \cdot \frac{\color{blue}{\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot {D}^{2}}}{{d}^{2} \cdot \ell} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \frac{\color{blue}{\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot {D}^{2}}}{{d}^{2} \cdot \ell} \]

       6. *-commutative N/A

          \[\leadsto w0 \cdot \frac{\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot \frac{-1}{8}\right)} \cdot {D}^{2}}{{d}^{2} \cdot \ell} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \frac{\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot \frac{-1}{8}\right)} \cdot {D}^{2}}{{d}^{2} \cdot \ell} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \frac{\left(\color{blue}{\left({M}^{2} \cdot h\right)} \cdot \frac{-1}{8}\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell} \]

       9. unpow2 N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot \frac{-1}{8}\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto w0 \cdot \frac{\left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot \frac{-1}{8}\right) \cdot {D}^{2}}{{d}^{2} \cdot \ell} \]

       11. unpow2 N/A

           \[\leadsto w0 \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2} \cdot \ell} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto w0 \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot \color{blue}{\left(D \cdot D\right)}}{{d}^{2} \cdot \ell} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto w0 \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot \left(D \cdot D\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

       14. unpow2 N/A

           \[\leadsto w0 \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell} \]

       15. *-lowering-*.f64 13.7

           \[\leadsto w0 \cdot \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot -0.125\right) \cdot \left(D \cdot D\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell} \]

   11. Simplified 13.7%

       \[\leadsto w0 \cdot \color{blue}{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot -0.125\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto w0 \cdot \frac{\color{blue}{\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D}}{\left(d \cdot d\right) \cdot \ell} \]

       2. *-commutative N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D}{\color{blue}{\ell \cdot \left(d \cdot d\right)}} \]

       3. associate-/l* N/A

          \[\leadsto w0 \cdot \color{blue}{\left(\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot \frac{D}{\ell \cdot \left(d \cdot d\right)}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \color{blue}{\left(\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot \frac{D}{\ell \cdot \left(d \cdot d\right)}\right)} \]

       5. associate-*l* N/A

          \[\leadsto w0 \cdot \left(\left(\color{blue}{\left(\left(M \cdot M\right) \cdot \left(h \cdot \frac{-1}{8}\right)\right)} \cdot D\right) \cdot \frac{D}{\ell \cdot \left(d \cdot d\right)}\right) \]

       6. associate-*l* N/A

          \[\leadsto w0 \cdot \left(\color{blue}{\left(\left(M \cdot M\right) \cdot \left(\left(h \cdot \frac{-1}{8}\right) \cdot D\right)\right)} \cdot \frac{D}{\ell \cdot \left(d \cdot d\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \left(\color{blue}{\left(\left(M \cdot M\right) \cdot \left(\left(h \cdot \frac{-1}{8}\right) \cdot D\right)\right)} \cdot \frac{D}{\ell \cdot \left(d \cdot d\right)}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \left(\left(h \cdot \frac{-1}{8}\right) \cdot D\right)\right) \cdot \frac{D}{\ell \cdot \left(d \cdot d\right)}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto w0 \cdot \left(\left(\left(M \cdot M\right) \cdot \color{blue}{\left(\left(h \cdot \frac{-1}{8}\right) \cdot D\right)}\right) \cdot \frac{D}{\ell \cdot \left(d \cdot d\right)}\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto w0 \cdot \left(\left(\left(M \cdot M\right) \cdot \left(\color{blue}{\left(h \cdot \frac{-1}{8}\right)} \cdot D\right)\right) \cdot \frac{D}{\ell \cdot \left(d \cdot d\right)}\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto w0 \cdot \left(\left(\left(M \cdot M\right) \cdot \left(\left(h \cdot \frac{-1}{8}\right) \cdot D\right)\right) \cdot \color{blue}{\frac{D}{\ell \cdot \left(d \cdot d\right)}}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto w0 \cdot \left(\left(\left(M \cdot M\right) \cdot \left(\left(h \cdot \frac{-1}{8}\right) \cdot D\right)\right) \cdot \frac{D}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right) \]

       13. *-lowering-*.f64 17.1

           \[\leadsto w0 \cdot \left(\left(\left(M \cdot M\right) \cdot \left(\left(h \cdot -0.125\right) \cdot D\right)\right) \cdot \frac{D}{\ell \cdot \color{blue}{\left(d \cdot d\right)}}\right) \]

   13. Applied egg-rr 17.1%

       \[\leadsto w0 \cdot \color{blue}{\left(\left(\left(M \cdot M\right) \cdot \left(\left(h \cdot -0.125\right) \cdot D\right)\right) \cdot \frac{D}{\ell \cdot \left(d \cdot d\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{+74}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\left(\left(M \cdot M\right) \cdot \left(D \cdot \left(h \cdot -0.125\right)\right)\right) \cdot \frac{D}{\ell \cdot \left(d \cdot d\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 68.6% accurate, 157.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ w0 \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d) :precision binary64 w0)

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    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
    code = w0
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	return w0

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	return w0
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp = code(w0, M_m, D_m, h, l, d)
	tmp = w0;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := w0

Derivation

1. Initial program 82.1%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
2. Add Preprocessing

3. Taylor expanded in M around 0

   \[\leadsto \color{blue}{w0} \]
4. Step-by-step derivation

5. Simplified 70.7%

   \[\leadsto \color{blue}{w0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024197 
(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))))))