Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.0% → 88.4%

Time: 8.8s

Alternatives: 15

Speedup: 2.0×

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: 80.0% 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: 88.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d} \cdot \frac{D\_m}{2}\\ \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{d \cdot 2}\right)}^{2} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{1 - \left(t\_0 \cdot \frac{h}{\ell}\right) \cdot t\_0} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\frac{D\_m}{d} \cdot 0.5\right) \cdot \frac{\left(0.5 \cdot D\_m\right) \cdot \left(\left(h \cdot M\_m\right) \cdot \frac{M\_m}{d}\right)}{\ell}} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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) (/ D_m 2.0))))
   (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* d 2.0)) 2.0)) 5e-14)
     (* (sqrt (- 1.0 (* (* t_0 (/ h l)) t_0))) w0)
     (*
      (sqrt
       (-
        1.0
        (* (* (/ D_m d) 0.5) (/ (* (* 0.5 D_m) (* (* h M_m) (/ M_m d))) l))))
      w0))))

C

D_m = fabs(D);
M_m = fabs(M);
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) * (D_m / 2.0);
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= 5e-14) {
		tmp = sqrt((1.0 - ((t_0 * (h / l)) * t_0))) * w0;
	} else {
		tmp = sqrt((1.0 - (((D_m / d) * 0.5) * (((0.5 * D_m) * ((h * M_m) * (M_m / d))) / l)))) * w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
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) * (d_m / 2.0d0)
    if (((h / l) * (((d_m * m_m) / (d * 2.0d0)) ** 2.0d0)) <= 5d-14) then
        tmp = sqrt((1.0d0 - ((t_0 * (h / l)) * t_0))) * w0
    else
        tmp = sqrt((1.0d0 - (((d_m / d) * 0.5d0) * (((0.5d0 * d_m) * ((h * m_m) * (m_m / d))) / l)))) * w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (M_m / d) * (D_m / 2.0);
	double tmp;
	if (((h / l) * Math.pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= 5e-14) {
		tmp = Math.sqrt((1.0 - ((t_0 * (h / l)) * t_0))) * w0;
	} else {
		tmp = Math.sqrt((1.0 - (((D_m / d) * 0.5) * (((0.5 * D_m) * ((h * M_m) * (M_m / d))) / l)))) * w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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) * (D_m / 2.0)
	tmp = 0
	if ((h / l) * math.pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= 5e-14:
		tmp = math.sqrt((1.0 - ((t_0 * (h / l)) * t_0))) * w0
	else:
		tmp = math.sqrt((1.0 - (((D_m / d) * 0.5) * (((0.5 * D_m) * ((h * M_m) * (M_m / d))) / l)))) * w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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) * Float64(D_m / 2.0))
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0)) <= 5e-14)
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(t_0 * Float64(h / l)) * t_0))) * w0);
	else
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(Float64(D_m / d) * 0.5) * Float64(Float64(Float64(0.5 * D_m) * Float64(Float64(h * M_m) * Float64(M_m / d))) / l)))) * w0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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) * (D_m / 2.0);
	tmp = 0.0;
	if (((h / l) * (((D_m * M_m) / (d * 2.0)) ^ 2.0)) <= 5e-14)
		tmp = sqrt((1.0 - ((t_0 * (h / l)) * t_0))) * w0;
	else
		tmp = sqrt((1.0 - (((D_m / d) * 0.5) * (((0.5 * D_m) * ((h * M_m) * (M_m / d))) / l)))) * w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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), $MachinePrecision] * N[(D$95$m / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 5e-14], N[(N[Sqrt[N[(1.0 - N[(N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(1.0 - N[(N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(0.5 * D$95$m), $MachinePrecision] * N[(N[(h * M$95$m), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 5.0000000000000002e-14

   1. Initial program 88.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. unpow-prod-down N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-*.f64 N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-/.f64 70.3

          \[\leadsto w0 \cdot \sqrt{1 - \left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left({\color{blue}{\left(\frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)} \]

   4. Applied rewrites 70.3%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. unpow-prod-down N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow-prod-down N/A

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

      13. lift-/.f64 N/A

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

      14. times-frac N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 89.1%

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

   if 5.0000000000000002e-14 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 4.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 13.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \left(\left(D \cdot 0.5\right) \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \left(\frac{D}{d} \cdot 0.5\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 73.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 73.7

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

   6. Applied rewrites 73.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(h \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(0.5 \cdot D\right)}{\ell}} \cdot \left(\frac{D}{d} \cdot 0.5\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 73.6

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

   8. Applied rewrites 73.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{1 - \left(\left(\frac{M}{d} \cdot \frac{D}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\frac{D}{d} \cdot 0.5\right) \cdot \frac{\left(0.5 \cdot D\right) \cdot \left(\left(h \cdot M\right) \cdot \frac{M}{d}\right)}{\ell}} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{d \cdot 2}\right)}^{2} \leq 1.000005:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \frac{\left(\left(\left(D\_m \cdot M\_m\right) \cdot M\_m\right) \cdot h\right) \cdot 0.5}{\ell \cdot d} \cdot \left(\frac{D\_m}{d} \cdot 0.5\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= (- 1.0 (* (/ h l) (pow (/ (* D_m M_m) (* d 2.0)) 2.0))) 1.000005)
   (* 1.0 w0)
   (*
    (sqrt
     (-
      1.0
      (* (/ (* (* (* (* D_m M_m) M_m) h) 0.5) (* l d)) (* (/ D_m d) 0.5))))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
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 ((1.0 - ((h / l) * pow(((D_m * M_m) / (d * 2.0)), 2.0))) <= 1.000005) {
		tmp = 1.0 * w0;
	} else {
		tmp = sqrt((1.0 - ((((((D_m * M_m) * M_m) * h) * 0.5) / (l * d)) * ((D_m / d) * 0.5)))) * w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
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 ((1.0d0 - ((h / l) * (((d_m * m_m) / (d * 2.0d0)) ** 2.0d0))) <= 1.000005d0) then
        tmp = 1.0d0 * w0
    else
        tmp = sqrt((1.0d0 - ((((((d_m * m_m) * m_m) * h) * 0.5d0) / (l * d)) * ((d_m / d) * 0.5d0)))) * w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((1.0 - ((h / l) * Math.pow(((D_m * M_m) / (d * 2.0)), 2.0))) <= 1.000005) {
		tmp = 1.0 * w0;
	} else {
		tmp = Math.sqrt((1.0 - ((((((D_m * M_m) * M_m) * h) * 0.5) / (l * d)) * ((D_m / d) * 0.5)))) * w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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 (1.0 - ((h / l) * math.pow(((D_m * M_m) / (d * 2.0)), 2.0))) <= 1.000005:
		tmp = 1.0 * w0
	else:
		tmp = math.sqrt((1.0 - ((((((D_m * M_m) * M_m) * h) * 0.5) / (l * d)) * ((D_m / d) * 0.5)))) * w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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(1.0 - Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0))) <= 1.000005)
		tmp = Float64(1.0 * w0);
	else
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(D_m * M_m) * M_m) * h) * 0.5) / Float64(l * d)) * Float64(Float64(D_m / d) * 0.5)))) * w0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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 ((1.0 - ((h / l) * (((D_m * M_m) / (d * 2.0)) ^ 2.0))) <= 1.000005)
		tmp = 1.0 * w0;
	else
		tmp = sqrt((1.0 - ((((((D_m * M_m) * M_m) * h) * 0.5) / (l * d)) * ((D_m / d) * 0.5)))) * w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.000005], N[(1.0 * w0), $MachinePrecision], N[(N[Sqrt[N[(1.0 - N[(N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * h), $MachinePrecision] * 0.5), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 1.00000500000000003

   1. Initial program 98.8%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 98.7%

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

   if 1.00000500000000003 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 47.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 48.8%

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

   5. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 55.3

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

   7. Applied rewrites 55.3%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{0.5 \cdot \left(\left(\left(M \cdot M\right) \cdot D\right) \cdot h\right)}{\ell \cdot d}} \cdot \left(\frac{D}{d} \cdot 0.5\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 58.5%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \leq 1.000005:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \frac{\left(\left(\left(D \cdot M\right) \cdot M\right) \cdot h\right) \cdot 0.5}{\ell \cdot d} \cdot \left(\frac{D}{d} \cdot 0.5\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{d \cdot 2}\right)}^{2} \leq -1 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{1 - \frac{\left(h \cdot D\_m\right) \cdot M\_m}{\ell} \cdot \left(\left(\frac{D\_m}{d} \cdot 0.5\right) \cdot M\_m\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= (* (/ h l) (pow (/ (* D_m M_m) (* d 2.0)) 2.0)) -1e+114)
   (* (sqrt (- 1.0 (* (/ (* (* h D_m) M_m) l) (* (* (/ D_m d) 0.5) M_m)))) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
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 (((h / l) * pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -1e+114) {
		tmp = sqrt((1.0 - ((((h * D_m) * M_m) / l) * (((D_m / d) * 0.5) * M_m)))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
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 (((h / l) * (((d_m * m_m) / (d * 2.0d0)) ** 2.0d0)) <= (-1d+114)) then
        tmp = sqrt((1.0d0 - ((((h * d_m) * m_m) / l) * (((d_m / d) * 0.5d0) * m_m)))) * w0
    else
        tmp = 1.0d0 * w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * Math.pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -1e+114) {
		tmp = Math.sqrt((1.0 - ((((h * D_m) * M_m) / l) * (((D_m / d) * 0.5) * M_m)))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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 ((h / l) * math.pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -1e+114:
		tmp = math.sqrt((1.0 - ((((h * D_m) * M_m) / l) * (((D_m / d) * 0.5) * M_m)))) * w0
	else:
		tmp = 1.0 * w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0)) <= -1e+114)
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(h * D_m) * M_m) / l) * Float64(Float64(Float64(D_m / d) * 0.5) * M_m)))) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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 (((h / l) * (((D_m * M_m) / (d * 2.0)) ^ 2.0)) <= -1e+114)
		tmp = sqrt((1.0 - ((((h * D_m) * M_m) / l) * (((D_m / d) * 0.5) * M_m)))) * w0;
	else
		tmp = 1.0 * w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e+114], N[(N[Sqrt[N[(1.0 - N[(N[(N[(N[(h * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1e114

   1. Initial program 59.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r/ N/A

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

   4. Applied rewrites 19.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 19.1%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\left(D \cdot h\right) \cdot M\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)}{\ell}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. times-frac N/A

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

      13. associate-*r/ N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

   8. Applied rewrites 19.2%

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

   if -1e114 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 92.6%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \leq -1 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{1 - \frac{\left(h \cdot D\right) \cdot M}{\ell} \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D\_m \cdot M\_m}{d \cdot 2}\right)}^{2} \leq 5 \cdot 10^{-24}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \frac{\left(\left(\frac{M\_m}{d} \cdot M\_m\right) \cdot 0.5\right) \cdot D\_m}{\ell \cdot d} \cdot \left(\left(0.5 \cdot D\_m\right) \cdot h\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= (pow (/ (* D_m M_m) (* d 2.0)) 2.0) 5e-24)
   (* 1.0 w0)
   (*
    (sqrt
     (-
      1.0
      (* (/ (* (* (* (/ M_m d) M_m) 0.5) D_m) (* l d)) (* (* 0.5 D_m) h))))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
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 (pow(((D_m * M_m) / (d * 2.0)), 2.0) <= 5e-24) {
		tmp = 1.0 * w0;
	} else {
		tmp = sqrt((1.0 - ((((((M_m / d) * M_m) * 0.5) * D_m) / (l * d)) * ((0.5 * D_m) * h)))) * w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
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 ((((d_m * m_m) / (d * 2.0d0)) ** 2.0d0) <= 5d-24) then
        tmp = 1.0d0 * w0
    else
        tmp = sqrt((1.0d0 - ((((((m_m / d) * m_m) * 0.5d0) * d_m) / (l * d)) * ((0.5d0 * d_m) * h)))) * w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (Math.pow(((D_m * M_m) / (d * 2.0)), 2.0) <= 5e-24) {
		tmp = 1.0 * w0;
	} else {
		tmp = Math.sqrt((1.0 - ((((((M_m / d) * M_m) * 0.5) * D_m) / (l * d)) * ((0.5 * D_m) * h)))) * w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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 math.pow(((D_m * M_m) / (d * 2.0)), 2.0) <= 5e-24:
		tmp = 1.0 * w0
	else:
		tmp = math.sqrt((1.0 - ((((((M_m / d) * M_m) * 0.5) * D_m) / (l * d)) * ((0.5 * D_m) * h)))) * w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0) <= 5e-24)
		tmp = Float64(1.0 * w0);
	else
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M_m / d) * M_m) * 0.5) * D_m) / Float64(l * d)) * Float64(Float64(0.5 * D_m) * h)))) * w0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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 ((((D_m * M_m) / (d * 2.0)) ^ 2.0) <= 5e-24)
		tmp = 1.0 * w0;
	else
		tmp = sqrt((1.0 - ((((((M_m / d) * M_m) * 0.5) * D_m) / (l * d)) * ((0.5 * D_m) * h)))) * w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 5e-24], N[(1.0 * w0), $MachinePrecision], N[(N[Sqrt[N[(1.0 - N[(N[(N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * D$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 4.9999999999999998e-24

   1. Initial program 87.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 94.8%

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

   if 4.9999999999999998e-24 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 68.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 65.1%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \left(\left(D \cdot 0.5\right) \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \left(\frac{D}{d} \cdot 0.5\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 65.4

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 65.4

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

   6. Applied rewrites 65.4%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(h \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(0.5 \cdot D\right)}{\ell}} \cdot \left(\frac{D}{d} \cdot 0.5\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 66.4

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

   8. Applied rewrites 66.4%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\frac{M}{d} \cdot \left(M \cdot h\right)\right)} \cdot \left(0.5 \cdot D\right)}{\ell} \cdot \left(\frac{D}{d} \cdot 0.5\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

   10. Applied rewrites 64.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \leq 5 \cdot 10^{-24}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \frac{\left(\left(\frac{M}{d} \cdot M\right) \cdot 0.5\right) \cdot D}{\ell \cdot d} \cdot \left(\left(0.5 \cdot D\right) \cdot h\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{d \cdot 2}\right)}^{2} \leq -1 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\left(h \cdot D\_m\right) \cdot D\_m}{\ell \cdot d} \cdot \left(\frac{M\_m}{d} \cdot M\_m\right), 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= (* (/ h l) (pow (/ (* D_m M_m) (* d 2.0)) 2.0)) -1e-13)
   (* (fma -0.125 (* (/ (* (* h D_m) D_m) (* l d)) (* (/ M_m d) M_m)) 1.0) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
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 (((h / l) * pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -1e-13) {
		tmp = fma(-0.125, ((((h * D_m) * D_m) / (l * d)) * ((M_m / d) * M_m)), 1.0) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0)) <= -1e-13)
		tmp = Float64(fma(-0.125, Float64(Float64(Float64(Float64(h * D_m) * D_m) / Float64(l * d)) * Float64(Float64(M_m / d) * M_m)), 1.0) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e-13], N[(N[(-0.125 * N[(N[(N[(N[(h * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1e-13

   1. Initial program 63.6%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.5

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

   5. Applied rewrites 33.5%

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

   7. Applied rewrites 45.8%

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

   if -1e-13 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.1%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \leq -1 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\left(h \cdot D\right) \cdot D}{\ell \cdot d} \cdot \left(\frac{M}{d} \cdot M\right), 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{d \cdot 2}\right)}^{2} \leq -2 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{D\_m \cdot D\_m}{\ell \cdot d} \cdot \left(\left(\frac{M\_m}{d} \cdot h\right) \cdot M\_m\right), 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= (* (/ h l) (pow (/ (* D_m M_m) (* d 2.0)) 2.0)) -2e+144)
   (* (fma -0.125 (* (/ (* D_m D_m) (* l d)) (* (* (/ M_m d) h) M_m)) 1.0) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
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 (((h / l) * pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -2e+144) {
		tmp = fma(-0.125, (((D_m * D_m) / (l * d)) * (((M_m / d) * h) * M_m)), 1.0) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0)) <= -2e+144)
		tmp = Float64(fma(-0.125, Float64(Float64(Float64(D_m * D_m) / Float64(l * d)) * Float64(Float64(Float64(M_m / d) * h) * M_m)), 1.0) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -2e+144], N[(N[(-0.125 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m / d), $MachinePrecision] * h), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.00000000000000005e144

   1. Initial program 57.8%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 37.7

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

   5. Applied rewrites 37.7%

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

   7. Applied rewrites 44.9%

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

   if -2.00000000000000005e144 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 91.2%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \leq -2 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(\frac{M}{d} \cdot h\right) \cdot M\right), 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{d \cdot 2}\right)}^{2} \leq -2 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \left(\left(\frac{D\_m}{\ell} \cdot \frac{D\_m}{d \cdot d}\right) \cdot \left(h \cdot M\_m\right)\right) \cdot M\_m, 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= (* (/ h l) (pow (/ (* D_m M_m) (* d 2.0)) 2.0)) -2e+144)
   (* (fma -0.125 (* (* (* (/ D_m l) (/ D_m (* d d))) (* h M_m)) M_m) 1.0) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
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 (((h / l) * pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -2e+144) {
		tmp = fma(-0.125, ((((D_m / l) * (D_m / (d * d))) * (h * M_m)) * M_m), 1.0) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0)) <= -2e+144)
		tmp = Float64(fma(-0.125, Float64(Float64(Float64(Float64(D_m / l) * Float64(D_m / Float64(d * d))) * Float64(h * M_m)) * M_m), 1.0) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -2e+144], N[(N[(-0.125 * N[(N[(N[(N[(D$95$m / l), $MachinePrecision] * N[(D$95$m / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.00000000000000005e144

   1. Initial program 57.8%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 37.7

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

   5. Applied rewrites 37.7%

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

   7. Applied rewrites 48.0%

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

   if -2.00000000000000005e144 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 91.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \leq -2 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \left(\left(\frac{D}{\ell} \cdot \frac{D}{d \cdot d}\right) \cdot \left(h \cdot M\right)\right) \cdot M, 1\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.6% accurate, 0.8× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= (* (/ h l) (pow (/ (* D_m M_m) (* d 2.0)) 2.0)) (- INFINITY))
   (* (fma -0.125 (/ (* (* (* h M_m) (* D_m D_m)) M_m) (* (* d d) l)) 1.0) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
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 (((h / l) * pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -((double) INFINITY)) {
		tmp = fma(-0.125, ((((h * M_m) * (D_m * D_m)) * M_m) / ((d * d) * l)), 1.0) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0)) <= Float64(-Inf))
		tmp = Float64(fma(-0.125, Float64(Float64(Float64(Float64(h * M_m) * Float64(D_m * D_m)) * M_m) / Float64(Float64(d * d) * l)), 1.0) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(-0.125 * N[(N[(N[(N[(h * M$95$m), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 54.1%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 41.0

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

   5. Applied rewrites 41.0%

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

   7. Applied rewrites 43.0%

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

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

   1. Initial program 87.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.1%

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

4. Final simplification 79.0%

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

Alternative 9: 77.9% accurate, 0.8× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= (* (/ h l) (pow (/ (* D_m M_m) (* d 2.0)) 2.0)) (- INFINITY))
   (* (fma -0.125 (/ (* (* (* (* h D_m) D_m) M_m) M_m) (* (* d d) l)) 1.0) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
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 (((h / l) * pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -((double) INFINITY)) {
		tmp = fma(-0.125, (((((h * D_m) * D_m) * M_m) * M_m) / ((d * d) * l)), 1.0) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0)) <= Float64(-Inf))
		tmp = Float64(fma(-0.125, Float64(Float64(Float64(Float64(Float64(h * D_m) * D_m) * M_m) * M_m) / Float64(Float64(d * d) * l)), 1.0) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(-0.125 * N[(N[(N[(N[(N[(h * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 54.1%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 41.0

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

   5. Applied rewrites 41.0%

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

   7. Applied rewrites 48.4%

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

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

   1. Initial program 87.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.1%

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

4. Final simplification 80.2%

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

Alternative 10: 76.1% accurate, 0.8× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= (* (/ h l) (pow (/ (* D_m M_m) (* d 2.0)) 2.0)) (- INFINITY))
   (* (fma -0.125 (* (/ D_m (* (* d d) l)) (* (* (* M_m M_m) h) D_m)) 1.0) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
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 (((h / l) * pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -((double) INFINITY)) {
		tmp = fma(-0.125, ((D_m / ((d * d) * l)) * (((M_m * M_m) * h) * D_m)), 1.0) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0)) <= Float64(-Inf))
		tmp = Float64(fma(-0.125, Float64(Float64(D_m / Float64(Float64(d * d) * l)) * Float64(Float64(Float64(M_m * M_m) * h) * D_m)), 1.0) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(-0.125 * N[(N[(D$95$m / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 54.1%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 41.0

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

   5. Applied rewrites 41.0%

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

   7. Applied rewrites 48.3%

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

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

   1. Initial program 87.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.1%

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

4. Final simplification 80.2%

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

Alternative 11: 76.0% accurate, 0.8× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (<= (* (/ h l) (pow (/ (* D_m M_m) (* d 2.0)) 2.0)) (- INFINITY))
   (fma (/ (* (* (* (* M_m M_m) h) w0) (* D_m D_m)) (* (* d d) l)) -0.125 w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
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 (((h / l) * pow(((D_m * M_m) / (d * 2.0)), 2.0)) <= -((double) INFINITY)) {
		tmp = fma((((((M_m * M_m) * h) * w0) * (D_m * D_m)) / ((d * d) * l)), -0.125, w0);
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
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(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0)) <= Float64(-Inf))
		tmp = fma(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * w0) * Float64(D_m * D_m)) / Float64(Float64(d * d) * l)), -0.125, w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * w0), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125 + w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 54.1%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 40.8

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

   5. Applied rewrites 40.8%

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

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

   1. Initial program 87.5%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.1%

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

4. Final simplification 78.5%

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

Alternative 12: 87.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{d} \cdot M\_m\\ \mathbf{if}\;\frac{D\_m \cdot M\_m}{d \cdot 2} \leq 10^{-37}:\\ \;\;\;\;\sqrt{1 - \frac{\left(\left(0.25 \cdot h\right) \cdot t\_0\right) \cdot t\_0}{\ell}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \frac{0.5 \cdot D\_m}{\frac{\frac{\ell}{\frac{M\_m}{d}}}{\left(\left(h \cdot D\_m\right) \cdot M\_m\right) \cdot 0.5} \cdot d}} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (* (/ D_m d) M_m)))
   (if (<= (/ (* D_m M_m) (* d 2.0)) 1e-37)
     (* (sqrt (- 1.0 (/ (* (* (* 0.25 h) t_0) t_0) l))) w0)
     (*
      (sqrt
       (-
        1.0
        (/ (* 0.5 D_m) (* (/ (/ l (/ M_m d)) (* (* (* h D_m) M_m) 0.5)) d))))
      w0))))

C

D_m = fabs(D);
M_m = fabs(M);
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 = (D_m / d) * M_m;
	double tmp;
	if (((D_m * M_m) / (d * 2.0)) <= 1e-37) {
		tmp = sqrt((1.0 - ((((0.25 * h) * t_0) * t_0) / l))) * w0;
	} else {
		tmp = sqrt((1.0 - ((0.5 * D_m) / (((l / (M_m / d)) / (((h * D_m) * M_m) * 0.5)) * d)))) * w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
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 = (d_m / d) * m_m
    if (((d_m * m_m) / (d * 2.0d0)) <= 1d-37) then
        tmp = sqrt((1.0d0 - ((((0.25d0 * h) * t_0) * t_0) / l))) * w0
    else
        tmp = sqrt((1.0d0 - ((0.5d0 * d_m) / (((l / (m_m / d)) / (((h * d_m) * m_m) * 0.5d0)) * d)))) * w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (D_m / d) * M_m;
	double tmp;
	if (((D_m * M_m) / (d * 2.0)) <= 1e-37) {
		tmp = Math.sqrt((1.0 - ((((0.25 * h) * t_0) * t_0) / l))) * w0;
	} else {
		tmp = Math.sqrt((1.0 - ((0.5 * D_m) / (((l / (M_m / d)) / (((h * D_m) * M_m) * 0.5)) * d)))) * w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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 = (D_m / d) * M_m
	tmp = 0
	if ((D_m * M_m) / (d * 2.0)) <= 1e-37:
		tmp = math.sqrt((1.0 - ((((0.25 * h) * t_0) * t_0) / l))) * w0
	else:
		tmp = math.sqrt((1.0 - ((0.5 * D_m) / (((l / (M_m / d)) / (((h * D_m) * M_m) * 0.5)) * d)))) * w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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(D_m / d) * M_m)
	tmp = 0.0
	if (Float64(Float64(D_m * M_m) / Float64(d * 2.0)) <= 1e-37)
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(0.25 * h) * t_0) * t_0) / l))) * w0);
	else
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(0.5 * D_m) / Float64(Float64(Float64(l / Float64(M_m / d)) / Float64(Float64(Float64(h * D_m) * M_m) * 0.5)) * d)))) * w0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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 = (D_m / d) * M_m;
	tmp = 0.0;
	if (((D_m * M_m) / (d * 2.0)) <= 1e-37)
		tmp = sqrt((1.0 - ((((0.25 * h) * t_0) * t_0) / l))) * w0;
	else
		tmp = sqrt((1.0 - ((0.5 * D_m) / (((l / (M_m / d)) / (((h * D_m) * M_m) * 0.5)) * d)))) * w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 1e-37], N[(N[Sqrt[N[(1.0 - N[(N[(N[(N[(0.25 * h), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(1.0 - N[(N[(0.5 * D$95$m), $MachinePrecision] / N[(N[(N[(l / N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(h * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1.00000000000000007e-37

   1. Initial program 82.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. unpow-prod-down N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-*.f64 N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-/.f64 67.3

          \[\leadsto w0 \cdot \sqrt{1 - \left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left({\color{blue}{\left(\frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)} \]

   4. Applied rewrites 67.3%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. div-inv N/A

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

      12. unpow-1 N/A

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

      13. lift-pow.f64 N/A

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

      14. div-inv N/A

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

      15. frac-times N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l/ N/A

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

      18. lower-/.f64 N/A

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

   6. Applied rewrites 89.6%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right) \cdot h}{\ell}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 89.6

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot \color{blue}{\left(0.25 \cdot h\right)}}{\ell}} \]

   8. Applied rewrites 89.6%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot \left(0.25 \cdot h\right)}}{\ell}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 91.0

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

   10. Applied rewrites 91.0%

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

   if 1.00000000000000007e-37 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 73.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 71.8%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \left(\left(D \cdot 0.5\right) \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \left(\frac{D}{d} \cdot 0.5\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 73.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 73.7

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

   6. Applied rewrites 73.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(h \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \left(0.5 \cdot D\right)}{\ell}} \cdot \left(\frac{D}{d} \cdot 0.5\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 75.4

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

   8. Applied rewrites 75.4%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\frac{M}{d} \cdot \left(M \cdot h\right)\right)} \cdot \left(0.5 \cdot D\right)}{\ell} \cdot \left(\frac{D}{d} \cdot 0.5\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

   10. Applied rewrites 77.2%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{D \cdot M}{d \cdot 2} \leq 10^{-37}:\\ \;\;\;\;\sqrt{1 - \frac{\left(\left(0.25 \cdot h\right) \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot M\right)}{\ell}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \frac{0.5 \cdot D}{\frac{\frac{\ell}{\frac{M}{d}}}{\left(\left(h \cdot D\right) \cdot M\right) \cdot 0.5} \cdot d}} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \cdot 2 \leq 5 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{1 - \frac{\frac{\left(D\_m \cdot M\_m\right) \cdot 0.5}{d} \cdot \left(\left(h \cdot D\_m\right) \cdot M\_m\right)}{\ell}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot 0.5}{\ell \cdot d} \cdot \left(\frac{D\_m}{d} \cdot 0.5\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 2.0) 5e-123)
   (* (sqrt (- 1.0 (/ (* (/ (* (* D_m M_m) 0.5) d) (* (* h D_m) M_m)) l))) w0)
   (*
    (sqrt
     (-
      1.0
      (* (/ (* (* (* (* M_m M_m) h) D_m) 0.5) (* l d)) (* (/ D_m d) 0.5))))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
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 * 2.0) <= 5e-123) {
		tmp = sqrt((1.0 - (((((D_m * M_m) * 0.5) / d) * ((h * D_m) * M_m)) / l))) * w0;
	} else {
		tmp = sqrt((1.0 - ((((((M_m * M_m) * h) * D_m) * 0.5) / (l * d)) * ((D_m / d) * 0.5)))) * w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
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 ((d * 2.0d0) <= 5d-123) then
        tmp = sqrt((1.0d0 - (((((d_m * m_m) * 0.5d0) / d) * ((h * d_m) * m_m)) / l))) * w0
    else
        tmp = sqrt((1.0d0 - ((((((m_m * m_m) * h) * d_m) * 0.5d0) / (l * d)) * ((d_m / d) * 0.5d0)))) * w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((d * 2.0) <= 5e-123) {
		tmp = Math.sqrt((1.0 - (((((D_m * M_m) * 0.5) / d) * ((h * D_m) * M_m)) / l))) * w0;
	} else {
		tmp = Math.sqrt((1.0 - ((((((M_m * M_m) * h) * D_m) * 0.5) / (l * d)) * ((D_m / d) * 0.5)))) * w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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 (d * 2.0) <= 5e-123:
		tmp = math.sqrt((1.0 - (((((D_m * M_m) * 0.5) / d) * ((h * D_m) * M_m)) / l))) * w0
	else:
		tmp = math.sqrt((1.0 - ((((((M_m * M_m) * h) * D_m) * 0.5) / (l * d)) * ((D_m / d) * 0.5)))) * w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
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(d * 2.0) <= 5e-123)
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D_m * M_m) * 0.5) / d) * Float64(Float64(h * D_m) * M_m)) / l))) * w0);
	else
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * 0.5) / Float64(l * d)) * Float64(Float64(D_m / d) * 0.5)))) * w0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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 ((d * 2.0) <= 5e-123)
		tmp = sqrt((1.0 - (((((D_m * M_m) * 0.5) / d) * ((h * D_m) * M_m)) / l))) * w0;
	else
		tmp = sqrt((1.0 - ((((((M_m * M_m) * h) * D_m) * 0.5) / (l * d)) * ((D_m / d) * 0.5)))) * w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(d * 2.0), $MachinePrecision], 5e-123], N[(N[Sqrt[N[(1.0 - N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / d), $MachinePrecision] * N[(N[(h * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(1.0 - N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) d) < 5.0000000000000003e-123

   1. Initial program 77.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r/ N/A

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

   4. Applied rewrites 59.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 62.8%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\left(D \cdot h\right) \cdot M\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right)}{\ell}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. times-frac N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 62.3

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

   8. Applied rewrites 62.3%

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

   if 5.0000000000000003e-123 < (*.f64 #s(literal 2 binary64) d)

   1. Initial program 84.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 78.9%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{h}{\ell} \cdot \left(\left(D \cdot 0.5\right) \cdot \left(\frac{M}{d} \cdot M\right)\right)\right) \cdot \left(\frac{D}{d} \cdot 0.5\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 81.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 81.3

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

   6. Applied rewrites 81.3%

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

   7. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 70.5

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

   9. Applied rewrites 70.5%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot 2 \leq 5 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{1 - \frac{\frac{\left(D \cdot M\right) \cdot 0.5}{d} \cdot \left(\left(h \cdot D\right) \cdot M\right)}{\ell}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot 0.5}{\ell \cdot d} \cdot \left(\frac{D}{d} \cdot 0.5\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 86.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{d} \cdot M\_m\\ \sqrt{1 - \frac{\left(\left(0.25 \cdot h\right) \cdot t\_0\right) \cdot t\_0}{\ell}} \cdot w0 \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (* (/ D_m d) M_m)))
   (* (sqrt (- 1.0 (/ (* (* (* 0.25 h) t_0) t_0) l))) w0)))

C

D_m = fabs(D);
M_m = fabs(M);
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 = (D_m / d) * M_m;
	return sqrt((1.0 - ((((0.25 * h) * t_0) * t_0) / l))) * w0;
}

Fortran

D_m = abs(d)
M_m = abs(m)
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
    t_0 = (d_m / d) * m_m
    code = sqrt((1.0d0 - ((((0.25d0 * h) * t_0) * t_0) / l))) * w0
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = (D_m / d) * M_m;
	return Math.sqrt((1.0 - ((((0.25 * h) * t_0) * t_0) / l))) * w0;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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 = (D_m / d) * M_m
	return math.sqrt((1.0 - ((((0.25 * h) * t_0) * t_0) / l))) * w0

Julia

D_m = abs(D)
M_m = abs(M)
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(D_m / d) * M_m)
	return Float64(sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(0.25 * h) * t_0) * t_0) / l))) * w0)
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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)
	t_0 = (D_m / d) * M_m;
	tmp = sqrt((1.0 - ((((0.25 * h) * t_0) * t_0) / l))) * w0;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, N[(N[Sqrt[N[(1.0 - N[(N[(N[(N[(0.25 * h), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Initial program 80.2%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. unpow-prod-down N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. unpow-prod-down N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. pow2 N/A

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

   17. lower-*.f64 N/A

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

   18. metadata-eval N/A

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

   19. lower-*.f64 N/A

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

   20. lower-pow.f64 N/A

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

   21. lower-/.f64 64.3

       \[\leadsto w0 \cdot \sqrt{1 - \left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left({\color{blue}{\left(\frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)} \]

4. Applied rewrites 64.3%

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left(M \cdot M\right) \cdot 0.25\right) \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. clear-num N/A

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

   11. div-inv N/A

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

   12. unpow-1 N/A

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

   13. lift-pow.f64 N/A

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

   14. div-inv N/A

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

   15. frac-times N/A

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

   16. lift-/.f64 N/A

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

   17. associate-*l/ N/A

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

   18. lower-/.f64 N/A

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

6. Applied rewrites 85.5%

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right) \cdot h}{\ell}}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 85.5

      \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot \color{blue}{\left(0.25 \cdot h\right)}}{\ell}} \]

8. Applied rewrites 85.5%

   \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{{\left(\frac{D}{d} \cdot M\right)}^{2} \cdot \left(0.25 \cdot h\right)}}{\ell}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 88.8

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

10. Applied rewrites 88.8%

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

11. Final simplification 88.8%

    \[\leadsto \sqrt{1 - \frac{\left(\left(0.25 \cdot h\right) \cdot \left(\frac{D}{d} \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot M\right)}{\ell}} \cdot w0 \]
12. Add Preprocessing

Alternative 15: 67.2% accurate, 26.2× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
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 (* 1.0 w0))

C

D_m = fabs(D);
M_m = fabs(M);
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 1.0 * w0;
}

Fortran

D_m = abs(d)
M_m = abs(m)
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 = 1.0d0 * w0
end function

Java

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

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[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 1.0 * w0

Julia

D_m = abs(D)
M_m = abs(M)
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 Float64(1.0 * w0)
end

MATLAB

D_m = abs(D);
M_m = abs(M);
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 = 1.0 * w0;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $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_] := N[(1.0 * w0), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 70.5%

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

6. Final simplification 70.5%

   \[\leadsto 1 \cdot w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024312 
(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))))))