Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 90.8%

Time: 14.5s

Alternatives: 9

Speedup: 0.8×

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: 80.8% 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: 90.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+253}:\\ \;\;\;\;\left(D\_m \cdot \sqrt{\left(M\_m \cdot -0.25\right) \cdot \frac{M\_m \cdot h}{\ell}}\right) \cdot \frac{w0}{d\_m}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l))))
   (if (<= t_0 -2e+253)
     (* (* D_m (sqrt (* (* M_m -0.25) (/ (* M_m h) l)))) (/ w0 d_m))
     (if (<= t_0 2e-7) (* w0 (sqrt (- 1.0 t_0))) w0))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -2e+253) {
		tmp = (D_m * sqrt(((M_m * -0.25) * ((M_m * h) / l)))) * (w0 / d_m);
	} else if (t_0 <= 2e-7) {
		tmp = w0 * sqrt((1.0 - t_0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)
    if (t_0 <= (-2d+253)) then
        tmp = (d_m * sqrt(((m_m * (-0.25d0)) * ((m_m * h) / l)))) * (w0 / d_m_1)
    else if (t_0 <= 2d-7) then
        tmp = w0 * sqrt((1.0d0 - t_0))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -2e+253) {
		tmp = (D_m * Math.sqrt(((M_m * -0.25) * ((M_m * h) / l)))) * (w0 / d_m);
	} else if (t_0 <= 2e-7) {
		tmp = w0 * Math.sqrt((1.0 - t_0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)
	tmp = 0
	if t_0 <= -2e+253:
		tmp = (D_m * math.sqrt(((M_m * -0.25) * ((M_m * h) / l)))) * (w0 / d_m)
	elif t_0 <= 2e-7:
		tmp = w0 * math.sqrt((1.0 - t_0))
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= -2e+253)
		tmp = Float64(Float64(D_m * sqrt(Float64(Float64(M_m * -0.25) * Float64(Float64(M_m * h) / l)))) * Float64(w0 / d_m));
	elseif (t_0 <= 2e-7)
		tmp = Float64(w0 * sqrt(Float64(1.0 - t_0)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = (((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_0 <= -2e+253)
		tmp = (D_m * sqrt(((M_m * -0.25) * ((M_m * h) / l)))) * (w0 / d_m);
	elseif (t_0 <= 2e-7)
		tmp = w0 * sqrt((1.0 - t_0));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+253], N[(N[(D$95$m * N[Sqrt[N[(N[(M$95$m * -0.25), $MachinePrecision] * N[(N[(M$95$m * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(w0 / d$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-7], N[(w0 * N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.9999999999999999e253

   1. Initial program 65.5%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 54.4

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

   5. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. sqrt-div N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. sqrt-pow1 N/A

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

      13. metadata-eval N/A

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

      14. unpow1 N/A

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

      15. lower-/.f64 N/A

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

   7. Applied rewrites 26.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. div-inv N/A

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

      12. associate-*l* N/A

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

   9. Applied rewrites 34.4%

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

       1. associate-*l/ N/A

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

       2. *-lft-identity N/A

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

       3. lower-/.f64 34.4

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

   11. Applied rewrites 34.4%

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

   if -1.9999999999999999e253 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 1.9999999999999999e-7

   1. Initial program 100.0%

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

   if 1.9999999999999999e-7 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 0.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 89.0%

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

      1. *-rgt-identity 89.0

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

   7. Applied rewrites 89.0%

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

Alternative 2: 89.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+168}:\\ \;\;\;\;\left(D\_m \cdot \sqrt{\left(M\_m \cdot -0.25\right) \cdot \frac{M\_m \cdot h}{\ell}}\right) \cdot \frac{w0}{d\_m}\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M\_m \cdot D\_m}{d\_m \cdot -2}, \left(M\_m \cdot D\_m\right) \cdot \left(\frac{h}{\ell} \cdot \frac{0.5}{d\_m}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l))))
   (if (<= t_0 -1e+168)
     (* (* D_m (sqrt (* (* M_m -0.25) (/ (* M_m h) l)))) (/ w0 d_m))
     (if (<= t_0 -0.01)
       (*
        w0
        (sqrt
         (fma
          (/ (* M_m D_m) (* d_m -2.0))
          (* (* M_m D_m) (* (/ h l) (/ 0.5 d_m)))
          1.0)))
       w0))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -1e+168) {
		tmp = (D_m * sqrt(((M_m * -0.25) * ((M_m * h) / l)))) * (w0 / d_m);
	} else if (t_0 <= -0.01) {
		tmp = w0 * sqrt(fma(((M_m * D_m) / (d_m * -2.0)), ((M_m * D_m) * ((h / l) * (0.5 / d_m))), 1.0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= -1e+168)
		tmp = Float64(Float64(D_m * sqrt(Float64(Float64(M_m * -0.25) * Float64(Float64(M_m * h) / l)))) * Float64(w0 / d_m));
	elseif (t_0 <= -0.01)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M_m * D_m) / Float64(d_m * -2.0)), Float64(Float64(M_m * D_m) * Float64(Float64(h / l) * Float64(0.5 / d_m))), 1.0)));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+168], N[(N[(D$95$m * N[Sqrt[N[(N[(M$95$m * -0.25), $MachinePrecision] * N[(N[(M$95$m * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(w0 / d$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(w0 * N[Sqrt[N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -9.9999999999999993e167

   1. Initial program 66.9%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 52.2

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

   5. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. sqrt-div N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. sqrt-pow1 N/A

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

      13. metadata-eval N/A

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

      14. unpow1 N/A

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

      15. lower-/.f64 N/A

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

   7. Applied rewrites 26.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. div-inv N/A

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

      12. associate-*l* N/A

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

   9. Applied rewrites 33.2%

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

       1. associate-*l/ N/A

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

       2. *-lft-identity N/A

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

       3. lower-/.f64 33.1

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

   11. Applied rewrites 33.1%

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

   if -9.9999999999999993e167 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -0.0100000000000000002

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. distribute-lft-neg-in N/A

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

      14. *-commutative N/A

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

   4. Applied rewrites 99.7%

      \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot -2}, \left(M \cdot D\right) \cdot \left(\frac{0.5}{d} \cdot \frac{h}{\ell}\right), 1\right)}} \]

   if -0.0100000000000000002 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 98.4%

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

      1. *-rgt-identity 98.4

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

   7. Applied rewrites 98.4%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+168}:\\ \;\;\;\;\left(D \cdot \sqrt{\left(M \cdot -0.25\right) \cdot \frac{M \cdot h}{\ell}}\right) \cdot \frac{w0}{d}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.01:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot -2}, \left(M \cdot D\right) \cdot \left(\frac{h}{\ell} \cdot \frac{0.5}{d}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+253}:\\ \;\;\;\;\left(D\_m \cdot \sqrt{\left(M\_m \cdot -0.25\right) \cdot \frac{M\_m \cdot h}{\ell}}\right) \cdot \frac{w0}{d\_m}\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \frac{M\_m \cdot D\_m}{\left(d\_m \cdot d\_m\right) \cdot 4}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l))))
   (if (<= t_0 -2e+253)
     (* (* D_m (sqrt (* (* M_m -0.25) (/ (* M_m h) l)))) (/ w0 d_m))
     (if (<= t_0 -0.01)
       (*
        w0
        (sqrt
         (-
          1.0
          (* (/ h l) (* (* M_m D_m) (/ (* M_m D_m) (* (* d_m d_m) 4.0)))))))
       w0))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -2e+253) {
		tmp = (D_m * sqrt(((M_m * -0.25) * ((M_m * h) / l)))) * (w0 / d_m);
	} else if (t_0 <= -0.01) {
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M_m * D_m) * ((M_m * D_m) / ((d_m * d_m) * 4.0))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)
    if (t_0 <= (-2d+253)) then
        tmp = (d_m * sqrt(((m_m * (-0.25d0)) * ((m_m * h) / l)))) * (w0 / d_m_1)
    else if (t_0 <= (-0.01d0)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((m_m * d_m) * ((m_m * d_m) / ((d_m_1 * d_m_1) * 4.0d0))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -2e+253) {
		tmp = (D_m * Math.sqrt(((M_m * -0.25) * ((M_m * h) / l)))) * (w0 / d_m);
	} else if (t_0 <= -0.01) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * ((M_m * D_m) * ((M_m * D_m) / ((d_m * d_m) * 4.0))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)
	tmp = 0
	if t_0 <= -2e+253:
		tmp = (D_m * math.sqrt(((M_m * -0.25) * ((M_m * h) / l)))) * (w0 / d_m)
	elif t_0 <= -0.01:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * ((M_m * D_m) * ((M_m * D_m) / ((d_m * d_m) * 4.0))))))
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= -2e+253)
		tmp = Float64(Float64(D_m * sqrt(Float64(Float64(M_m * -0.25) * Float64(Float64(M_m * h) / l)))) * Float64(w0 / d_m));
	elseif (t_0 <= -0.01)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(M_m * D_m) * Float64(Float64(M_m * D_m) / Float64(Float64(d_m * d_m) * 4.0)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = (((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_0 <= -2e+253)
		tmp = (D_m * sqrt(((M_m * -0.25) * ((M_m * h) / l)))) * (w0 / d_m);
	elseif (t_0 <= -0.01)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M_m * D_m) * ((M_m * D_m) / ((d_m * d_m) * 4.0))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+253], N[(N[(D$95$m * N[Sqrt[N[(N[(M$95$m * -0.25), $MachinePrecision] * N[(N[(M$95$m * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(w0 / d$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(N[(d$95$m * d$95$m), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.9999999999999999e253

   1. Initial program 65.5%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 54.4

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

   5. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. sqrt-div N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. sqrt-pow1 N/A

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

      13. metadata-eval N/A

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

      14. unpow1 N/A

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

      15. lower-/.f64 N/A

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

   7. Applied rewrites 26.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. div-inv N/A

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

      12. associate-*l* N/A

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

   9. Applied rewrites 34.4%

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

       1. associate-*l/ N/A

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

       2. *-lft-identity N/A

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

       3. lower-/.f64 34.4

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

   11. Applied rewrites 34.4%

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

   if -1.9999999999999999e253 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -0.0100000000000000002

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. swap-sqr N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 85.4

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

   4. Applied rewrites 85.4%

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

   if -0.0100000000000000002 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 98.4%

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

      1. *-rgt-identity 98.4

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

   7. Applied rewrites 98.4%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+253}:\\ \;\;\;\;\left(D \cdot \sqrt{\left(M \cdot -0.25\right) \cdot \frac{M \cdot h}{\ell}}\right) \cdot \frac{w0}{d}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0.01:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\left(M \cdot D\right) \cdot \frac{M \cdot D}{\left(d \cdot d\right) \cdot 4}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+15}:\\ \;\;\;\;w0 \cdot \frac{D\_m \cdot \sqrt{\left(M\_m \cdot -0.25\right) \cdot \frac{M\_m \cdot h}{\ell}}}{d\_m}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -2e+15)
   (* w0 (/ (* D_m (sqrt (* (* M_m -0.25) (/ (* M_m h) l)))) d_m))
   w0))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+15) {
		tmp = w0 * ((D_m * sqrt(((M_m * -0.25) * ((M_m * h) / l)))) / d_m);
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)) <= (-2d+15)) then
        tmp = w0 * ((d_m * sqrt(((m_m * (-0.25d0)) * ((m_m * h) / l)))) / d_m_1)
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+15) {
		tmp = w0 * ((D_m * Math.sqrt(((M_m * -0.25) * ((M_m * h) / l)))) / d_m);
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+15:
		tmp = w0 * ((D_m * math.sqrt(((M_m * -0.25) * ((M_m * h) / l)))) / d_m)
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e+15)
		tmp = Float64(w0 * Float64(Float64(D_m * sqrt(Float64(Float64(M_m * -0.25) * Float64(Float64(M_m * h) / l)))) / d_m));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -2e+15)
		tmp = w0 * ((D_m * sqrt(((M_m * -0.25) * ((M_m * h) / l)))) / d_m);
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+15], N[(w0 * N[(N[(D$95$m * N[Sqrt[N[(N[(M$95$m * -0.25), $MachinePrecision] * N[(N[(M$95$m * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision], w0]

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)) < -2e15

   1. Initial program 70.2%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.4

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

   5. Applied rewrites 48.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. sqrt-div N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. sqrt-pow1 N/A

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

      13. metadata-eval N/A

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

      14. unpow1 N/A

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

      15. lower-/.f64 N/A

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

   7. Applied rewrites 24.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 24.3

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

   9. Applied rewrites 31.3%

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

   if -2e15 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 97.6%

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

      1. *-rgt-identity 97.6

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

   7. Applied rewrites 97.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+15}:\\ \;\;\;\;w0 \cdot \frac{D \cdot \sqrt{\left(M \cdot -0.25\right) \cdot \frac{M \cdot h}{\ell}}}{d}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\left(D\_m \cdot \sqrt{\left(M\_m \cdot -0.25\right) \cdot \frac{M\_m \cdot h}{\ell}}\right) \cdot \frac{w0}{d\_m}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -2e+15)
   (* (* D_m (sqrt (* (* M_m -0.25) (/ (* M_m h) l)))) (/ w0 d_m))
   w0))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+15) {
		tmp = (D_m * sqrt(((M_m * -0.25) * ((M_m * h) / l)))) * (w0 / d_m);
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)) <= (-2d+15)) then
        tmp = (d_m * sqrt(((m_m * (-0.25d0)) * ((m_m * h) / l)))) * (w0 / d_m_1)
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+15) {
		tmp = (D_m * Math.sqrt(((M_m * -0.25) * ((M_m * h) / l)))) * (w0 / d_m);
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+15:
		tmp = (D_m * math.sqrt(((M_m * -0.25) * ((M_m * h) / l)))) * (w0 / d_m)
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e+15)
		tmp = Float64(Float64(D_m * sqrt(Float64(Float64(M_m * -0.25) * Float64(Float64(M_m * h) / l)))) * Float64(w0 / d_m));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -2e+15)
		tmp = (D_m * sqrt(((M_m * -0.25) * ((M_m * h) / l)))) * (w0 / d_m);
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+15], N[(N[(D$95$m * N[Sqrt[N[(N[(M$95$m * -0.25), $MachinePrecision] * N[(N[(M$95$m * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(w0 / d$95$m), $MachinePrecision]), $MachinePrecision], w0]

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)) < -2e15

   1. Initial program 70.2%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.4

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

   5. Applied rewrites 48.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. sqrt-div N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. sqrt-pow1 N/A

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

      13. metadata-eval N/A

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

      14. unpow1 N/A

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

      15. lower-/.f64 N/A

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

   7. Applied rewrites 24.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. div-inv N/A

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

      12. associate-*l* N/A

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

   9. Applied rewrites 30.1%

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

       1. associate-*l/ N/A

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

       2. *-lft-identity N/A

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

       3. lower-/.f64 30.0

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

   11. Applied rewrites 30.0%

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

   if -2e15 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 97.6%

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

      1. *-rgt-identity 97.6

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

   7. Applied rewrites 97.6%

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

Alternative 6: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m \cdot D\_m, \left(M\_m \cdot -0.125\right) \cdot \left(M\_m \cdot \frac{h}{d\_m \cdot \left(d\_m \cdot \ell\right)}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -5e+264)
   (*
    w0
    (fma (* D_m D_m) (* (* M_m -0.125) (* M_m (/ h (* d_m (* d_m l))))) 1.0))
   w0))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -5e+264) {
		tmp = w0 * fma((D_m * D_m), ((M_m * -0.125) * (M_m * (h / (d_m * (d_m * l))))), 1.0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -5e+264)
		tmp = Float64(w0 * fma(Float64(D_m * D_m), Float64(Float64(M_m * -0.125) * Float64(M_m * Float64(h / Float64(d_m * Float64(d_m * l))))), 1.0));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+264], N[(w0 * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * -0.125), $MachinePrecision] * N[(M$95$m * N[(h / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], w0]

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)) < -5.00000000000000033e264

   1. Initial program 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 53.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 57.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 61.7

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

   9. Applied rewrites 61.7%

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

   if -5.00000000000000033e264 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 91.8%

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

      1. *-rgt-identity 91.8

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

   7. Applied rewrites 91.8%

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

Alternative 7: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;\frac{-0.125 \cdot \left(\left(D\_m \cdot D\_m\right) \cdot \left(\left(M\_m \cdot M\_m\right) \cdot \left(h \cdot w0\right)\right)\right)}{\ell \cdot \left(d\_m \cdot d\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -5e+264)
   (/ (* -0.125 (* (* D_m D_m) (* (* M_m M_m) (* h w0)))) (* l (* d_m d_m)))
   w0))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -5e+264) {
		tmp = (-0.125 * ((D_m * D_m) * ((M_m * M_m) * (h * w0)))) / (l * (d_m * d_m));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)) <= (-5d+264)) then
        tmp = ((-0.125d0) * ((d_m * d_m) * ((m_m * m_m) * (h * w0)))) / (l * (d_m_1 * d_m_1))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -5e+264) {
		tmp = (-0.125 * ((D_m * D_m) * ((M_m * M_m) * (h * w0)))) / (l * (d_m * d_m));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -5e+264:
		tmp = (-0.125 * ((D_m * D_m) * ((M_m * M_m) * (h * w0)))) / (l * (d_m * d_m))
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -5e+264)
		tmp = Float64(Float64(-0.125 * Float64(Float64(D_m * D_m) * Float64(Float64(M_m * M_m) * Float64(h * w0)))) / Float64(l * Float64(d_m * d_m)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -5e+264)
		tmp = (-0.125 * ((D_m * D_m) * ((M_m * M_m) * (h * w0)))) / (l * (d_m * d_m));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+264], N[(N[(-0.125 * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(h * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

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)) < -5.00000000000000033e264

   1. Initial program 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in D around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 55.3

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

   10. Applied rewrites 55.3%

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

   if -5.00000000000000033e264 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 91.8%

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

      1. *-rgt-identity 91.8

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

   7. Applied rewrites 91.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;\frac{-0.125 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot w0\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;w0 \cdot \left(-0.125 \cdot \frac{\left(D\_m \cdot D\_m\right) \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)}{\ell \cdot \left(d\_m \cdot d\_m\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -5e+264)
   (* w0 (* -0.125 (/ (* (* D_m D_m) (* h (* M_m M_m))) (* l (* d_m d_m)))))
   w0))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -5e+264) {
		tmp = w0 * (-0.125 * (((D_m * D_m) * (h * (M_m * M_m))) / (l * (d_m * d_m))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)) <= (-5d+264)) then
        tmp = w0 * ((-0.125d0) * (((d_m * d_m) * (h * (m_m * m_m))) / (l * (d_m_1 * d_m_1))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -5e+264) {
		tmp = w0 * (-0.125 * (((D_m * D_m) * (h * (M_m * M_m))) / (l * (d_m * d_m))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -5e+264:
		tmp = w0 * (-0.125 * (((D_m * D_m) * (h * (M_m * M_m))) / (l * (d_m * d_m))))
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -5e+264)
		tmp = Float64(w0 * Float64(-0.125 * Float64(Float64(Float64(D_m * D_m) * Float64(h * Float64(M_m * M_m))) / Float64(l * Float64(d_m * d_m)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -5e+264)
		tmp = w0 * (-0.125 * (((D_m * D_m) * (h * (M_m * M_m))) / (l * (d_m * d_m))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+264], N[(w0 * N[(-0.125 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

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)) < -5.00000000000000033e264

   1. Initial program 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in D around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 53.8

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

   10. Applied rewrites 53.8%

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

   if -5.00000000000000033e264 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 90.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 91.8%

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

      1. *-rgt-identity 91.8

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

   7. Applied rewrites 91.8%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;w0 \cdot \left(-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.7% accurate, 157.0× speedup

Math

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

FPCore

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

C

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

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 68.9%

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

   1. *-rgt-identity 68.9

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

7. Applied rewrites 68.9%

   \[\leadsto \color{blue}{w0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))))))