Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.3% → 86.8%

Time: 14.0s

Alternatives: 13

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.3% 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: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\\ \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq 10^{+278}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{-0.5 \cdot \left(D\_m \cdot M\_m\right)}{d}, t\_0, 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(h \cdot -0.5\right) \cdot D\_m\right) \cdot M\_m, \frac{t\_0}{\ell \cdot d}, 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* D_m (* M_m (/ 0.5 d)))))
   (if (<= (- 1.0 (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0))) 1e+278)
     (* (sqrt (fma (* (/ h l) (/ (* -0.5 (* D_m M_m)) d)) t_0 1.0)) w0)
     (* (sqrt (fma (* (* (* h -0.5) D_m) M_m) (/ t_0 (* l d)) 1.0)) w0))))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M_m * (0.5 / d));
	double tmp;
	if ((1.0 - ((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0))) <= 1e+278) {
		tmp = sqrt(fma(((h / l) * ((-0.5 * (D_m * M_m)) / d)), t_0, 1.0)) * w0;
	} else {
		tmp = sqrt(fma((((h * -0.5) * D_m) * M_m), (t_0 / (l * d)), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(D_m * Float64(M_m * Float64(0.5 / d)))
	tmp = 0.0
	if (Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0))) <= 1e+278)
		tmp = Float64(sqrt(fma(Float64(Float64(h / l) * Float64(Float64(-0.5 * Float64(D_m * M_m)) / d)), t_0, 1.0)) * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(h * -0.5) * D_m) * M_m), Float64(t_0 / Float64(l * d)), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+278], N[(N[Sqrt[N[(N[(N[(h / l), $MachinePrecision] * N[(N[(-0.5 * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(h * -0.5), $MachinePrecision] * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(t$95$0 / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 9.99999999999999964e277

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 97.5%

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

   if 9.99999999999999964e277 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 44.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-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. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 61.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lower-*.f64 61.9

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 61.9

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

   6. Applied rewrites 61.9%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. times-frac N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-/l* N/A

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

   8. Applied rewrites 60.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq 10^{+278}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{-0.5 \cdot \left(D \cdot M\right)}{d}, D \cdot \left(M \cdot \frac{0.5}{d}\right), 1\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(h \cdot -0.5\right) \cdot D\right) \cdot M, \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell \cdot d}, 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\left(-0.25 \cdot h\right) \cdot \left(\left(\left(\frac{D\_m}{d} \cdot M\_m\right) \cdot \frac{M\_m}{\ell \cdot d}\right) \cdot D\_m\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -1e+19)
   (* (sqrt (* (* -0.25 h) (* (* (* (/ D_m d) M_m) (/ M_m (* l d))) D_m))) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -1e+19) {
		tmp = sqrt(((-0.25 * h) * ((((D_m / d) * M_m) * (M_m / (l * d))) * D_m))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((h / l) * (((d_m * m_m) / (2.0d0 * d)) ** 2.0d0)) <= (-1d+19)) then
        tmp = sqrt((((-0.25d0) * h) * ((((d_m / d) * m_m) * (m_m / (l * d))) * d_m))) * w0
    else
        tmp = 1.0d0 * w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * Math.pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -1e+19) {
		tmp = Math.sqrt(((-0.25 * h) * ((((D_m / d) * M_m) * (M_m / (l * d))) * D_m))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if ((h / l) * math.pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -1e+19:
		tmp = math.sqrt(((-0.25 * h) * ((((D_m / d) * M_m) * (M_m / (l * d))) * D_m))) * w0
	else:
		tmp = 1.0 * w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -1e+19)
		tmp = Float64(sqrt(Float64(Float64(-0.25 * h) * Float64(Float64(Float64(Float64(D_m / d) * M_m) * Float64(M_m / Float64(l * d))) * D_m))) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((h / l) * (((D_m * M_m) / (2.0 * d)) ^ 2.0)) <= -1e+19)
		tmp = sqrt(((-0.25 * h) * ((((D_m / d) * M_m) * (M_m / (l * d))) * D_m))) * w0;
	else
		tmp = 1.0 * w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e+19], N[(N[Sqrt[N[(N[(-0.25 * h), $MachinePrecision] * N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(M$95$m / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1e19

   1. Initial program 64.8%

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

   3. Taylor expanded in h 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 42.2%

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

   7. Applied rewrites 43.4%

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

   9. Applied rewrites 57.8%

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

   if -1e19 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.0%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 94.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\left(-0.25 \cdot h\right) \cdot \left(\left(\left(\frac{D}{d} \cdot M\right) \cdot \frac{M}{\ell \cdot d}\right) \cdot D\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\left(\left(\frac{D\_m \cdot M\_m}{\left(d \cdot d\right) \cdot \ell} \cdot M\_m\right) \cdot D\_m\right) \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -1e+19)
   (* (sqrt (* (* (* (/ (* D_m M_m) (* (* d d) l)) M_m) D_m) (* -0.25 h))) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -1e+19) {
		tmp = sqrt((((((D_m * M_m) / ((d * d) * l)) * M_m) * D_m) * (-0.25 * h))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((h / l) * (((d_m * m_m) / (2.0d0 * d)) ** 2.0d0)) <= (-1d+19)) then
        tmp = sqrt((((((d_m * m_m) / ((d * d) * l)) * m_m) * d_m) * ((-0.25d0) * h))) * w0
    else
        tmp = 1.0d0 * w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * Math.pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -1e+19) {
		tmp = Math.sqrt((((((D_m * M_m) / ((d * d) * l)) * M_m) * D_m) * (-0.25 * h))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if ((h / l) * math.pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -1e+19:
		tmp = math.sqrt((((((D_m * M_m) / ((d * d) * l)) * M_m) * D_m) * (-0.25 * h))) * w0
	else:
		tmp = 1.0 * w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -1e+19)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(D_m * M_m) / Float64(Float64(d * d) * l)) * M_m) * D_m) * Float64(-0.25 * h))) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((h / l) * (((D_m * M_m) / (2.0 * d)) ^ 2.0)) <= -1e+19)
		tmp = sqrt((((((D_m * M_m) / ((d * d) * l)) * M_m) * D_m) * (-0.25 * h))) * w0;
	else
		tmp = 1.0 * w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e+19], N[(N[Sqrt[N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(-0.25 * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1e19

   1. Initial program 64.8%

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

   3. Taylor expanded in h 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 42.2%

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

   7. Applied rewrites 43.4%

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

   9. Applied rewrites 49.6%

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

   if -1e19 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.0%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 94.1%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\left(\left(\frac{D \cdot M}{\left(d \cdot d\right) \cdot \ell} \cdot M\right) \cdot D\right) \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\frac{\left(\left(D\_m \cdot D\_m\right) \cdot M\_m\right) \cdot M\_m}{\left(d \cdot d\right) \cdot \ell} \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -1e+19)
   (* (sqrt (* (/ (* (* (* D_m D_m) M_m) M_m) (* (* d d) l)) (* -0.25 h))) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -1e+19) {
		tmp = sqrt((((((D_m * D_m) * M_m) * M_m) / ((d * d) * l)) * (-0.25 * h))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((h / l) * (((d_m * m_m) / (2.0d0 * d)) ** 2.0d0)) <= (-1d+19)) then
        tmp = sqrt((((((d_m * d_m) * m_m) * m_m) / ((d * d) * l)) * ((-0.25d0) * h))) * w0
    else
        tmp = 1.0d0 * w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
M_m = Math.abs(M);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * Math.pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -1e+19) {
		tmp = Math.sqrt((((((D_m * D_m) * M_m) * M_m) / ((d * d) * l)) * (-0.25 * h))) * w0;
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
M_m = math.fabs(M)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if ((h / l) * math.pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -1e+19:
		tmp = math.sqrt((((((D_m * D_m) * M_m) * M_m) / ((d * d) * l)) * (-0.25 * h))) * w0
	else:
		tmp = 1.0 * w0
	return tmp

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -1e+19)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(D_m * D_m) * M_m) * M_m) / Float64(Float64(d * d) * l)) * Float64(-0.25 * h))) * w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
M_m = abs(M);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((h / l) * (((D_m * M_m) / (2.0 * d)) ^ 2.0)) <= -1e+19)
		tmp = sqrt((((((D_m * D_m) * M_m) * M_m) / ((d * d) * l)) * (-0.25 * h))) * w0;
	else
		tmp = 1.0 * w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1e+19], N[(N[Sqrt[N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1e19

   1. Initial program 64.8%

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

   3. Taylor expanded in h 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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 42.2%

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

   6. Taylor expanded in M around 0

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

   8. Applied rewrites 46.1%

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

   if -1e19 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.0%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 94.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\frac{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}{\left(d \cdot d\right) \cdot \ell} \cdot \left(-0.25 \cdot h\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D\_m \cdot D\_m\right), \frac{\left(\frac{w0}{\ell \cdot d} \cdot M\_m\right) \cdot \left(h \cdot M\_m\right)}{d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -2e+130)
   (fma (* -0.125 (* D_m D_m)) (/ (* (* (/ w0 (* l d)) M_m) (* h M_m)) d) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -2e+130) {
		tmp = fma((-0.125 * (D_m * D_m)), ((((w0 / (l * d)) * M_m) * (h * M_m)) / d), w0);
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -2e+130)
		tmp = fma(Float64(-0.125 * Float64(D_m * D_m)), Float64(Float64(Float64(Float64(w0 / Float64(l * d)) * M_m) * Float64(h * M_m)) / d), w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -2e+130], N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(w0 / N[(l * d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(h * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] + w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.0000000000000001e130

   1. Initial program 61.2%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 37.2%

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

   10. Applied rewrites 40.8%

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

   12. Applied rewrites 46.7%

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

   if -2.0000000000000001e130 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.7%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 89.5%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{\left(\frac{w0}{\ell \cdot d} \cdot M\right) \cdot \left(h \cdot M\right)}{d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D\_m \cdot D\_m\right), \frac{w0}{\ell \cdot d} \cdot \frac{\left(h \cdot M\_m\right) \cdot M\_m}{d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -2e+130)
   (fma (* -0.125 (* D_m D_m)) (* (/ w0 (* l d)) (/ (* (* h M_m) M_m) d)) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -2e+130) {
		tmp = fma((-0.125 * (D_m * D_m)), ((w0 / (l * d)) * (((h * M_m) * M_m) / d)), w0);
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -2e+130)
		tmp = fma(Float64(-0.125 * Float64(D_m * D_m)), Float64(Float64(w0 / Float64(l * d)) * Float64(Float64(Float64(h * M_m) * M_m) / d)), w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -2e+130], N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.0000000000000001e130

   1. Initial program 61.2%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 37.2%

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

   10. Applied rewrites 40.8%

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

   12. Applied rewrites 45.4%

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

   if -2.0000000000000001e130 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.7%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 89.5%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{w0}{\ell \cdot d} \cdot \frac{\left(h \cdot M\right) \cdot M}{d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D\_m \cdot D\_m\right), \frac{\left(\left(h \cdot M\_m\right) \cdot M\_m\right) \cdot w0}{\left(\ell \cdot d\right) \cdot d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -2e+130)
   (fma (* -0.125 (* D_m D_m)) (/ (* (* (* h M_m) M_m) w0) (* (* l d) d)) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -2e+130) {
		tmp = fma((-0.125 * (D_m * D_m)), ((((h * M_m) * M_m) * w0) / ((l * d) * d)), w0);
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -2e+130)
		tmp = fma(Float64(-0.125 * Float64(D_m * D_m)), Float64(Float64(Float64(Float64(h * M_m) * M_m) * w0) / Float64(Float64(l * d) * d)), w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -2e+130], N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(h * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * w0), $MachinePrecision] / N[(N[(l * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.0000000000000001e130

   1. Initial program 61.2%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 37.2%

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

   10. Applied rewrites 44.1%

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

   if -2.0000000000000001e130 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.7%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 89.5%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), \frac{\left(\left(h \cdot M\right) \cdot M\right) \cdot w0}{\left(\ell \cdot d\right) \cdot d}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot w0, \frac{\left(\left(h \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)\right) \cdot D\_m}{\left(d \cdot d\right) \cdot \ell}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -2e+99)
   (fma (* -0.125 w0) (/ (* (* (* h M_m) (* D_m M_m)) D_m) (* (* d d) l)) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -2e+99) {
		tmp = fma((-0.125 * w0), ((((h * M_m) * (D_m * M_m)) * D_m) / ((d * d) * l)), w0);
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -2e+99)
		tmp = fma(Float64(-0.125 * w0), Float64(Float64(Float64(Float64(h * M_m) * Float64(D_m * M_m)) * D_m) / Float64(Float64(d * d) * l)), w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -2e+99], N[(N[(-0.125 * w0), $MachinePrecision] * N[(N[(N[(N[(h * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -1.9999999999999999e99

   1. Initial program 62.1%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 36.4%

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

   9. Taylor expanded in w0 around 0

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

   11. Applied rewrites 39.5%

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

   13. Applied rewrites 42.7%

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

   if -1.9999999999999999e99 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.5%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 90.5%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot w0, \frac{\left(\left(h \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D\_m \cdot M\_m}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot w0, \frac{\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m}{\left(d \cdot d\right) \cdot \ell} \cdot D\_m, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* D_m M_m) (* 2.0 d)) 2.0)) -2e+130)
   (fma (* -0.125 w0) (* (/ (* (* (* M_m M_m) h) D_m) (* (* d d) l)) D_m) w0)
   (* 1.0 w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((h / l) * pow(((D_m * M_m) / (2.0 * d)), 2.0)) <= -2e+130) {
		tmp = fma((-0.125 * w0), (((((M_m * M_m) * h) * D_m) / ((d * d) * l)) * D_m), w0);
	} else {
		tmp = 1.0 * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) ^ 2.0)) <= -2e+130)
		tmp = fma(Float64(-0.125 * w0), Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) / Float64(Float64(d * d) * l)) * D_m), w0);
	else
		tmp = Float64(1.0 * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -2e+130], N[(N[(-0.125 * w0), $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] + w0), $MachinePrecision], N[(1.0 * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -2.0000000000000001e130

   1. Initial program 61.2%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in h around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 37.2%

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

   9. Taylor expanded in w0 around 0

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

   11. Applied rewrites 40.2%

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

   13. Applied rewrites 41.4%

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

   if -2.0000000000000001e130 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.7%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 89.5%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \leq -2 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot w0, \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot D, w0\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \sqrt{\mathsf{fma}\left(\frac{D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)}{\ell}, \frac{\frac{-0.5 \cdot \left(D\_m \cdot M\_m\right)}{d}}{{h}^{-1}}, 1\right)} \cdot w0 \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (*
  (sqrt
   (fma
    (/ (* D_m (* M_m (/ 0.5 d))) l)
    (/ (/ (* -0.5 (* D_m M_m)) d) (pow h -1.0))
    1.0))
  w0))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return sqrt(fma(((D_m * (M_m * (0.5 / d))) / l), (((-0.5 * (D_m * M_m)) / d) / pow(h, -1.0)), 1.0)) * w0;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	return Float64(sqrt(fma(Float64(Float64(D_m * Float64(M_m * Float64(0.5 / d))) / l), Float64(Float64(Float64(-0.5 * Float64(D_m * M_m)) / d) / (h ^ -1.0)), 1.0)) * w0)
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := N[(N[Sqrt[N[(N[(N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(-0.5 * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-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. lift-*.f64 N/A

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

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

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. div-inv N/A

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

   13. times-frac N/A

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 84.4%

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

5. Final simplification 84.4%

   \[\leadsto \sqrt{\mathsf{fma}\left(\frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}, \frac{\frac{-0.5 \cdot \left(D \cdot M\right)}{d}}{{h}^{-1}}, 1\right)} \cdot w0 \]
6. Add Preprocessing

Alternative 11: 76.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{D\_m \cdot M\_m}{2 \cdot d} \leq 2 \cdot 10^{-202}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(-0.5 \cdot D\_m\right) \cdot M\_m}{\ell \cdot d} \cdot h, D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right), 1\right)} \cdot w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (/ (* D_m M_m) (* 2.0 d)) 2e-202)
   (* 1.0 w0)
   (*
    (sqrt
     (fma
      (* (/ (* (* -0.5 D_m) M_m) (* l d)) h)
      (* D_m (* M_m (/ 0.5 d)))
      1.0))
    w0)))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (((D_m * M_m) / (2.0 * d)) <= 2e-202) {
		tmp = 1.0 * w0;
	} else {
		tmp = sqrt(fma(((((-0.5 * D_m) * M_m) / (l * d)) * h), (D_m * (M_m * (0.5 / d))), 1.0)) * w0;
	}
	return tmp;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(Float64(D_m * M_m) / Float64(2.0 * d)) <= 2e-202)
		tmp = Float64(1.0 * w0);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(Float64(-0.5 * D_m) * M_m) / Float64(l * d)) * h), Float64(D_m * Float64(M_m * Float64(0.5 / d))), 1.0)) * w0);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2e-202], N[(1.0 * w0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(-0.5 * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 2.0000000000000001e-202

   1. Initial program 80.6%

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

   3. Taylor expanded in h around 0

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

   5. Applied rewrites 68.1%

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

   if 2.0000000000000001e-202 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 76.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 78.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 80.8

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

   6. Applied rewrites 80.8%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{D \cdot M}{2 \cdot d} \leq 2 \cdot 10^{-202}:\\ \;\;\;\;1 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(-0.5 \cdot D\right) \cdot M}{\ell \cdot d} \cdot h, D \cdot \left(M \cdot \frac{0.5}{d}\right), 1\right)} \cdot w0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 88.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ M_m = \left|M\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \sqrt{\mathsf{fma}\left(\frac{D\_m \cdot \left(M\_m \cdot \frac{0.5}{d}\right)}{\ell}, \left(\frac{-0.5}{d} \cdot \left(D\_m \cdot M\_m\right)\right) \cdot h, 1\right)} \cdot w0 \end{array} \]

FPCore

D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (*
  (sqrt
   (fma (/ (* D_m (* M_m (/ 0.5 d))) l) (* (* (/ -0.5 d) (* D_m M_m)) h) 1.0))
  w0))

C

D_m = fabs(D);
M_m = fabs(M);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return sqrt(fma(((D_m * (M_m * (0.5 / d))) / l), (((-0.5 / d) * (D_m * M_m)) * h), 1.0)) * w0;
}

Julia

D_m = abs(D)
M_m = abs(M)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	return Float64(sqrt(fma(Float64(Float64(D_m * Float64(M_m * Float64(0.5 / d))) / l), Float64(Float64(Float64(-0.5 / d) * Float64(D_m * M_m)) * h), 1.0)) * w0)
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := N[(N[Sqrt[N[(N[(N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(-0.5 / d), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-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. lift-*.f64 N/A

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

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

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. div-inv N/A

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

   13. times-frac N/A

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 84.4%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow-1 N/A

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

   5. remove-double-div N/A

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

   6. lower-*.f64 84.3

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-*.f64 84.3

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

6. Applied rewrites 84.3%

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

7. Final simplification 84.3%

   \[\leadsto \sqrt{\mathsf{fma}\left(\frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}, \left(\frac{-0.5}{d} \cdot \left(D \cdot M\right)\right) \cdot h, 1\right)} \cdot w0 \]
8. Add Preprocessing

Alternative 13: 67.0% accurate, 26.2× speedup

Math

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

FPCore

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

C

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

Fortran

D_m = abs(d)
M_m = abs(m)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = 1.0d0 * w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Taylor expanded in h around 0

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

5. Applied rewrites 60.3%

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

6. Final simplification 60.3%

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

Reproduce

herbie shell --seed 2024249 
(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))))))