Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.7% → 88.8%

Time: 13.3s

Alternatives: 13

Speedup: 1.6×

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: 80.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{M \cdot D}{d \cdot 2}\\ \mathbf{if}\;1 - {t\_0}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{\frac{M \cdot D}{d \cdot -2}}{\frac{\ell}{h}}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{M}{d} \cdot \left(D \cdot -0.5\right)}{\ell}, \left(M \cdot 0.5\right) \cdot \left(h \cdot \frac{D}{d}\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* d 2.0))))
   (if (<= (- 1.0 (* (pow t_0 2.0) (/ h l))) 5e+293)
     (* w0 (sqrt (fma t_0 (/ (/ (* M D) (* d -2.0)) (/ l h)) 1.0)))
     (*
      w0
      (sqrt
       (fma (/ (* (/ M d) (* D -0.5)) l) (* (* M 0.5) (* h (/ D d))) 1.0))))))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / (d * 2.0);
	double tmp;
	if ((1.0 - (pow(t_0, 2.0) * (h / l))) <= 5e+293) {
		tmp = w0 * sqrt(fma(t_0, (((M * D) / (d * -2.0)) / (l / h)), 1.0));
	} else {
		tmp = w0 * sqrt(fma((((M / d) * (D * -0.5)) / l), ((M * 0.5) * (h * (D / d))), 1.0));
	}
	return tmp;
}

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * D) / Float64(d * 2.0))
	tmp = 0.0
	if (Float64(1.0 - Float64((t_0 ^ 2.0) * Float64(h / l))) <= 5e+293)
		tmp = Float64(w0 * sqrt(fma(t_0, Float64(Float64(Float64(M * D) / Float64(d * -2.0)) / Float64(l / h)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(M / d) * Float64(D * -0.5)) / l), Float64(Float64(M * 0.5) * Float64(h * Float64(D / d))), 1.0)));
	end
	return tmp
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 - N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+293], N[(w0 * N[Sqrt[N[(t$95$0 * N[(N[(N[(M * D), $MachinePrecision] / N[(d * -2.0), $MachinePrecision]), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(N[(M / d), $MachinePrecision] * N[(D * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M * 0.5), $MachinePrecision] * N[(h * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $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))) < 5.00000000000000033e293

   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

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 99.9%

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

   if 5.00000000000000033e293 < (-.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 43.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied rewrites 70.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. /-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 67.6

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

   6. Applied rewrites 67.6%

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 71.4

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

   8. Applied rewrites 71.4%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{\frac{M \cdot D}{d \cdot -2}}{\frac{\ell}{h}}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{M}{d} \cdot \left(D \cdot -0.5\right)}{\ell}, \left(M \cdot 0.5\right) \cdot \left(h \cdot \frac{D}{d}\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{M \cdot D}{d \cdot 2}\\ \mathbf{if}\;1 - {t\_0}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{M \cdot D}{d \cdot -2} \cdot \frac{h}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{M}{d} \cdot \left(D \cdot -0.5\right)}{\ell}, \left(M \cdot 0.5\right) \cdot \left(h \cdot \frac{D}{d}\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* d 2.0))))
   (if (<= (- 1.0 (* (pow t_0 2.0) (/ h l))) 5e+293)
     (* w0 (sqrt (fma t_0 (* (/ (* M D) (* d -2.0)) (/ h l)) 1.0)))
     (*
      w0
      (sqrt
       (fma (/ (* (/ M d) (* D -0.5)) l) (* (* M 0.5) (* h (/ D d))) 1.0))))))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / (d * 2.0);
	double tmp;
	if ((1.0 - (pow(t_0, 2.0) * (h / l))) <= 5e+293) {
		tmp = w0 * sqrt(fma(t_0, (((M * D) / (d * -2.0)) * (h / l)), 1.0));
	} else {
		tmp = w0 * sqrt(fma((((M / d) * (D * -0.5)) / l), ((M * 0.5) * (h * (D / d))), 1.0));
	}
	return tmp;
}

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * D) / Float64(d * 2.0))
	tmp = 0.0
	if (Float64(1.0 - Float64((t_0 ^ 2.0) * Float64(h / l))) <= 5e+293)
		tmp = Float64(w0 * sqrt(fma(t_0, Float64(Float64(Float64(M * D) / Float64(d * -2.0)) * Float64(h / l)), 1.0)));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(M / d) * Float64(D * -0.5)) / l), Float64(Float64(M * 0.5) * Float64(h * Float64(D / d))), 1.0)));
	end
	return tmp
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 - N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+293], N[(w0 * N[Sqrt[N[(t$95$0 * N[(N[(N[(M * D), $MachinePrecision] / N[(d * -2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(N[(M / d), $MachinePrecision] * N[(D * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M * 0.5), $MachinePrecision] * N[(h * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $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))) < 5.00000000000000033e293

   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

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 99.9%

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

   if 5.00000000000000033e293 < (-.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 43.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied rewrites 70.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. /-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 67.6

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

   6. Applied rewrites 67.6%

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 71.4

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

   8. Applied rewrites 71.4%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, \frac{M \cdot D}{d \cdot -2} \cdot \frac{h}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{M}{d} \cdot \left(D \cdot -0.5\right)}{\ell}, \left(M \cdot 0.5\right) \cdot \left(h \cdot \frac{D}{d}\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 - N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(w0 * N[Sqrt[N[(t$95$0 * N[(N[(N[(M * D), $MachinePrecision] / N[(d * -2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(M * D), $MachinePrecision] / N[(N[(d * -2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(N[(M * D), $MachinePrecision] * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $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))) < +inf.0

   1. Initial program 90.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 91.9%

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

   if +inf.0 < (-.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 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied rewrites 76.7%

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

   6. Applied rewrites 76.3%

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

4. Final simplification 90.6%

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

Alternative 4: 86.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 1.0002:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{\left(d \cdot -2\right) \cdot \ell}, h \cdot \frac{\left(M \cdot D\right) \cdot 0.5}{d}, 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (- 1.0 (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l))) 1.0002)
   (* w0 1.0)
   (*
    w0
    (sqrt
     (fma (/ (* M D) (* (* d -2.0) l)) (* h (/ (* (* M D) 0.5) d)) 1.0)))))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((1.0 - (pow(((M * D) / (d * 2.0)), 2.0) * (h / l))) <= 1.0002) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * sqrt(fma(((M * D) / ((d * -2.0) * l)), (h * (((M * D) * 0.5) / d)), 1.0));
	}
	return tmp;
}

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l))) <= 1.0002)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M * D) / Float64(Float64(d * -2.0) * l)), Float64(h * Float64(Float64(Float64(M * D) * 0.5) / d)), 1.0)));
	end
	return tmp
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0002], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(M * D), $MachinePrecision] / N[(N[(d * -2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(N[(M * D), $MachinePrecision] * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 1.0002

   1. Initial program 100.0%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 99.8%

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

   if 1.0002 < (-.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 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied rewrites 75.9%

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

   6. Applied rewrites 72.3%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 1.0002:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{\left(d \cdot -2\right) \cdot \ell}, h \cdot \frac{\left(M \cdot D\right) \cdot 0.5}{d}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-11}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\left(M \cdot D\right) \cdot \frac{M \cdot D}{\left(d \cdot d\right) \cdot 4}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -1e-11)
   (* w0 (sqrt (- 1.0 (* (/ h l) (* (* M D) (/ (* M D) (* (* d d) 4.0)))))))
   (* w0 1.0)))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -1e-11) {
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M * D) * ((M * D) / ((d * d) * 4.0))))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (((((m * d) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l)) <= (-1d-11)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((m * d) * ((m * d) / ((d_1 * d_1) * 4.0d0))))))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -1e-11) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * ((M * D) * ((M * D) / ((d * d) * 4.0))))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -1e-11:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * ((M * D) * ((M * D) / ((d * d) * 4.0))))))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= -1e-11)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(M * D) * Float64(Float64(M * D) / Float64(Float64(d * d) * 4.0)))))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (d * 2.0)) ^ 2.0) * (h / l)) <= -1e-11)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M * D) * ((M * D) / ((d * d) * 4.0))))));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e-11], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -9.99999999999999939e-12

   1. Initial program 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. swap-sqr N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 61.8

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

   4. Applied rewrites 61.8%

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

   if -9.99999999999999939e-12 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.5%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{-11}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\left(M \cdot D\right) \cdot \frac{M \cdot D}{\left(d \cdot d\right) \cdot 4}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+15}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot \left(h \cdot \frac{-1}{\left(d \cdot \ell\right) \cdot \left(d \cdot 4\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -2e+15)
   (*
    w0
    (sqrt
     (+ 1.0 (* (* M D) (* (* M D) (* h (/ -1.0 (* (* d l) (* d 4.0)))))))))
   (* w0 1.0)))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -2e+15) {
		tmp = w0 * sqrt((1.0 + ((M * D) * ((M * D) * (h * (-1.0 / ((d * l) * (d * 4.0))))))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (((((m * d) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l)) <= (-2d+15)) then
        tmp = w0 * sqrt((1.0d0 + ((m * d) * ((m * d) * (h * ((-1.0d0) / ((d_1 * l) * (d_1 * 4.0d0))))))))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -2e+15) {
		tmp = w0 * Math.sqrt((1.0 + ((M * D) * ((M * D) * (h * (-1.0 / ((d * l) * (d * 4.0))))))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -2e+15:
		tmp = w0 * math.sqrt((1.0 + ((M * D) * ((M * D) * (h * (-1.0 / ((d * l) * (d * 4.0))))))))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= -2e+15)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(M * D) * Float64(Float64(M * D) * Float64(h * Float64(-1.0 / Float64(Float64(d * l) * Float64(d * 4.0)))))))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (d * 2.0)) ^ 2.0) * (h / l)) <= -2e+15)
		tmp = w0 * sqrt((1.0 + ((M * D) * ((M * D) * (h * (-1.0 / ((d * l) * (d * 4.0))))))));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+15], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(M * D), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(h * N[(-1.0 / N[(N[(d * l), $MachinePrecision] * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 70.9

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

   4. Applied rewrites 61.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

   6. Applied rewrites 61.9%

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

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

   1. Initial program 88.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.2%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+15}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot \left(h \cdot \frac{-1}{\left(d \cdot \ell\right) \cdot \left(d \cdot 4\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+15}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(M \cdot D\right) \cdot \left(-M \cdot D\right), \frac{h}{d \cdot \left(4 \cdot \left(d \cdot \ell\right)\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -2e+15)
   (* w0 (sqrt (fma (* (* M D) (- (* M D))) (/ h (* d (* 4.0 (* d l)))) 1.0)))
   (* w0 1.0)))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -2e+15) {
		tmp = w0 * sqrt(fma(((M * D) * -(M * D)), (h / (d * (4.0 * (d * l)))), 1.0));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= -2e+15)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(M * D) * Float64(-Float64(M * D))), Float64(h / Float64(d * Float64(4.0 * Float64(d * l)))), 1.0)));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+15], N[(w0 * N[Sqrt[N[(N[(N[(M * D), $MachinePrecision] * (-N[(M * D), $MachinePrecision])), $MachinePrecision] * N[(h / N[(d * N[(4.0 * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 70.9

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

   4. Applied rewrites 61.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

   6. Applied rewrites 61.9%

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

   8. Applied rewrites 59.9%

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

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

   1. Initial program 88.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.2%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+15}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(M \cdot D\right) \cdot \left(-M \cdot D\right), \frac{h}{d \cdot \left(4 \cdot \left(d \cdot \ell\right)\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -500000000:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot \ell\right) \cdot \left(d \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -500000000.0)
   (* w0 (sqrt (- 1.0 (* h (/ (* M (* D (* M D))) (* (* d l) (* d 4.0)))))))
   (* w0 1.0)))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -500000000.0) {
		tmp = w0 * sqrt((1.0 - (h * ((M * (D * (M * D))) / ((d * l) * (d * 4.0))))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (((((m * d) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l)) <= (-500000000.0d0)) then
        tmp = w0 * sqrt((1.0d0 - (h * ((m * (d * (m * d))) / ((d_1 * l) * (d_1 * 4.0d0))))))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -500000000.0) {
		tmp = w0 * Math.sqrt((1.0 - (h * ((M * (D * (M * D))) / ((d * l) * (d * 4.0))))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -500000000.0:
		tmp = w0 * math.sqrt((1.0 - (h * ((M * (D * (M * D))) / ((d * l) * (d * 4.0))))))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= -500000000.0)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(M * Float64(D * Float64(M * D))) / Float64(Float64(d * l) * Float64(d * 4.0)))))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (d * 2.0)) ^ 2.0) * (h / l)) <= -500000000.0)
		tmp = w0 * sqrt((1.0 - (h * ((M * (D * (M * D))) / ((d * l) * (d * 4.0))))));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -500000000.0], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(M * N[(D * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * l), $MachinePrecision] * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5e8

   1. Initial program 71.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 71.3

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

   4. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 55.4%

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

   if -5e8 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.6%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -500000000:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot \ell\right) \cdot \left(d \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.8% accurate, 0.8× speedup

Math

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

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -20.0)
   (* w0 (sqrt (/ (* (* M (* M D)) (* D (* h -0.25))) (* d (* d l)))))
   (* w0 1.0)))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -20.0) {
		tmp = w0 * sqrt((((M * (M * D)) * (D * (h * -0.25))) / (d * (d * l))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (((((m * d) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l)) <= (-20.0d0)) then
        tmp = w0 * sqrt((((m * (m * d)) * (d * (h * (-0.25d0)))) / (d_1 * (d_1 * l))))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -20.0) {
		tmp = w0 * Math.sqrt((((M * (M * D)) * (D * (h * -0.25))) / (d * (d * l))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -20.0:
		tmp = w0 * math.sqrt((((M * (M * D)) * (D * (h * -0.25))) / (d * (d * l))))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= -20.0)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(M * Float64(M * D)) * Float64(D * Float64(h * -0.25))) / Float64(d * Float64(d * l)))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (d * 2.0)) ^ 2.0) * (h / l)) <= -20.0)
		tmp = w0 * sqrt((((M * (M * D)) * (D * (h * -0.25))) / (d * (d * l))));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -20.0], N[(w0 * N[Sqrt[N[(N[(N[(M * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(D * N[(h * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -20

   1. Initial program 71.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 71.6

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

   4. Applied rewrites 59.6%

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

   5. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 46.1

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

   7. Applied rewrites 46.1%

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

   9. Applied rewrites 49.9%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-approx N/A

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

       3. lift-*.f64 49.9

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

   11. Applied rewrites 47.8%

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

   if -20 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.1%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -20:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\left(M \cdot \left(M \cdot D\right)\right) \cdot \left(D \cdot \left(h \cdot -0.25\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -500000000:\\ \;\;\;\;w0 \cdot \sqrt{\left(D \cdot D\right) \cdot \frac{-0.25 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -500000000.0)
   (* w0 (sqrt (* (* D D) (/ (* -0.25 (* h (* M M))) (* d (* d l))))))
   (* w0 1.0)))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -500000000.0) {
		tmp = w0 * sqrt(((D * D) * ((-0.25 * (h * (M * M))) / (d * (d * l)))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (((((m * d) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l)) <= (-500000000.0d0)) then
        tmp = w0 * sqrt(((d * d) * (((-0.25d0) * (h * (m * m))) / (d_1 * (d_1 * l)))))
    else
        tmp = w0 * 1.0d0
    end if
    code = tmp
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -500000000.0) {
		tmp = w0 * Math.sqrt(((D * D) * ((-0.25 * (h * (M * M))) / (d * (d * l)))));
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -500000000.0:
		tmp = w0 * math.sqrt(((D * D) * ((-0.25 * (h * (M * M))) / (d * (d * l)))))
	else:
		tmp = w0 * 1.0
	return tmp

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= -500000000.0)
		tmp = Float64(w0 * sqrt(Float64(Float64(D * D) * Float64(Float64(-0.25 * Float64(h * Float64(M * M))) / Float64(d * Float64(d * l))))));
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

MATLAB

w0, M, D, h, l, d = num2cell(sort([w0, M, D, h, l, d])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (d * 2.0)) ^ 2.0) * (h / l)) <= -500000000.0)
		tmp = w0 * sqrt(((D * D) * ((-0.25 * (h * (M * M))) / (d * (d * l)))));
	else
		tmp = w0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -500000000.0], N[(w0 * N[Sqrt[N[(N[(D * D), $MachinePrecision] * N[(N[(-0.25 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -5e8

   1. Initial program 71.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 71.3

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 46.7

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

   7. Applied rewrites 46.7%

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

   if -5e8 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 88.3%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.6%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -500000000:\\ \;\;\;\;w0 \cdot \sqrt{\left(D \cdot D\right) \cdot \frac{-0.25 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [w0, M, D, h, l, d] = \mathsf{sort}([w0, M, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{-0.125 \cdot \left(h \cdot \left(M \cdot \left(w0 \cdot M\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)) -2e+15)
   (fma (* D D) (/ (* -0.125 (* h (* M (* w0 M)))) (* l (* d d))) w0)
   (* w0 1.0)))

C

assert(w0 < M && M < D && D < h && h < l && l < d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (d * 2.0)), 2.0) * (h / l)) <= -2e+15) {
		tmp = fma((D * D), ((-0.125 * (h * (M * (w0 * M)))) / (l * (d * d))), w0);
	} else {
		tmp = w0 * 1.0;
	}
	return tmp;
}

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= -2e+15)
		tmp = fma(Float64(D * D), Float64(Float64(-0.125 * Float64(h * Float64(M * Float64(w0 * M)))) / Float64(l * Float64(d * d))), w0);
	else
		tmp = Float64(w0 * 1.0);
	end
	return tmp
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+15], N[(N[(D * D), $MachinePrecision] * N[(N[(-0.125 * N[(h * N[(M * N[(w0 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 70.9%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. 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. lower-fma.f64 N/A

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

   5. Applied rewrites 42.7%

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

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

   1. Initial program 88.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 95.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{-0.125 \cdot \left(h \cdot \left(M \cdot \left(w0 \cdot M\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 88.8% accurate, 1.6× speedup

Math

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

FPCore

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (*
  w0
  (sqrt
   (fma
    (/ (/ (* M D) (* d -2.0)) l)
    (/ (/ (* M D) (* d 2.0)) (/ 1.0 h))
    1.0))))

C

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

Julia

w0, M, D, h, l, d = sort([w0, M, D, h, l, d])
function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(fma(Float64(Float64(Float64(M * D) / Float64(d * -2.0)) / l), Float64(Float64(Float64(M * D) / Float64(d * 2.0)) / Float64(1.0 / h)), 1.0)))
end

Wolfram

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(N[(N[(N[(M * D), $MachinePrecision] / N[(d * -2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 / h), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

4. Applied rewrites 90.7%

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

5. Final simplification 90.7%

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

Alternative 13: 68.2% accurate, 26.2× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: w0, M, D, h, l, and d should be sorted in increasing order before calling this function.
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 * 1.0d0
end function

Java

assert w0 < M && M < D && D < h && h < l && l < d;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * 1.0;
}

Python

[w0, M, D, h, l, d] = sort([w0, M, D, h, l, d])
def code(w0, M, D, h, l, d):
	return w0 * 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 68.9%

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

Reproduce

herbie shell --seed 2024212 
(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))))))