Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 89.2%

Time: 14.5s

Alternatives: 15

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{2 \cdot d\_m}\\ \mathbf{if}\;t\_0 \leq 10^{-151}:\\ \;\;\;\;w0\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+257}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{M\_m \cdot D\_m}{d\_m \cdot -2} \cdot \frac{h}{\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|M\_m\right| \cdot \left(D\_m \cdot \left(w0 \cdot \sqrt{\frac{h \cdot -0.25}{d\_m \cdot \left(d\_m \cdot \ell\right)}}\right)\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ (* M_m D_m) (* 2.0 d_m))))
   (if (<= t_0 1e-151)
     w0
     (if (<= t_0 5e+257)
       (* w0 (sqrt (fma t_0 (* (/ (* M_m D_m) (* d_m -2.0)) (/ h l)) 1.0)))
       (*
        (fabs M_m)
        (* D_m (* w0 (sqrt (/ (* h -0.25) (* d_m (* d_m l)))))))))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * D_m) / (2.0 * d_m);
	double tmp;
	if (t_0 <= 1e-151) {
		tmp = w0;
	} else if (t_0 <= 5e+257) {
		tmp = w0 * sqrt(fma(t_0, (((M_m * D_m) / (d_m * -2.0)) * (h / l)), 1.0));
	} else {
		tmp = fabs(M_m) * (D_m * (w0 * sqrt(((h * -0.25) / (d_m * (d_m * l))))));
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(M_m * D_m) / Float64(2.0 * d_m))
	tmp = 0.0
	if (t_0 <= 1e-151)
		tmp = w0;
	elseif (t_0 <= 5e+257)
		tmp = Float64(w0 * sqrt(fma(t_0, Float64(Float64(Float64(M_m * D_m) / Float64(d_m * -2.0)) * Float64(h / l)), 1.0)));
	else
		tmp = Float64(abs(M_m) * Float64(D_m * Float64(w0 * sqrt(Float64(Float64(h * -0.25) / Float64(d_m * Float64(d_m * l)))))));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-151], w0, If[LessEqual[t$95$0, 5e+257], N[(w0 * N[Sqrt[N[(t$95$0 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[M$95$m], $MachinePrecision] * N[(D$95$m * N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 9.9999999999999994e-152

   1. Initial program 85.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 80.6%

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

      1. *-rgt-identity 80.6

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

   7. Applied rewrites 80.6%

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

   if 9.9999999999999994e-152 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 5.00000000000000028e257

   1. Initial program 92.1%

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

      1. lift-*.f64 N/A

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

      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

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

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

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

      14. associate-*l* N/A

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

   4. Applied rewrites 95.0%

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

   if 5.00000000000000028e257 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 58.3%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 54.0

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

   5. Applied rewrites 54.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 54.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 54.0

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

   9. Applied rewrites 63.1%

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

       1. lift-*.f64 N/A

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

       2. lift-fabs.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. associate-*l* N/A

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

       9. lift-fabs.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. fabs-mul N/A

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

       12. associate-*l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-fabs.f64 N/A

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

   11. Applied rewrites 48.2%

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

Alternative 2: 71.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2}} \leq 4 \cdot 10^{+270}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D\_m \cdot D\_m, \left(M\_m \cdot \left(M\_m \cdot h\right)\right) \cdot \frac{-0.125}{d\_m \cdot \left(d\_m \cdot \ell\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<=
      (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0)))))
      4e+270)
   w0
   (*
    w0
    (fma (* D_m D_m) (* (* M_m (* M_m h)) (/ -0.125 (* d_m (* d_m l)))) 1.0))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((w0 * sqrt((1.0 - ((h / l) * pow(((M_m * D_m) / (2.0 * d_m)), 2.0))))) <= 4e+270) {
		tmp = w0;
	} else {
		tmp = w0 * fma((D_m * D_m), ((M_m * (M_m * h)) * (-0.125 / (d_m * (d_m * l)))), 1.0);
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0))))) <= 4e+270)
		tmp = w0;
	else
		tmp = Float64(w0 * fma(Float64(D_m * D_m), Float64(Float64(M_m * Float64(M_m * h)) * Float64(-0.125 / Float64(d_m * Float64(d_m * l)))), 1.0));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e+270], w0, N[(w0 * N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision] * N[(-0.125 / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.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))))) < 4.0000000000000002e270

   1. Initial program 96.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 81.5%

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

      1. *-rgt-identity 81.5

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

   7. Applied rewrites 81.5%

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

   if 4.0000000000000002e270 < (*.f64 w0 (sqrt.f64 (-.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 40.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. sub-neg N/A

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

      8. +-commutative N/A

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 70.4

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

   7. Applied rewrites 70.4%

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

   8. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   10. Applied rewrites 59.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*l* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-/.f64 57.5

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

   12. Applied rewrites 57.5%

       \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \color{blue}{\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{-0.125}{d \cdot \left(d \cdot \ell\right)}}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \leq 4 \cdot 10^{+270}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D \cdot D, \left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{-0.125}{d \cdot \left(d \cdot \ell\right)}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \leq -40000:\\ \;\;\;\;w0 \cdot \left(\left|M\_m \cdot D\_m\right| \cdot \sqrt{-0.25 \cdot \frac{\frac{h}{d\_m \cdot \ell}}{d\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0)) -40000.0)
   (* w0 (* (fabs (* M_m D_m)) (sqrt (* -0.25 (/ (/ h (* d_m l)) d_m)))))
   w0))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) <= -40000.0) {
		tmp = w0 * (fabs((M_m * D_m)) * sqrt((-0.25 * ((h / (d_m * l)) / d_m))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((h / l) * (((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0)) <= (-40000.0d0)) then
        tmp = w0 * (abs((m_m * d_m)) * sqrt(((-0.25d0) * ((h / (d_m_1 * l)) / d_m_1))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
M_m = Math.abs(M);
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) <= -40000.0) {
		tmp = w0 * (Math.abs((M_m * D_m)) * Math.sqrt((-0.25 * ((h / (d_m * l)) / d_m))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if ((h / l) * math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) <= -40000.0:
		tmp = w0 * (math.fabs((M_m * D_m)) * math.sqrt((-0.25 * ((h / (d_m * l)) / d_m))))
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0)) <= -40000.0)
		tmp = Float64(w0 * Float64(abs(Float64(M_m * D_m)) * sqrt(Float64(-0.25 * Float64(Float64(h / Float64(d_m * l)) / d_m)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((h / l) * (((M_m * D_m) / (2.0 * d_m)) ^ 2.0)) <= -40000.0)
		tmp = w0 * (abs((M_m * D_m)) * sqrt((-0.25 * ((h / (d_m * l)) / d_m))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -40000.0], N[(w0 * N[(N[Abs[N[(M$95$m * D$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(-0.25 * N[(N[(h / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4e4

   1. Initial program 77.9%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.3

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

   5. Applied rewrites 48.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 56.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 56.8

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

   9. Applied rewrites 70.4%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 75.3

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

   11. Applied rewrites 75.3%

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

   if -4e4 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 91.5%

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

      1. *-rgt-identity 91.5

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

   7. Applied rewrites 91.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -40000:\\ \;\;\;\;w0 \cdot \left(\left|M \cdot D\right| \cdot \sqrt{-0.25 \cdot \frac{\frac{h}{d \cdot \ell}}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.4% accurate, 0.8× speedup

Math

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

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (- 1.0 (* (/ h l) (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0))) 2.0)
   w0
   (* (fabs M_m) (* D_m (* w0 (sqrt (/ (* h -0.25) (* d_m (* d_m l)))))))))

C

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

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if ((1.0d0 - ((h / l) * (((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0))) <= 2.0d0) then
        tmp = w0
    else
        tmp = abs(m_m) * (d_m * (w0 * sqrt(((h * (-0.25d0)) / (d_m_1 * (d_m_1 * l))))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((1.0 - ((h / l) * (((M_m * D_m) / (2.0 * d_m)) ^ 2.0))) <= 2.0)
		tmp = w0;
	else
		tmp = abs(M_m) * (D_m * (w0 * sqrt(((h * -0.25) / (d_m * (d_m * l))))));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], w0, N[(N[Abs[M$95$m], $MachinePrecision] * N[(D$95$m * N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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))) < 2

   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 99.1%

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

      1. *-rgt-identity 99.1

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

   7. Applied rewrites 99.1%

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

   if 2 < (-.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 54.0%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 38.7

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

   5. Applied rewrites 38.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 46.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 46.8

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

   9. Applied rewrites 60.5%

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

       1. lift-*.f64 N/A

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

       2. lift-fabs.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. associate-*l* N/A

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

       9. lift-fabs.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. fabs-mul N/A

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

       12. associate-*l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-fabs.f64 N/A

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

   11. Applied rewrites 29.7%

       \[\leadsto \color{blue}{\left|M\right| \cdot \left(D \cdot \left(w0 \cdot \sqrt{\frac{h \cdot -0.25}{d \cdot \left(d \cdot \ell\right)}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.3%

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

Alternative 5: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \leq -40000:\\ \;\;\;\;\left(w0 \cdot \sqrt{\frac{h \cdot -0.25}{d\_m \cdot \left(d\_m \cdot \ell\right)}}\right) \cdot \left|M\_m \cdot D\_m\right|\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0)) -40000.0)
   (* (* w0 (sqrt (/ (* h -0.25) (* d_m (* d_m l))))) (fabs (* M_m D_m)))
   w0))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) <= -40000.0) {
		tmp = (w0 * sqrt(((h * -0.25) / (d_m * (d_m * l))))) * fabs((M_m * D_m));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((h / l) * (((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0)) <= (-40000.0d0)) then
        tmp = (w0 * sqrt(((h * (-0.25d0)) / (d_m_1 * (d_m_1 * l))))) * abs((m_m * d_m))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
M_m = Math.abs(M);
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) <= -40000.0) {
		tmp = (w0 * Math.sqrt(((h * -0.25) / (d_m * (d_m * l))))) * Math.abs((M_m * D_m));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if ((h / l) * math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) <= -40000.0:
		tmp = (w0 * math.sqrt(((h * -0.25) / (d_m * (d_m * l))))) * math.fabs((M_m * D_m))
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0)) <= -40000.0)
		tmp = Float64(Float64(w0 * sqrt(Float64(Float64(h * -0.25) / Float64(d_m * Float64(d_m * l))))) * abs(Float64(M_m * D_m)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((h / l) * (((M_m * D_m) / (2.0 * d_m)) ^ 2.0)) <= -40000.0)
		tmp = (w0 * sqrt(((h * -0.25) / (d_m * (d_m * l))))) * abs((M_m * D_m));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -40000.0], N[(N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(M$95$m * D$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4e4

   1. Initial program 77.9%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.3

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

   5. Applied rewrites 48.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 56.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 56.8

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

   9. Applied rewrites 70.4%

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

       1. lift-*.f64 N/A

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

       2. lift-fabs.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

   11. Applied rewrites 75.0%

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

   if -4e4 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 91.5%

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

      1. *-rgt-identity 91.5

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

   7. Applied rewrites 91.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -40000:\\ \;\;\;\;\left(w0 \cdot \sqrt{\frac{h \cdot -0.25}{d \cdot \left(d \cdot \ell\right)}}\right) \cdot \left|M \cdot D\right|\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \leq -40000:\\ \;\;\;\;\sqrt{\frac{h \cdot -0.25}{d\_m \cdot \left(d\_m \cdot \ell\right)}} \cdot \left(w0 \cdot \left|M\_m \cdot D\_m\right|\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (/ h l) (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0)) -40000.0)
   (* (sqrt (/ (* h -0.25) (* d_m (* d_m l)))) (* w0 (fabs (* M_m D_m))))
   w0))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) <= -40000.0) {
		tmp = sqrt(((h * -0.25) / (d_m * (d_m * l)))) * (w0 * fabs((M_m * D_m)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((h / l) * (((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0)) <= (-40000.0d0)) then
        tmp = sqrt(((h * (-0.25d0)) / (d_m_1 * (d_m_1 * l)))) * (w0 * abs((m_m * d_m)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
M_m = Math.abs(M);
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((h / l) * Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) <= -40000.0) {
		tmp = Math.sqrt(((h * -0.25) / (d_m * (d_m * l)))) * (w0 * Math.abs((M_m * D_m)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if ((h / l) * math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) <= -40000.0:
		tmp = math.sqrt(((h * -0.25) / (d_m * (d_m * l)))) * (w0 * math.fabs((M_m * D_m)))
	else:
		tmp = w0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0)) <= -40000.0)
		tmp = Float64(sqrt(Float64(Float64(h * -0.25) / Float64(d_m * Float64(d_m * l)))) * Float64(w0 * abs(Float64(M_m * D_m))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((h / l) * (((M_m * D_m) / (2.0 * d_m)) ^ 2.0)) <= -40000.0)
		tmp = sqrt(((h * -0.25) / (d_m * (d_m * l)))) * (w0 * abs((M_m * D_m)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -40000.0], N[(N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(w0 * N[Abs[N[(M$95$m * D$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4e4

   1. Initial program 77.9%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.3

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

   5. Applied rewrites 48.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 56.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 56.8

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

   9. Applied rewrites 70.4%

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

       1. lift-*.f64 N/A

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

       2. lift-fabs.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 73.0

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

       14. lift-*.f64 N/A

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

   11. Applied rewrites 73.0%

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

   if -4e4 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.1%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 91.5%

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

      1. *-rgt-identity 91.5

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

   7. Applied rewrites 91.5%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq -40000:\\ \;\;\;\;\sqrt{\frac{h \cdot -0.25}{d \cdot \left(d \cdot \ell\right)}} \cdot \left(w0 \cdot \left|M \cdot D\right|\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{2 \cdot d\_m}\\ \mathbf{if}\;t\_0 \leq 10^{-117}:\\ \;\;\;\;w0\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+257}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{-0.5 \cdot \left(\left(M\_m \cdot D\_m\right) \cdot h\right)}{d\_m \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|M\_m\right| \cdot \left(D\_m \cdot \left(w0 \cdot \sqrt{\frac{h \cdot -0.25}{d\_m \cdot \left(d\_m \cdot \ell\right)}}\right)\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ (* M_m D_m) (* 2.0 d_m))))
   (if (<= t_0 1e-117)
     w0
     (if (<= t_0 5e+257)
       (* w0 (sqrt (fma t_0 (/ (* -0.5 (* (* M_m D_m) h)) (* d_m l)) 1.0)))
       (*
        (fabs M_m)
        (* D_m (* w0 (sqrt (/ (* h -0.25) (* d_m (* d_m l)))))))))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * D_m) / (2.0 * d_m);
	double tmp;
	if (t_0 <= 1e-117) {
		tmp = w0;
	} else if (t_0 <= 5e+257) {
		tmp = w0 * sqrt(fma(t_0, ((-0.5 * ((M_m * D_m) * h)) / (d_m * l)), 1.0));
	} else {
		tmp = fabs(M_m) * (D_m * (w0 * sqrt(((h * -0.25) / (d_m * (d_m * l))))));
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(M_m * D_m) / Float64(2.0 * d_m))
	tmp = 0.0
	if (t_0 <= 1e-117)
		tmp = w0;
	elseif (t_0 <= 5e+257)
		tmp = Float64(w0 * sqrt(fma(t_0, Float64(Float64(-0.5 * Float64(Float64(M_m * D_m) * h)) / Float64(d_m * l)), 1.0)));
	else
		tmp = Float64(abs(M_m) * Float64(D_m * Float64(w0 * sqrt(Float64(Float64(h * -0.25) / Float64(d_m * Float64(d_m * l)))))));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-117], w0, If[LessEqual[t$95$0, 5e+257], N[(w0 * N[Sqrt[N[(t$95$0 * N[(N[(-0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[M$95$m], $MachinePrecision] * N[(D$95$m * N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1.00000000000000003e-117

   1. Initial program 85.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 80.9%

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

      1. *-rgt-identity 80.9

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

   7. Applied rewrites 80.9%

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

   if 1.00000000000000003e-117 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 5.00000000000000028e257

   1. Initial program 91.0%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 96.9%

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

   5. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 94.2

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

   7. Applied rewrites 94.2%

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

   if 5.00000000000000028e257 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 58.3%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 54.0

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

   5. Applied rewrites 54.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 54.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 54.0

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

   9. Applied rewrites 63.1%

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

       1. lift-*.f64 N/A

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

       2. lift-fabs.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. associate-*l* N/A

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

       9. lift-fabs.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. fabs-mul N/A

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

       12. associate-*l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-fabs.f64 N/A

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

   11. Applied rewrites 48.2%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{-117}:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+257}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{M \cdot D}{2 \cdot d}, \frac{-0.5 \cdot \left(\left(M \cdot D\right) \cdot h\right)}{d \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|M\right| \cdot \left(D \cdot \left(w0 \cdot \sqrt{\frac{h \cdot -0.25}{d \cdot \left(d \cdot \ell\right)}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{2 \cdot d\_m}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-74}:\\ \;\;\;\;w0\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+257}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot 0.25\right)}{d\_m}, \frac{D\_m \cdot h}{-d\_m \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|M\_m\right| \cdot \left(D\_m \cdot \left(w0 \cdot \sqrt{\frac{h \cdot -0.25}{d\_m \cdot \left(d\_m \cdot \ell\right)}}\right)\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ (* M_m D_m) (* 2.0 d_m))))
   (if (<= t_0 5e-74)
     w0
     (if (<= t_0 5e+257)
       (*
        w0
        (sqrt
         (fma
          (/ (* (* M_m D_m) (* M_m 0.25)) d_m)
          (/ (* D_m h) (- (* d_m l)))
          1.0)))
       (*
        (fabs M_m)
        (* D_m (* w0 (sqrt (/ (* h -0.25) (* d_m (* d_m l)))))))))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * D_m) / (2.0 * d_m);
	double tmp;
	if (t_0 <= 5e-74) {
		tmp = w0;
	} else if (t_0 <= 5e+257) {
		tmp = w0 * sqrt(fma((((M_m * D_m) * (M_m * 0.25)) / d_m), ((D_m * h) / -(d_m * l)), 1.0));
	} else {
		tmp = fabs(M_m) * (D_m * (w0 * sqrt(((h * -0.25) / (d_m * (d_m * l))))));
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(M_m * D_m) / Float64(2.0 * d_m))
	tmp = 0.0
	if (t_0 <= 5e-74)
		tmp = w0;
	elseif (t_0 <= 5e+257)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(M_m * D_m) * Float64(M_m * 0.25)) / d_m), Float64(Float64(D_m * h) / Float64(-Float64(d_m * l))), 1.0)));
	else
		tmp = Float64(abs(M_m) * Float64(D_m * Float64(w0 * sqrt(Float64(Float64(h * -0.25) / Float64(d_m * Float64(d_m * l)))))));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-74], w0, If[LessEqual[t$95$0, 5e+257], N[(w0 * N[Sqrt[N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * 0.25), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(N[(D$95$m * h), $MachinePrecision] / (-N[(d$95$m * l), $MachinePrecision])), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[M$95$m], $MachinePrecision] * N[(D$95$m * N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 4.99999999999999998e-74

   1. Initial program 85.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 80.8%

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

      1. *-rgt-identity 80.8

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

   7. Applied rewrites 80.8%

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

   if 4.99999999999999998e-74 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 5.00000000000000028e257

   1. Initial program 90.2%

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

   4. Applied rewrites 77.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 77.4

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

   6. Applied rewrites 71.7%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 80.9

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

   8. Applied rewrites 80.9%

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

   if 5.00000000000000028e257 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 58.3%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 54.0

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

   5. Applied rewrites 54.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 54.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 54.0

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

   9. Applied rewrites 63.1%

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

       1. lift-*.f64 N/A

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

       2. lift-fabs.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. associate-*l* N/A

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

       9. lift-fabs.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. fabs-mul N/A

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

       12. associate-*l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-fabs.f64 N/A

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

   11. Applied rewrites 48.2%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{-74}:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+257}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot 0.25\right)}{d}, \frac{D \cdot h}{-d \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|M\right| \cdot \left(D \cdot \left(w0 \cdot \sqrt{\frac{h \cdot -0.25}{d \cdot \left(d \cdot \ell\right)}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{2 \cdot d\_m}\\ \mathbf{if}\;t\_0 \leq 10^{-104}:\\ \;\;\;\;w0\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+217}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D\_m}{d\_m} \cdot \left(0.25 \cdot \left(M\_m \cdot M\_m\right)\right), -D\_m \cdot \frac{h}{d\_m \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|M\_m\right| \cdot \left(D\_m \cdot \left(w0 \cdot \sqrt{\frac{h \cdot -0.25}{d\_m \cdot \left(d\_m \cdot \ell\right)}}\right)\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ (* M_m D_m) (* 2.0 d_m))))
   (if (<= t_0 1e-104)
     w0
     (if (<= t_0 5e+217)
       (*
        w0
        (sqrt
         (fma
          (* (/ D_m d_m) (* 0.25 (* M_m M_m)))
          (- (* D_m (/ h (* d_m l))))
          1.0)))
       (*
        (fabs M_m)
        (* D_m (* w0 (sqrt (/ (* h -0.25) (* d_m (* d_m l)))))))))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * D_m) / (2.0 * d_m);
	double tmp;
	if (t_0 <= 1e-104) {
		tmp = w0;
	} else if (t_0 <= 5e+217) {
		tmp = w0 * sqrt(fma(((D_m / d_m) * (0.25 * (M_m * M_m))), -(D_m * (h / (d_m * l))), 1.0));
	} else {
		tmp = fabs(M_m) * (D_m * (w0 * sqrt(((h * -0.25) / (d_m * (d_m * l))))));
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(M_m * D_m) / Float64(2.0 * d_m))
	tmp = 0.0
	if (t_0 <= 1e-104)
		tmp = w0;
	elseif (t_0 <= 5e+217)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(D_m / d_m) * Float64(0.25 * Float64(M_m * M_m))), Float64(-Float64(D_m * Float64(h / Float64(d_m * l)))), 1.0)));
	else
		tmp = Float64(abs(M_m) * Float64(D_m * Float64(w0 * sqrt(Float64(Float64(h * -0.25) / Float64(d_m * Float64(d_m * l)))))));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-104], w0, If[LessEqual[t$95$0, 5e+217], N[(w0 * N[Sqrt[N[(N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(0.25 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[(D$95$m * N[(h / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[M$95$m], $MachinePrecision] * N[(D$95$m * N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 9.99999999999999927e-105

   1. Initial program 85.7%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 80.9%

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

      1. *-rgt-identity 80.9

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

   7. Applied rewrites 80.9%

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

   if 9.99999999999999927e-105 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 5.00000000000000041e217

   1. Initial program 93.8%

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

   4. Applied rewrites 81.8%

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

   5. Taylor expanded in D around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 79.1

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

   7. Applied rewrites 79.1%

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

   if 5.00000000000000041e217 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 55.6%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 51.5

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

   5. Applied rewrites 51.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 51.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 51.5

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

   9. Applied rewrites 60.3%

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

       1. lift-*.f64 N/A

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

       2. lift-fabs.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. associate-*l* N/A

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

       9. lift-fabs.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. fabs-mul N/A

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

       12. associate-*l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-fabs.f64 N/A

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

   11. Applied rewrites 46.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{-104}:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+217}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D}{d} \cdot \left(0.25 \cdot \left(M \cdot M\right)\right), -D \cdot \frac{h}{d \cdot \ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|M\right| \cdot \left(D \cdot \left(w0 \cdot \sqrt{\frac{h \cdot -0.25}{d \cdot \left(d \cdot \ell\right)}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{2 \cdot d\_m}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+257}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{\frac{\left(M\_m \cdot D\_m\right) \cdot h}{2 \cdot d\_m}}{-\ell}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|M\_m\right| \cdot \left(D\_m \cdot \left(w0 \cdot \sqrt{\frac{h \cdot -0.25}{d\_m \cdot \left(d\_m \cdot \ell\right)}}\right)\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ (* M_m D_m) (* 2.0 d_m))))
   (if (<= t_0 5e+257)
     (* w0 (sqrt (fma t_0 (/ (/ (* (* M_m D_m) h) (* 2.0 d_m)) (- l)) 1.0)))
     (* (fabs M_m) (* D_m (* w0 (sqrt (/ (* h -0.25) (* d_m (* d_m l))))))))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * D_m) / (2.0 * d_m);
	double tmp;
	if (t_0 <= 5e+257) {
		tmp = w0 * sqrt(fma(t_0, ((((M_m * D_m) * h) / (2.0 * d_m)) / -l), 1.0));
	} else {
		tmp = fabs(M_m) * (D_m * (w0 * sqrt(((h * -0.25) / (d_m * (d_m * l))))));
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(M_m * D_m) / Float64(2.0 * d_m))
	tmp = 0.0
	if (t_0 <= 5e+257)
		tmp = Float64(w0 * sqrt(fma(t_0, Float64(Float64(Float64(Float64(M_m * D_m) * h) / Float64(2.0 * d_m)) / Float64(-l)), 1.0)));
	else
		tmp = Float64(abs(M_m) * Float64(D_m * Float64(w0 * sqrt(Float64(Float64(h * -0.25) / Float64(d_m * Float64(d_m * l)))))));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+257], N[(w0 * N[Sqrt[N[(t$95$0 * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[M$95$m], $MachinePrecision] * N[(D$95$m * N[(w0 * N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 5.00000000000000028e257

   1. Initial program 86.5%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Add Preprocessing
   3. 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 93.5%

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

   if 5.00000000000000028e257 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 58.3%

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

   3. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 54.0

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

   5. Applied rewrites 54.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 54.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 54.0

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

   9. Applied rewrites 63.1%

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

       1. lift-*.f64 N/A

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

       2. lift-fabs.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. associate-*l* N/A

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

       9. lift-fabs.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. fabs-mul N/A

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

       12. associate-*l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-fabs.f64 N/A

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

   11. Applied rewrites 48.2%

       \[\leadsto \color{blue}{\left|M\right| \cdot \left(D \cdot \left(w0 \cdot \sqrt{\frac{h \cdot -0.25}{d \cdot \left(d \cdot \ell\right)}}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 76.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 2 \cdot 10^{+176}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D\_m \cdot D\_m, \frac{-0.125 \cdot \left(h \cdot \left(w0 \cdot \left(M\_m \cdot M\_m\right)\right)\right)}{d\_m \cdot \left(d\_m \cdot \ell\right)}, w0\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ (* M_m D_m) (* 2.0 d_m)) 2e+176)
   w0
   (fma
    (* D_m D_m)
    (/ (* -0.125 (* h (* w0 (* M_m M_m)))) (* d_m (* d_m l)))
    w0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 2e+176) {
		tmp = w0;
	} else {
		tmp = fma((D_m * D_m), ((-0.125 * (h * (w0 * (M_m * M_m)))) / (d_m * (d_m * l))), w0);
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) <= 2e+176)
		tmp = w0;
	else
		tmp = fma(Float64(D_m * D_m), Float64(Float64(-0.125 * Float64(h * Float64(w0 * Float64(M_m * M_m)))) / Float64(d_m * Float64(d_m * l))), w0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2e+176], w0, N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(-0.125 * N[(h * N[(w0 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 2e176

   1. Initial program 87.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 77.8%

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

      1. *-rgt-identity 77.8

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

   7. Applied rewrites 77.8%

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

   if 2e176 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 55.4%

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

      1. lift-*.f64 N/A

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

      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 64.5%

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

   5. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 60.1

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

   7. Applied rewrites 60.1%

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

   8. Taylor expanded in M around 0

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

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

   10. Applied rewrites 51.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(D \cdot D, \frac{-0.125 \cdot \left(h \cdot \left(\left(M \cdot M\right) \cdot w0\right)\right)}{d \cdot \left(d \cdot \ell\right)}, w0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{+176}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{-0.125 \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}, w0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 5 \cdot 10^{+297}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(-0.125 \cdot \frac{\left(D\_m \cdot D\_m\right) \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)}{\ell \cdot \left(d\_m \cdot d\_m\right)}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ (* M_m D_m) (* 2.0 d_m)) 5e+297)
   w0
   (* w0 (* -0.125 (/ (* (* D_m D_m) (* h (* M_m M_m))) (* l (* d_m d_m)))))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+297) {
		tmp = w0;
	} else {
		tmp = w0 * (-0.125 * (((D_m * D_m) * (h * (M_m * M_m))) / (l * (d_m * d_m))));
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((m_m * d_m) / (2.0d0 * d_m_1)) <= 5d+297) then
        tmp = w0
    else
        tmp = w0 * ((-0.125d0) * (((d_m * d_m) * (h * (m_m * m_m))) / (l * (d_m_1 * d_m_1))))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
M_m = Math.abs(M);
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+297) {
		tmp = w0;
	} else {
		tmp = w0 * (-0.125 * (((D_m * D_m) * (h * (M_m * M_m))) / (l * (d_m * d_m))));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if ((M_m * D_m) / (2.0 * d_m)) <= 5e+297:
		tmp = w0
	else:
		tmp = w0 * (-0.125 * (((D_m * D_m) * (h * (M_m * M_m))) / (l * (d_m * d_m))))
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) <= 5e+297)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(-0.125 * Float64(Float64(Float64(D_m * D_m) * Float64(h * Float64(M_m * M_m))) / Float64(l * Float64(d_m * d_m)))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+297)
		tmp = w0;
	else
		tmp = w0 * (-0.125 * (((D_m * D_m) * (h * (M_m * M_m))) / (l * (d_m * d_m))));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 5e+297], w0, N[(w0 * N[(-0.125 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 4.9999999999999998e297

   1. Initial program 86.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 76.3%

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

      1. *-rgt-identity 76.3

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

   7. Applied rewrites 76.3%

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

   if 4.9999999999999998e297 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 58.9%

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

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

   4. Applied rewrites 65.2%

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

   5. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 59.6

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

   7. Applied rewrites 59.6%

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

   8. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   10. Applied rewrites 54.1%

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

   11. Taylor expanded in D around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 54.1

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

   13. Applied rewrites 54.1%

       \[\leadsto w0 \cdot \color{blue}{\left(\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)} \cdot -0.125\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+297}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 5 \cdot 10^{+297}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(D\_m \cdot D\_m\right) \cdot \left(w0 \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)\right)}{\ell \cdot \left(d\_m \cdot d\_m\right)}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ (* M_m D_m) (* 2.0 d_m)) 5e+297)
   w0
   (* -0.125 (/ (* (* D_m D_m) (* w0 (* h (* M_m M_m)))) (* l (* d_m d_m))))))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+297) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D_m * D_m) * (w0 * (h * (M_m * M_m)))) / (l * (d_m * d_m)));
	}
	return tmp;
}

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((m_m * d_m) / (2.0d0 * d_m_1)) <= 5d+297) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d_m * d_m) * (w0 * (h * (m_m * m_m)))) / (l * (d_m_1 * d_m_1)))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
M_m = Math.abs(M);
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+297) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D_m * D_m) * (w0 * (h * (M_m * M_m)))) / (l * (d_m * d_m)));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
M_m = math.fabs(M)
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if ((M_m * D_m) / (2.0 * d_m)) <= 5e+297:
		tmp = w0
	else:
		tmp = -0.125 * (((D_m * D_m) * (w0 * (h * (M_m * M_m)))) / (l * (d_m * d_m)))
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) <= 5e+297)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D_m * D_m) * Float64(w0 * Float64(h * Float64(M_m * M_m)))) / Float64(l * Float64(d_m * d_m))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+297)
		tmp = w0;
	else
		tmp = -0.125 * (((D_m * D_m) * (w0 * (h * (M_m * M_m)))) / (l * (d_m * d_m)));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 5e+297], w0, N[(-0.125 * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(w0 * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 4.9999999999999998e297

   1. Initial program 86.2%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 76.3%

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

      1. *-rgt-identity 76.3

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

   7. Applied rewrites 76.3%

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

   if 4.9999999999999998e297 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 58.9%

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

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

   4. Applied rewrites 65.2%

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

   5. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 59.6

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

   7. Applied rewrites 59.6%

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

   8. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   10. Applied rewrites 54.1%

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

   11. Taylor expanded in D around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 54.1

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

   13. Applied rewrites 54.1%

       \[\leadsto \color{blue}{-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot w0\right)}{\ell \cdot \left(d \cdot d\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+297}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \left(w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 68.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 6.3 \cdot 10^{-118}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D\_m, w0 \cdot \left(D\_m \cdot \frac{h \cdot \left(\left(M\_m \cdot M\_m\right) \cdot -0.125\right)}{d\_m \cdot \left(d\_m \cdot \ell\right)}\right), w0\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 6.3e-118)
   w0
   (fma
    D_m
    (* w0 (* D_m (/ (* h (* (* M_m M_m) -0.125)) (* d_m (* d_m l)))))
    w0)))

C

d_m = fabs(d);
D_m = fabs(D);
M_m = fabs(M);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 6.3e-118) {
		tmp = w0;
	} else {
		tmp = fma(D_m, (w0 * (D_m * ((h * ((M_m * M_m) * -0.125)) / (d_m * (d_m * l))))), w0);
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
M_m = abs(M)
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 6.3e-118)
		tmp = w0;
	else
		tmp = fma(D_m, Float64(w0 * Float64(D_m * Float64(Float64(h * Float64(Float64(M_m * M_m) * -0.125)) / Float64(d_m * Float64(d_m * l))))), w0);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 6.3e-118], w0, N[(D$95$m * N[(w0 * N[(D$95$m * N[(N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 6.2999999999999997e-118

   1. Initial program 84.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 74.1%

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

      1. *-rgt-identity 74.1

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

   7. Applied rewrites 74.1%

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

   if 6.2999999999999997e-118 < M

   1. Initial program 83.7%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 91.6%

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

   5. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 90.2

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

   7. Applied rewrites 90.2%

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

   8. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   10. Applied rewrites 59.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. distribute-rgt-in N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*l* N/A

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

       11. associate-*l* N/A

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

       12. *-lft-identity N/A

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

   12. Applied rewrites 63.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(D, \left(D \cdot \frac{h \cdot \left(-0.125 \cdot \left(M \cdot M\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right) \cdot w0, w0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.3 \cdot 10^{-118}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(D, w0 \cdot \left(D \cdot \frac{h \cdot \left(\left(M \cdot M\right) \cdot -0.125\right)}{d \cdot \left(d \cdot \ell\right)}\right), w0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.9% accurate, 157.0× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ M_m = \left|M\right| \\ w0 \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
M_m = (fabs.f64 M)
(FPCore (w0 M_m D_m h l d_m) :precision binary64 w0)

C

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

Fortran

d_m = abs(d)
D_m = abs(d)
M_m = abs(m)
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

d_m = abs(d);
D_m = abs(D);
M_m = abs(M);
function tmp = code(w0, M_m, D_m, h, l, d_m)
	tmp = w0;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := w0

Derivation

1. Initial program 84.2%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 71.0%

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

   1. *-rgt-identity 71.0

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

7. Applied rewrites 71.0%

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

Reproduce

herbie shell --seed 2024216 
(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))))))