Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.7% → 88.2%

Time: 19.1s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+239}:\\ \;\;\;\;w0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{\ell}\right) \cdot \left(M \cdot h\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
   (if (<= t_0 2e+239)
     (* w0 (sqrt t_0))
     (*
      w0
      (sqrt (+ 1.0 (* -0.25 (* (/ D d) (* (* (/ D d) (/ M l)) (* M h))))))))))

C

D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = 1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 2e+239) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / d) * (((D / d) * (M / l)) * (M * h))))));
	}
	return tmp;
}

Fortran

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))
    if (t_0 <= 2d+239) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((d / d_1) * (((d / d_1) * (m / l)) * (m * h))))))
    end if
    code = tmp
end function

Java

D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = 1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 2e+239) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((D / d) * (((D / d) * (M / l)) * (M * h))))));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = 1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 2e+239:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((D / d) * (((D / d) * (M / l)) * (M * h))))))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 2e+239)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D / d) * Float64(Float64(Float64(D / d) * Float64(M / l)) * Float64(M * h)))))));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = 1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 2e+239)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / d) * (((D / d) * (M / l)) * (M * h))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 2e+239], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D / d), $MachinePrecision] * N[(N[(N[(D / d), $MachinePrecision] * N[(M / l), $MachinePrecision]), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 1.99999999999999998e239

   1. Initial program 99.9%

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

   if 1.99999999999999998e239 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 44.0%

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

   2. Taylor expanded in w0 around 0 45.7%

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

   4. Simplified 44.5%

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

      1. *-un-lft-identity 44.5%

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

      2. *-commutative 44.5%

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

      3. times-frac 51.2%

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

      4. associate-/l* 45.6%

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

      5. associate-/r/ 52.4%

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

   6. Applied egg-rr 52.4%

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

      1. *-lft-identity 52.4%

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

      2. *-commutative 52.4%

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

      3. associate-*l* 55.0%

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

      4. *-commutative 55.0%

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

      5. associate-/l* 59.0%

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

   8. Simplified 59.0%

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

   9. Taylor expanded in h around 0 53.8%

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

       1. unpow2 53.8%

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

       2. associate-*r* 61.3%

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

       3. associate-*l/ 63.9%

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

       4. *-commutative 63.9%

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

       5. associate-/l* 63.9%

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

   11. Simplified 63.9%

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

   12. Taylor expanded in D around 0 55.8%

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

       1. associate-/l* 53.5%

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

       2. unpow2 53.5%

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

       3. associate-/l* 55.8%

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

       4. *-commutative 55.8%

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

       5. associate-/r* 57.3%

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

       6. associate-*l/ 57.3%

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

       7. associate-*r* 60.5%

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

       8. associate-/l* 58.3%

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

       9. *-commutative 58.3%

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

       10. associate-*l/ 54.9%

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

       11. associate-/r/ 47.0%

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

       12. associate-*l* 55.3%

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

       13. associate-*l/ 63.9%

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

       14. associate-*r/ 63.9%

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

       15. *-commutative 63.9%

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

       16. associate-*r* 64.7%

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

       17. *-commutative 64.7%

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

       18. associate-*l* 70.1%

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

   14. Simplified 70.1%

       \[\leadsto \sqrt{1 + -0.25 \cdot \left(\frac{D}{d} \cdot \color{blue}{\left(\left(h \cdot M\right) \cdot \left(\frac{M}{\ell} \cdot \frac{D}{d}\right)\right)}\right)} \cdot w0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{+239}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{\ell}\right) \cdot \left(M \cdot h\right)\right)\right)}\\ \end{array} \]

Alternative 2: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{-155}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{\ell}\right) \cdot \left(M \cdot h\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 2e-155)
   (* w0 (sqrt (- 1.0 (* h (/ (pow (* D (/ M (/ d 0.5))) 2.0) l)))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (* (/ D d) (* (* (/ D d) (/ M l)) (* M h)))))))))

C

D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2e-155) {
		tmp = w0 * sqrt((1.0 - (h * (pow((D * (M / (d / 0.5))), 2.0) / l))));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / d) * (((D / d) * (M / l)) * (M * h))))));
	}
	return tmp;
}

Fortran

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 2d-155) then
        tmp = w0 * sqrt((1.0d0 - (h * (((d * (m / (d_1 / 0.5d0))) ** 2.0d0) / l))))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((d / d_1) * (((d / d_1) * (m / l)) * (m * h))))))
    end if
    code = tmp
end function

Java

D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2e-155) {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow((D * (M / (d / 0.5))), 2.0) / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((D / d) * (((D / d) * (M / l)) * (M * h))))));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2e-155:
		tmp = w0 * math.sqrt((1.0 - (h * (math.pow((D * (M / (d / 0.5))), 2.0) / l))))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((D / d) * (((D / d) * (M / l)) * (M * h))))))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2e-155)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(D * Float64(M / Float64(d / 0.5))) ^ 2.0) / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D / d) * Float64(Float64(Float64(D / d) * Float64(M / l)) * Float64(M * h)))))));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2e-155)
		tmp = w0 * sqrt((1.0 - (h * (((D * (M / (d / 0.5))) ^ 2.0) / l))));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / d) * (((D / d) * (M / l)) * (M * h))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 2e-155], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(D * N[(M / N[(d / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D / d), $MachinePrecision] * N[(N[(N[(D / d), $MachinePrecision] * N[(M / l), $MachinePrecision]), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.00000000000000003e-155

   1. Initial program 88.1%

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

      1. clear-num 88.1%

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

      2. un-div-inv 88.1%

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

      3. div-inv 88.0%

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

      4. associate-*l* 86.9%

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

      5. associate-/r* 86.9%

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

      6. metadata-eval 86.9%

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

   3. Applied egg-rr 86.9%

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

      1. associate-/r/ 90.3%

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

      2. *-commutative 90.3%

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

      3. *-commutative 90.3%

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

      4. associate-*r* 91.5%

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

      5. *-commutative 91.5%

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

      6. associate-*r/ 91.5%

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

      7. associate-/l* 91.5%

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

   5. Simplified 91.5%

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

   if 2.00000000000000003e-155 < M

   1. Initial program 69.0%

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

   2. Taylor expanded in w0 around 0 51.7%

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

   4. Simplified 49.5%

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

      1. *-un-lft-identity 49.5%

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

      2. *-commutative 49.5%

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

      3. times-frac 61.5%

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

      4. associate-/l* 62.2%

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

      5. associate-/r/ 60.4%

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

   6. Applied egg-rr 60.4%

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

      1. *-lft-identity 60.4%

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

      2. *-commutative 60.4%

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

      3. associate-*l* 62.7%

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

      4. *-commutative 62.7%

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

      5. associate-/l* 65.8%

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

   8. Simplified 65.8%

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

   9. Taylor expanded in h around 0 63.0%

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

       1. unpow2 63.0%

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

       2. associate-*r* 67.3%

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

       3. associate-*l/ 73.0%

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

       4. *-commutative 73.0%

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

       5. associate-/l* 71.9%

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

   11. Simplified 71.9%

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

   12. Taylor expanded in D around 0 63.9%

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

       1. associate-/l* 63.6%

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

       2. unpow2 63.6%

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

       3. associate-/l* 63.9%

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

       4. *-commutative 63.9%

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

       5. associate-/r* 65.2%

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

       6. associate-*l/ 66.2%

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

       7. associate-*r* 70.1%

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

       8. associate-/l* 66.9%

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

       9. *-commutative 66.9%

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

       10. associate-*l/ 63.7%

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

       11. associate-/r/ 64.5%

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

       12. associate-*l* 71.9%

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

       13. associate-*l/ 73.0%

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

       14. associate-*r/ 71.9%

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

       15. *-commutative 71.9%

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

       16. associate-*r* 72.2%

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

       17. *-commutative 72.2%

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

       18. associate-*l* 77.4%

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

   14. Simplified 77.4%

       \[\leadsto \sqrt{1 + -0.25 \cdot \left(\frac{D}{d} \cdot \color{blue}{\left(\left(h \cdot M\right) \cdot \left(\frac{M}{\ell} \cdot \frac{D}{d}\right)\right)}\right)} \cdot w0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{-155}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \frac{M}{\frac{d}{0.5}}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{\ell}\right) \cdot \left(M \cdot h\right)\right)\right)}\\ \end{array} \]

Alternative 3: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4.5 \cdot 10^{+36}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(h \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 4.5e+36)
   (* w0 (sqrt (+ 1.0 (* -0.25 (* h (* (/ D (/ l D)) (* (/ M d) (/ M d))))))))
   (* w0 (+ 1.0 (* -0.125 (/ (* (* (/ D d) (/ D d)) (* M (* M h))) l))))))

C

D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 4.5e+36) {
		tmp = w0 * sqrt((1.0 + (-0.25 * (h * ((D / (l / D)) * ((M / d) * (M / d)))))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * (M * h))) / l)));
	}
	return tmp;
}

Fortran

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 4.5d+36) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (h * ((d / (l / d)) * ((m / d_1) * (m / d_1)))))))
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((((d / d_1) * (d / d_1)) * (m * (m * h))) / l)))
    end if
    code = tmp
end function

Java

D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 4.5e+36) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (h * ((D / (l / D)) * ((M / d) * (M / d)))))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * (M * h))) / l)));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 4.5e+36:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (h * ((D / (l / D)) * ((M / d) * (M / d)))))))
	else:
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * (M * h))) / l)))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 4.5e+36)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(h * Float64(Float64(D / Float64(l / D)) * Float64(Float64(M / d) * Float64(M / d))))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(M * Float64(M * h))) / l))));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 4.5e+36)
		tmp = w0 * sqrt((1.0 + (-0.25 * (h * ((D / (l / D)) * ((M / d) * (M / d)))))));
	else
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * (M * h))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 4.5e+36], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(h * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.49999999999999997e36

   1. Initial program 79.7%

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

   2. Taylor expanded in w0 around 0 53.1%

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

   4. Simplified 52.0%

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

   5. Taylor expanded in D around 0 53.1%

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

      1. unpow2 52.1%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}}\right) \]

      2. *-commutative 52.1%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}\right) \]

      3. times-frac 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \]

      4. unpow2 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) \]

      5. associate-*l/ 50.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \color{blue}{\left(\frac{M \cdot M}{\ell} \cdot h\right)}\right)\right) \]

      6. unpow2 50.5%

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

      7. times-frac 61.6%

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

      8. associate-*r* 63.2%

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

      9. *-commutative 63.2%

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

      10. times-frac 52.1%

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

      11. unpow2 52.1%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \left(\frac{\color{blue}{{D}^{2}}}{d \cdot d} \cdot \frac{M \cdot M}{\ell}\right)\right)\right) \]

      12. times-frac 53.2%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\frac{{D}^{2} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}\right)\right) \]

      13. *-commutative 53.2%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{{D}^{2} \cdot \left(M \cdot M\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)\right) \]

      14. times-frac 54.8%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{M \cdot M}{d \cdot d}\right)}\right)\right) \]

      15. unpow2 54.8%

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

      16. associate-/l* 58.6%

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

      17. times-frac 72.1%

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

   7. Simplified 76.4%

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

   if 4.49999999999999997e36 < d

   1. Initial program 85.3%

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

   2. Taylor expanded in M around 0 65.9%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)} \]
   3. Step-by-step derivation

      1. associate-/l* 65.8%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 65.8%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2}}{\frac{\color{blue}{{d}^{2} \cdot \ell}}{{M}^{2} \cdot h}}\right) \]

      3. *-commutative 65.8%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2}}{\frac{{d}^{2} \cdot \ell}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      4. associate-/l* 65.9%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}}\right) \]

      5. times-frac 64.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right) \]

      6. unpow2 64.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right) \]

      7. unpow2 64.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right) \]

      8. *-commutative 64.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right) \]

      9. unpow2 64.4%

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

   4. Simplified 64.4%

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

      1. associate-*r/ 65.9%

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

      2. times-frac 76.6%

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

      3. *-commutative 76.6%

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

   6. Applied egg-rr 76.6%

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

   7. Taylor expanded in h around 0 76.6%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell}\right) \]
   8. Step-by-step derivation

      1. *-commutative 76.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{\ell}\right) \]

      2. unpow2 76.6%

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

      3. associate-*r* 81.0%

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

      4. *-commutative 81.0%

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

      5. *-commutative 81.0%

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

   9. Simplified 81.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.5 \cdot 10^{+36}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(h \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)}{\ell}\right)\\ \end{array} \]

Alternative 4: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 3.15 \cdot 10^{+40}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(h \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 3.15e+40)
   (* w0 (sqrt (+ 1.0 (* -0.25 (* h (* (/ D (/ l D)) (* (/ M d) (/ M d))))))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (* (* (/ D d) (/ D d)) (/ (* h (* M M)) l))))))))

C

D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 3.15e+40) {
		tmp = w0 * sqrt((1.0 + (-0.25 * (h * ((D / (l / D)) * ((M / d) * (M / d)))))));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * (((D / d) * (D / d)) * ((h * (M * M)) / l)))));
	}
	return tmp;
}

Fortran

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 3.15d+40) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (h * ((d / (l / d)) * ((m / d_1) * (m / d_1)))))))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (((d / d_1) * (d / d_1)) * ((h * (m * m)) / l)))))
    end if
    code = tmp
end function

Java

D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 3.15e+40) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (h * ((D / (l / D)) * ((M / d) * (M / d)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (((D / d) * (D / d)) * ((h * (M * M)) / l)))));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 3.15e+40:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (h * ((D / (l / D)) * ((M / d) * (M / d)))))))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (((D / d) * (D / d)) * ((h * (M * M)) / l)))))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 3.15e+40)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(h * Float64(Float64(D / Float64(l / D)) * Float64(Float64(M / d) * Float64(M / d))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(h * Float64(M * M)) / l))))));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 3.15e+40)
		tmp = w0 * sqrt((1.0 + (-0.25 * (h * ((D / (l / D)) * ((M / d) * (M / d)))))));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * (((D / d) * (D / d)) * ((h * (M * M)) / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 3.15e+40], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(h * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 3.15000000000000003e40

   1. Initial program 79.8%

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

   2. Taylor expanded in w0 around 0 53.4%

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

   4. Simplified 52.3%

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

   5. Taylor expanded in D around 0 53.4%

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

      1. unpow2 52.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}}\right) \]

      2. *-commutative 52.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}\right) \]

      3. times-frac 51.3%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \]

      4. unpow2 51.3%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) \]

      5. associate-*l/ 50.8%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \color{blue}{\left(\frac{M \cdot M}{\ell} \cdot h\right)}\right)\right) \]

      6. unpow2 50.8%

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

      7. times-frac 61.8%

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

      8. associate-*r* 63.4%

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

      9. *-commutative 63.4%

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

      10. times-frac 52.4%

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

      11. unpow2 52.4%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \left(\frac{\color{blue}{{D}^{2}}}{d \cdot d} \cdot \frac{M \cdot M}{\ell}\right)\right)\right) \]

      12. times-frac 53.4%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\frac{{D}^{2} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}\right)\right) \]

      13. *-commutative 53.4%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{{D}^{2} \cdot \left(M \cdot M\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)\right) \]

      14. times-frac 55.1%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{M \cdot M}{d \cdot d}\right)}\right)\right) \]

      15. unpow2 55.1%

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

      16. associate-/l* 58.8%

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

      17. times-frac 72.2%

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

   7. Simplified 76.5%

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

   if 3.15000000000000003e40 < d

   1. Initial program 85.1%

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

   2. Taylor expanded in w0 around 0 65.4%

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

   4. Simplified 63.8%

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

   5. Taylor expanded in D around 0 63.8%

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

      1. unpow2 63.9%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right)\right) \]

      2. unpow2 63.9%

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

      3. times-frac 74.7%

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

   7. Simplified 77.5%

      \[\leadsto \sqrt{1 + -0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right)} \cdot w0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 3.15 \cdot 10^{+40}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(h \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)}\\ \end{array} \]

Alternative 5: 77.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4.2 \cdot 10^{-100}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(h \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot \frac{M}{\ell}\right) \cdot \left(h \cdot \frac{D}{d}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 4.2e-100)
   (* w0 (sqrt (+ 1.0 (* -0.25 (* h (* (/ D (/ l D)) (* (/ M d) (/ M d))))))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (* (/ D d) (* (* M (/ M l)) (* h (/ D d))))))))))

C

D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 4.2e-100) {
		tmp = w0 * sqrt((1.0 + (-0.25 * (h * ((D / (l / D)) * ((M / d) * (M / d)))))));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / d) * ((M * (M / l)) * (h * (D / d)))))));
	}
	return tmp;
}

Fortran

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 4.2d-100) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (h * ((d / (l / d)) * ((m / d_1) * (m / d_1)))))))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((d / d_1) * ((m * (m / l)) * (h * (d / d_1)))))))
    end if
    code = tmp
end function

Java

D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 4.2e-100) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (h * ((D / (l / D)) * ((M / d) * (M / d)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((D / d) * ((M * (M / l)) * (h * (D / d)))))));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 4.2e-100:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (h * ((D / (l / D)) * ((M / d) * (M / d)))))))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((D / d) * ((M * (M / l)) * (h * (D / d)))))))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 4.2e-100)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(h * Float64(Float64(D / Float64(l / D)) * Float64(Float64(M / d) * Float64(M / d))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D / d) * Float64(Float64(M * Float64(M / l)) * Float64(h * Float64(D / d))))))));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 4.2e-100)
		tmp = w0 * sqrt((1.0 + (-0.25 * (h * ((D / (l / D)) * ((M / d) * (M / d)))))));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / d) * ((M * (M / l)) * (h * (D / d)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 4.2e-100], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(h * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D / d), $MachinePrecision] * N[(N[(M * N[(M / l), $MachinePrecision]), $MachinePrecision] * N[(h * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.20000000000000019e-100

   1. Initial program 81.1%

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

   2. Taylor expanded in w0 around 0 53.5%

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

   4. Simplified 50.8%

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

   5. Taylor expanded in D around 0 53.5%

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

      1. unpow2 53.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}}\right) \]

      2. *-commutative 53.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}\right) \]

      3. times-frac 50.8%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \]

      4. unpow2 50.8%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) \]

      5. associate-*l/ 49.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \color{blue}{\left(\frac{M \cdot M}{\ell} \cdot h\right)}\right)\right) \]

      6. unpow2 49.5%

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

      7. times-frac 63.2%

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

      8. associate-*r* 65.2%

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

      9. *-commutative 65.2%

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

      10. times-frac 51.5%

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

      11. unpow2 51.5%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \left(\frac{\color{blue}{{D}^{2}}}{d \cdot d} \cdot \frac{M \cdot M}{\ell}\right)\right)\right) \]

      12. times-frac 53.5%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\frac{{D}^{2} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}\right)\right) \]

      13. *-commutative 53.5%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{{D}^{2} \cdot \left(M \cdot M\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)\right) \]

      14. times-frac 54.9%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{M \cdot M}{d \cdot d}\right)}\right)\right) \]

      15. unpow2 54.9%

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

      16. associate-/l* 59.5%

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

      17. times-frac 76.1%

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

   7. Simplified 79.6%

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

   if 4.20000000000000019e-100 < d

   1. Initial program 81.2%

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

   2. Taylor expanded in w0 around 0 60.9%

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

   4. Simplified 61.8%

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

      1. *-un-lft-identity 61.8%

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

      2. *-commutative 61.8%

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

      3. times-frac 70.7%

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

      4. associate-/l* 69.6%

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

      5. associate-/r/ 70.6%

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

   6. Applied egg-rr 70.6%

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

      1. *-lft-identity 70.6%

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

      2. *-commutative 70.6%

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

      3. associate-*l* 71.9%

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

      4. *-commutative 71.9%

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

      5. associate-/l* 74.7%

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

   8. Simplified 74.7%

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

   9. Taylor expanded in D around 0 70.0%

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

       1. *-commutative 70.0%

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

       2. *-commutative 70.0%

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

       3. associate-*r* 70.7%

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

       4. times-frac 70.8%

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

       5. associate-*l/ 71.8%

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

       6. unpow2 71.8%

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

       7. associate-/l* 74.6%

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

       8. *-commutative 74.6%

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

       9. associate-/r/ 74.6%

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

       10. *-commutative 74.6%

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

       11. *-commutative 74.6%

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

   11. Simplified 74.6%

       \[\leadsto \sqrt{1 + -0.25 \cdot \left(\frac{D}{d} \cdot \color{blue}{\left(\left(M \cdot \frac{M}{\ell}\right) \cdot \left(h \cdot \frac{D}{d}\right)\right)}\right)} \cdot w0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.2 \cdot 10^{-100}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(h \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot \frac{M}{\ell}\right) \cdot \left(h \cdot \frac{D}{d}\right)\right)\right)}\\ \end{array} \]

Alternative 6: 78.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 6 \cdot 10^{-131}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(h \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{\ell}\right) \cdot \left(M \cdot h\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 6e-131)
   (* w0 (sqrt (+ 1.0 (* -0.25 (* h (* (/ D (/ l D)) (* (/ M d) (/ M d))))))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (* (/ D d) (* (* (/ D d) (/ M l)) (* M h)))))))))

C

D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 6e-131) {
		tmp = w0 * sqrt((1.0 + (-0.25 * (h * ((D / (l / D)) * ((M / d) * (M / d)))))));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / d) * (((D / d) * (M / l)) * (M * h))))));
	}
	return tmp;
}

Fortran

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 6d-131) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (h * ((d / (l / d)) * ((m / d_1) * (m / d_1)))))))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((d / d_1) * (((d / d_1) * (m / l)) * (m * h))))))
    end if
    code = tmp
end function

Java

D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 6e-131) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (h * ((D / (l / D)) * ((M / d) * (M / d)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((D / d) * (((D / d) * (M / l)) * (M * h))))));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 6e-131:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (h * ((D / (l / D)) * ((M / d) * (M / d)))))))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((D / d) * (((D / d) * (M / l)) * (M * h))))))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 6e-131)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(h * Float64(Float64(D / Float64(l / D)) * Float64(Float64(M / d) * Float64(M / d))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D / d) * Float64(Float64(Float64(D / d) * Float64(M / l)) * Float64(M * h)))))));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 6e-131)
		tmp = w0 * sqrt((1.0 + (-0.25 * (h * ((D / (l / D)) * ((M / d) * (M / d)))))));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / d) * (((D / d) * (M / l)) * (M * h))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 6e-131], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(h * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D / d), $MachinePrecision] * N[(N[(N[(D / d), $MachinePrecision] * N[(M / l), $MachinePrecision]), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 5.99999999999999992e-131

   1. Initial program 83.1%

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

   2. Taylor expanded in w0 around 0 53.3%

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

   4. Simplified 50.5%

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

   5. Taylor expanded in D around 0 53.3%

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

      1. unpow2 53.3%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}}\right) \]

      2. *-commutative 53.3%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}\right) \]

      3. times-frac 50.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \]

      4. unpow2 50.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) \]

      5. associate-*l/ 49.1%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \color{blue}{\left(\frac{M \cdot M}{\ell} \cdot h\right)}\right)\right) \]

      6. unpow2 49.1%

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

      7. times-frac 63.8%

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

      8. associate-*r* 65.9%

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

      9. *-commutative 65.9%

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

      10. times-frac 51.3%

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

      11. unpow2 51.3%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \left(\frac{\color{blue}{{D}^{2}}}{d \cdot d} \cdot \frac{M \cdot M}{\ell}\right)\right)\right) \]

      12. times-frac 53.3%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\frac{{D}^{2} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}\right)\right) \]

      13. *-commutative 53.3%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{{D}^{2} \cdot \left(M \cdot M\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)\right) \]

      14. times-frac 54.2%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{M \cdot M}{d \cdot d}\right)}\right)\right) \]

      15. unpow2 54.2%

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

      16. associate-/l* 59.1%

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

      17. times-frac 77.0%

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

   7. Simplified 79.5%

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

   if 5.99999999999999992e-131 < d

   1. Initial program 78.7%

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

   2. Taylor expanded in w0 around 0 60.4%

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

   4. Simplified 61.1%

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

      1. *-un-lft-identity 61.1%

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

      2. *-commutative 61.1%

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

      3. times-frac 69.2%

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

      4. associate-/l* 67.1%

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

      5. associate-/r/ 69.1%

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

   6. Applied egg-rr 69.1%

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

      1. *-lft-identity 69.1%

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

      2. *-commutative 69.1%

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

      3. associate-*l* 72.9%

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

      4. *-commutative 72.9%

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

      5. associate-/l* 75.4%

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

   8. Simplified 75.4%

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

   9. Taylor expanded in h around 0 72.9%

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

       1. unpow2 72.9%

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

       2. associate-*r* 79.7%

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

       3. associate-*l/ 83.8%

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

       4. *-commutative 83.8%

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

       5. associate-/l* 83.8%

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

   11. Simplified 83.8%

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

   12. Taylor expanded in D around 0 70.2%

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

       1. associate-/l* 69.9%

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

       2. unpow2 69.9%

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

       3. associate-/l* 70.2%

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

       4. *-commutative 70.2%

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

       5. associate-/r* 72.9%

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

       6. associate-*l/ 73.8%

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

       7. associate-*r* 75.3%

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

       8. associate-/l* 72.7%

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

       9. *-commutative 72.7%

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

       10. associate-*l/ 73.7%

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

       11. associate-/r/ 70.8%

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

       12. associate-*l* 79.5%

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

       13. associate-*l/ 83.8%

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

       14. associate-*r/ 83.8%

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

       15. *-commutative 83.8%

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

       16. associate-*r* 84.0%

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

       17. *-commutative 84.0%

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

       18. associate-*l* 86.5%

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

   14. Simplified 86.5%

       \[\leadsto \sqrt{1 + -0.25 \cdot \left(\frac{D}{d} \cdot \color{blue}{\left(\left(h \cdot M\right) \cdot \left(\frac{M}{\ell} \cdot \frac{D}{d}\right)\right)}\right)} \cdot w0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6 \cdot 10^{-131}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(h \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{\ell}\right) \cdot \left(M \cdot h\right)\right)\right)}\\ \end{array} \]

Alternative 7: 75.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 8.4 \cdot 10^{+36}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{D \cdot {\left(\frac{M}{d}\right)}^{2}}{\frac{\ell}{D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 8.4e+36)
   (* w0 (+ 1.0 (* -0.125 (* h (/ (* D (pow (/ M d) 2.0)) (/ l D))))))
   (* w0 (+ 1.0 (* -0.125 (/ (* (* (/ D d) (/ D d)) (* M (* M h))) l))))))

C

D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 8.4e+36) {
		tmp = w0 * (1.0 + (-0.125 * (h * ((D * pow((M / d), 2.0)) / (l / D)))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * (M * h))) / l)));
	}
	return tmp;
}

Fortran

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 8.4d+36) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (h * ((d * ((m / d_1) ** 2.0d0)) / (l / d)))))
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((((d / d_1) * (d / d_1)) * (m * (m * h))) / l)))
    end if
    code = tmp
end function

Java

D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 8.4e+36) {
		tmp = w0 * (1.0 + (-0.125 * (h * ((D * Math.pow((M / d), 2.0)) / (l / D)))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * (M * h))) / l)));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 8.4e+36:
		tmp = w0 * (1.0 + (-0.125 * (h * ((D * math.pow((M / d), 2.0)) / (l / D)))))
	else:
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * (M * h))) / l)))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 8.4e+36)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(h * Float64(Float64(D * (Float64(M / d) ^ 2.0)) / Float64(l / D))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(M * Float64(M * h))) / l))));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 8.4e+36)
		tmp = w0 * (1.0 + (-0.125 * (h * ((D * ((M / d) ^ 2.0)) / (l / D)))));
	else
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * (M * h))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 8.4e+36], N[(w0 * N[(1.0 + N[(-0.125 * N[(h * N[(N[(D * N[Power[N[(M / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 8.40000000000000018e36

   1. Initial program 79.7%

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

   2. Taylor expanded in M around 0 52.1%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)} \]
   3. Step-by-step derivation

      1. associate-/l* 51.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 51.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2}}{\frac{\color{blue}{{d}^{2} \cdot \ell}}{{M}^{2} \cdot h}}\right) \]

      3. *-commutative 51.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2}}{\frac{{d}^{2} \cdot \ell}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      4. associate-/l* 52.1%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}}\right) \]

      5. times-frac 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right) \]

      6. unpow2 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right) \]

      7. unpow2 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right) \]

      8. *-commutative 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right) \]

      9. unpow2 51.0%

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

   4. Simplified 51.0%

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

   5. Taylor expanded in D around 0 52.1%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   6. Step-by-step derivation

      1. unpow2 52.1%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}}\right) \]

      2. *-commutative 52.1%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}\right) \]

      3. times-frac 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \]

      4. unpow2 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) \]

      5. associate-*l/ 50.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \color{blue}{\left(\frac{M \cdot M}{\ell} \cdot h\right)}\right)\right) \]

      6. unpow2 50.5%

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

      7. times-frac 61.6%

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

      8. associate-*r* 63.2%

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

      9. *-commutative 63.2%

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

      10. times-frac 52.1%

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

      11. unpow2 52.1%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \left(\frac{\color{blue}{{D}^{2}}}{d \cdot d} \cdot \frac{M \cdot M}{\ell}\right)\right)\right) \]

      12. times-frac 53.2%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\frac{{D}^{2} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}\right)\right) \]

      13. *-commutative 53.2%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{{D}^{2} \cdot \left(M \cdot M\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)\right) \]

      14. times-frac 54.8%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{M \cdot M}{d \cdot d}\right)}\right)\right) \]

      15. unpow2 54.8%

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

      16. associate-/l* 58.6%

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

      17. times-frac 72.1%

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

   7. Simplified 72.1%

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

      1. associate-*l/ 73.8%

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

      2. pow2 73.8%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{D \cdot \color{blue}{{\left(\frac{M}{d}\right)}^{2}}}{\frac{\ell}{D}}\right)\right) \]

   9. Applied egg-rr 73.8%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\frac{D \cdot {\left(\frac{M}{d}\right)}^{2}}{\frac{\ell}{D}}}\right)\right) \]

   if 8.40000000000000018e36 < d

   1. Initial program 85.3%

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

   2. Taylor expanded in M around 0 65.9%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)} \]
   3. Step-by-step derivation

      1. associate-/l* 65.8%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 65.8%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2}}{\frac{\color{blue}{{d}^{2} \cdot \ell}}{{M}^{2} \cdot h}}\right) \]

      3. *-commutative 65.8%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2}}{\frac{{d}^{2} \cdot \ell}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      4. associate-/l* 65.9%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}}\right) \]

      5. times-frac 64.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right) \]

      6. unpow2 64.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right) \]

      7. unpow2 64.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right) \]

      8. *-commutative 64.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right) \]

      9. unpow2 64.4%

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

   4. Simplified 64.4%

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

      1. associate-*r/ 65.9%

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

      2. times-frac 76.6%

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

      3. *-commutative 76.6%

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

   6. Applied egg-rr 76.6%

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

   7. Taylor expanded in h around 0 76.6%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell}\right) \]
   8. Step-by-step derivation

      1. *-commutative 76.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{\ell}\right) \]

      2. unpow2 76.6%

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

      3. associate-*r* 81.0%

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

      4. *-commutative 81.0%

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

      5. *-commutative 81.0%

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

   9. Simplified 81.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 8.4 \cdot 10^{+36}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{D \cdot {\left(\frac{M}{d}\right)}^{2}}{\frac{\ell}{D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)}{\ell}\right)\\ \end{array} \]

Alternative 8: 72.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4.5 \cdot 10^{+57}:\\ \;\;\;\;w0 \cdot \left(1 + \left(h \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 4.5e+57)
   (* w0 (+ 1.0 (* (* h (* (/ D (/ l D)) (* (/ M d) (/ M d)))) -0.125)))
   w0))

C

D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 4.5e+57) {
		tmp = w0 * (1.0 + ((h * ((D / (l / D)) * ((M / d) * (M / d)))) * -0.125));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 4.5d+57) then
        tmp = w0 * (1.0d0 + ((h * ((d / (l / d)) * ((m / d_1) * (m / d_1)))) * (-0.125d0)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 4.5e+57) {
		tmp = w0 * (1.0 + ((h * ((D / (l / D)) * ((M / d) * (M / d)))) * -0.125));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 4.5e+57:
		tmp = w0 * (1.0 + ((h * ((D / (l / D)) * ((M / d) * (M / d)))) * -0.125))
	else:
		tmp = w0
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 4.5e+57)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(h * Float64(Float64(D / Float64(l / D)) * Float64(Float64(M / d) * Float64(M / d)))) * -0.125)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 4.5e+57)
		tmp = w0 * (1.0 + ((h * ((D / (l / D)) * ((M / d) * (M / d)))) * -0.125));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 4.5e+57], N[(w0 * N[(1.0 + N[(N[(h * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 4.49999999999999996e57

   1. Initial program 80.3%

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

   2. Taylor expanded in M around 0 53.1%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)} \]
   3. Step-by-step derivation

      1. associate-/l* 52.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 52.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2}}{\frac{\color{blue}{{d}^{2} \cdot \ell}}{{M}^{2} \cdot h}}\right) \]

      3. *-commutative 52.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2}}{\frac{{d}^{2} \cdot \ell}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      4. associate-/l* 53.1%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}}\right) \]

      5. times-frac 52.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right) \]

      6. unpow2 52.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right) \]

      7. unpow2 52.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right) \]

      8. *-commutative 52.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right) \]

      9. unpow2 52.0%

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

   4. Simplified 52.0%

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

   5. Taylor expanded in D around 0 53.1%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   6. Step-by-step derivation

      1. unpow2 53.1%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}}\right) \]

      2. *-commutative 53.1%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}\right) \]

      3. times-frac 52.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \]

      4. unpow2 52.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) \]

      5. associate-*l/ 51.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \color{blue}{\left(\frac{M \cdot M}{\ell} \cdot h\right)}\right)\right) \]

      6. unpow2 51.5%

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

      7. times-frac 62.8%

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

      8. associate-*r* 64.3%

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

      9. *-commutative 64.3%

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

      10. times-frac 53.1%

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

      11. unpow2 53.1%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \left(\frac{\color{blue}{{D}^{2}}}{d \cdot d} \cdot \frac{M \cdot M}{\ell}\right)\right)\right) \]

      12. times-frac 54.1%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\frac{{D}^{2} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}\right)\right) \]

      13. *-commutative 54.1%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{{D}^{2} \cdot \left(M \cdot M\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)\right) \]

      14. times-frac 55.7%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{M \cdot M}{d \cdot d}\right)}\right)\right) \]

      15. unpow2 55.7%

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

      16. associate-/l* 59.4%

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

      17. times-frac 72.4%

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

   7. Simplified 72.4%

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

   if 4.49999999999999996e57 < d

   1. Initial program 83.8%

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

   2. Taylor expanded in M around 0 87.0%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.5 \cdot 10^{+57}:\\ \;\;\;\;w0 \cdot \left(1 + \left(h \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 9: 71.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 6.7 \cdot 10^{+36}:\\ \;\;\;\;w0 \cdot \left(1 + \left(h \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 6.7e+36)
   (* w0 (+ 1.0 (* (* h (* (/ D (/ l D)) (* (/ M d) (/ M d)))) -0.125)))
   (* w0 (+ 1.0 (* (* (* (/ D d) (/ D d)) (/ (* h (* M M)) l)) -0.125)))))

C

D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 6.7e+36) {
		tmp = w0 * (1.0 + ((h * ((D / (l / D)) * ((M / d) * (M / d)))) * -0.125));
	} else {
		tmp = w0 * (1.0 + ((((D / d) * (D / d)) * ((h * (M * M)) / l)) * -0.125));
	}
	return tmp;
}

Fortran

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 6.7d+36) then
        tmp = w0 * (1.0d0 + ((h * ((d / (l / d)) * ((m / d_1) * (m / d_1)))) * (-0.125d0)))
    else
        tmp = w0 * (1.0d0 + ((((d / d_1) * (d / d_1)) * ((h * (m * m)) / l)) * (-0.125d0)))
    end if
    code = tmp
end function

Java

D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 6.7e+36) {
		tmp = w0 * (1.0 + ((h * ((D / (l / D)) * ((M / d) * (M / d)))) * -0.125));
	} else {
		tmp = w0 * (1.0 + ((((D / d) * (D / d)) * ((h * (M * M)) / l)) * -0.125));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 6.7e+36:
		tmp = w0 * (1.0 + ((h * ((D / (l / D)) * ((M / d) * (M / d)))) * -0.125))
	else:
		tmp = w0 * (1.0 + ((((D / d) * (D / d)) * ((h * (M * M)) / l)) * -0.125))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 6.7e+36)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(h * Float64(Float64(D / Float64(l / D)) * Float64(Float64(M / d) * Float64(M / d)))) * -0.125)));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(h * Float64(M * M)) / l)) * -0.125)));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 6.7e+36)
		tmp = w0 * (1.0 + ((h * ((D / (l / D)) * ((M / d) * (M / d)))) * -0.125));
	else
		tmp = w0 * (1.0 + ((((D / d) * (D / d)) * ((h * (M * M)) / l)) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 6.7e+36], N[(w0 * N[(1.0 + N[(N[(h * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 6.6999999999999997e36

   1. Initial program 79.7%

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

   2. Taylor expanded in M around 0 52.1%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)} \]
   3. Step-by-step derivation

      1. associate-/l* 51.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 51.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2}}{\frac{\color{blue}{{d}^{2} \cdot \ell}}{{M}^{2} \cdot h}}\right) \]

      3. *-commutative 51.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2}}{\frac{{d}^{2} \cdot \ell}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      4. associate-/l* 52.1%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}}\right) \]

      5. times-frac 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right) \]

      6. unpow2 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right) \]

      7. unpow2 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right) \]

      8. *-commutative 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right) \]

      9. unpow2 51.0%

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

   4. Simplified 51.0%

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

   5. Taylor expanded in D around 0 52.1%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   6. Step-by-step derivation

      1. unpow2 52.1%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}}\right) \]

      2. *-commutative 52.1%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}\right) \]

      3. times-frac 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \]

      4. unpow2 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) \]

      5. associate-*l/ 50.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \color{blue}{\left(\frac{M \cdot M}{\ell} \cdot h\right)}\right)\right) \]

      6. unpow2 50.5%

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

      7. times-frac 61.6%

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

      8. associate-*r* 63.2%

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

      9. *-commutative 63.2%

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

      10. times-frac 52.1%

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

      11. unpow2 52.1%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \left(\frac{\color{blue}{{D}^{2}}}{d \cdot d} \cdot \frac{M \cdot M}{\ell}\right)\right)\right) \]

      12. times-frac 53.2%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\frac{{D}^{2} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}\right)\right) \]

      13. *-commutative 53.2%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{{D}^{2} \cdot \left(M \cdot M\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)\right) \]

      14. times-frac 54.8%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{M \cdot M}{d \cdot d}\right)}\right)\right) \]

      15. unpow2 54.8%

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

      16. associate-/l* 58.6%

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

      17. times-frac 72.1%

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

   7. Simplified 72.1%

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

   if 6.6999999999999997e36 < d

   1. Initial program 85.3%

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

   2. Taylor expanded in M around 0 65.9%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)} \]
   3. Step-by-step derivation

      1. associate-/l* 65.8%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 65.8%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2}}{\frac{\color{blue}{{d}^{2} \cdot \ell}}{{M}^{2} \cdot h}}\right) \]

      3. *-commutative 65.8%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2}}{\frac{{d}^{2} \cdot \ell}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      4. associate-/l* 65.9%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}}\right) \]

      5. times-frac 64.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right) \]

      6. unpow2 64.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right) \]

      7. unpow2 64.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right) \]

      8. *-commutative 64.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right) \]

      9. unpow2 64.4%

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

   4. Simplified 64.4%

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

   5. Taylor expanded in D around 0 64.4%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{{d}^{2}}} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 64.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right)\right) \]

      2. unpow2 64.4%

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

      3. times-frac 75.1%

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

   7. Simplified 75.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6.7 \cdot 10^{+36}:\\ \;\;\;\;w0 \cdot \left(1 + \left(h \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 10: 72.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 5.4 \cdot 10^{+36}:\\ \;\;\;\;w0 \cdot \left(1 + \left(h \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 5.4e+36)
   (* w0 (+ 1.0 (* (* h (* (/ D (/ l D)) (* (/ M d) (/ M d)))) -0.125)))
   (* w0 (+ 1.0 (* -0.125 (/ (* (* (/ D d) (/ D d)) (* M (* M h))) l))))))

C

D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 5.4e+36) {
		tmp = w0 * (1.0 + ((h * ((D / (l / D)) * ((M / d) * (M / d)))) * -0.125));
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * (M * h))) / l)));
	}
	return tmp;
}

Fortran

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 5.4d+36) then
        tmp = w0 * (1.0d0 + ((h * ((d / (l / d)) * ((m / d_1) * (m / d_1)))) * (-0.125d0)))
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((((d / d_1) * (d / d_1)) * (m * (m * h))) / l)))
    end if
    code = tmp
end function

Java

D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 5.4e+36) {
		tmp = w0 * (1.0 + ((h * ((D / (l / D)) * ((M / d) * (M / d)))) * -0.125));
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * (M * h))) / l)));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 5.4e+36:
		tmp = w0 * (1.0 + ((h * ((D / (l / D)) * ((M / d) * (M / d)))) * -0.125))
	else:
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * (M * h))) / l)))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 5.4e+36)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(h * Float64(Float64(D / Float64(l / D)) * Float64(Float64(M / d) * Float64(M / d)))) * -0.125)));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(M * Float64(M * h))) / l))));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 5.4e+36)
		tmp = w0 * (1.0 + ((h * ((D / (l / D)) * ((M / d) * (M / d)))) * -0.125));
	else
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * (M * h))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 5.4e+36], N[(w0 * N[(1.0 + N[(N[(h * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 5.4000000000000002e36

   1. Initial program 79.7%

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

   2. Taylor expanded in M around 0 52.1%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)} \]
   3. Step-by-step derivation

      1. associate-/l* 51.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 51.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2}}{\frac{\color{blue}{{d}^{2} \cdot \ell}}{{M}^{2} \cdot h}}\right) \]

      3. *-commutative 51.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2}}{\frac{{d}^{2} \cdot \ell}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      4. associate-/l* 52.1%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}}\right) \]

      5. times-frac 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right) \]

      6. unpow2 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right) \]

      7. unpow2 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right) \]

      8. *-commutative 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right) \]

      9. unpow2 51.0%

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

   4. Simplified 51.0%

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

   5. Taylor expanded in D around 0 52.1%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   6. Step-by-step derivation

      1. unpow2 52.1%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}}\right) \]

      2. *-commutative 52.1%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}\right) \]

      3. times-frac 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \]

      4. unpow2 51.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) \]

      5. associate-*l/ 50.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \color{blue}{\left(\frac{M \cdot M}{\ell} \cdot h\right)}\right)\right) \]

      6. unpow2 50.5%

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

      7. times-frac 61.6%

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

      8. associate-*r* 63.2%

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

      9. *-commutative 63.2%

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

      10. times-frac 52.1%

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

      11. unpow2 52.1%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \left(\frac{\color{blue}{{D}^{2}}}{d \cdot d} \cdot \frac{M \cdot M}{\ell}\right)\right)\right) \]

      12. times-frac 53.2%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\frac{{D}^{2} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}\right)\right) \]

      13. *-commutative 53.2%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{{D}^{2} \cdot \left(M \cdot M\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)\right) \]

      14. times-frac 54.8%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{M \cdot M}{d \cdot d}\right)}\right)\right) \]

      15. unpow2 54.8%

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

      16. associate-/l* 58.6%

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

      17. times-frac 72.1%

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

   7. Simplified 72.1%

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

   if 5.4000000000000002e36 < d

   1. Initial program 85.3%

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

   2. Taylor expanded in M around 0 65.9%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)} \]
   3. Step-by-step derivation

      1. associate-/l* 65.8%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 65.8%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2}}{\frac{\color{blue}{{d}^{2} \cdot \ell}}{{M}^{2} \cdot h}}\right) \]

      3. *-commutative 65.8%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2}}{\frac{{d}^{2} \cdot \ell}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      4. associate-/l* 65.9%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}}\right) \]

      5. times-frac 64.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right) \]

      6. unpow2 64.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right) \]

      7. unpow2 64.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right) \]

      8. *-commutative 64.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right) \]

      9. unpow2 64.4%

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

   4. Simplified 64.4%

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

      1. associate-*r/ 65.9%

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

      2. times-frac 76.6%

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

      3. *-commutative 76.6%

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

   6. Applied egg-rr 76.6%

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

   7. Taylor expanded in h around 0 76.6%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell}\right) \]
   8. Step-by-step derivation

      1. *-commutative 76.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{\ell}\right) \]

      2. unpow2 76.6%

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

      3. associate-*r* 81.0%

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

      4. *-commutative 81.0%

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

      5. *-commutative 81.0%

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

   9. Simplified 81.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5.4 \cdot 10^{+36}:\\ \;\;\;\;w0 \cdot \left(1 + \left(h \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)}{\ell}\right)\\ \end{array} \]

Alternative 11: 69.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 0.000136:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{\ell}{w0}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 0.000136)
   w0
   (* -0.125 (* (* (/ D d) (/ D d)) (/ (* M (* M h)) (/ l w0))))))

C

D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 0.000136) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (D / d)) * ((M * (M * h)) / (l / w0)));
	}
	return tmp;
}

Fortran

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 0.000136d0) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d / d_1) * (d / d_1)) * ((m * (m * h)) / (l / w0)))
    end if
    code = tmp
end function

Java

D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 0.000136) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (D / d)) * ((M * (M * h)) / (l / w0)));
	}
	return tmp;
}

Python

D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 0.000136:
		tmp = w0
	else:
		tmp = -0.125 * (((D / d) * (D / d)) * ((M * (M * h)) / (l / w0)))
	return tmp

Julia

D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 0.000136)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * Float64(M * h)) / Float64(l / w0))));
	end
	return tmp
end

MATLAB

D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 0.000136)
		tmp = w0;
	else
		tmp = -0.125 * (((D / d) * (D / d)) * ((M * (M * h)) / (l / w0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 0.000136], w0, N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(l / w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.36e-4

   1. Initial program 86.7%

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

   2. Taylor expanded in M around 0 72.8%

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

   if 1.36e-4 < M

   1. Initial program 62.5%

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

   2. Taylor expanded in M around 0 43.6%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)} \]
   3. Step-by-step derivation

      1. associate-/l* 43.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell \cdot {d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      2. *-commutative 43.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2}}{\frac{\color{blue}{{d}^{2} \cdot \ell}}{{M}^{2} \cdot h}}\right) \]

      3. *-commutative 43.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2}}{\frac{{d}^{2} \cdot \ell}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      4. associate-/l* 43.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}}\right) \]

      5. times-frac 40.3%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)}\right) \]

      6. unpow2 40.3%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right) \]

      7. unpow2 40.3%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right) \]

      8. *-commutative 40.3%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right) \]

      9. unpow2 40.3%

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

   4. Simplified 40.3%

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

   5. Taylor expanded in D around 0 43.6%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   6. Step-by-step derivation

      1. unpow2 43.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}}\right) \]

      2. *-commutative 43.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{\left(d \cdot d\right) \cdot \ell}}\right) \]

      3. times-frac 40.3%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{{M}^{2} \cdot h}{\ell}\right)}\right) \]

      4. unpow2 40.3%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) \]

      5. associate-*l/ 37.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{{D}^{2}}{d \cdot d} \cdot \color{blue}{\left(\frac{M \cdot M}{\ell} \cdot h\right)}\right)\right) \]

      6. unpow2 37.0%

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

      7. times-frac 45.7%

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

      8. associate-*r* 49.1%

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

      9. *-commutative 49.1%

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

      10. times-frac 40.3%

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

      11. unpow2 40.3%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \left(\frac{\color{blue}{{D}^{2}}}{d \cdot d} \cdot \frac{M \cdot M}{\ell}\right)\right)\right) \]

      12. times-frac 43.6%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\frac{{D}^{2} \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}\right)\right) \]

      13. *-commutative 43.6%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{{D}^{2} \cdot \left(M \cdot M\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)\right) \]

      14. times-frac 42.2%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{M \cdot M}{d \cdot d}\right)}\right)\right) \]

      15. unpow2 42.2%

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

      16. associate-/l* 46.6%

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

      17. times-frac 57.2%

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

   7. Simplified 57.2%

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

   8. Taylor expanded in h around inf 17.3%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 17.3%

         \[\leadsto \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)\right)}{\ell \cdot {d}^{2}}} \]

      2. *-commutative 17.3%

         \[\leadsto \frac{-0.125 \cdot \left({D}^{2} \cdot \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

      3. *-commutative 17.3%

         \[\leadsto \frac{-0.125 \cdot \left({D}^{2} \cdot \left(w0 \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right)}{{d}^{2} \cdot \ell} \]

      4. associate-*r/ 17.3%

         \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)}{{d}^{2} \cdot \ell}} \]

      5. times-frac 17.5%

         \[\leadsto -0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right)} \]

      6. unpow2 17.5%

         \[\leadsto -0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{{d}^{2}} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right) \]

      7. unpow2 17.5%

         \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\color{blue}{d \cdot d}} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right) \]

      8. times-frac 21.5%

         \[\leadsto -0.125 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right) \]

      9. unpow2 21.5%

         \[\leadsto -0.125 \cdot \left(\color{blue}{{\left(\frac{D}{d}\right)}^{2}} \cdot \frac{w0 \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right) \]

      10. *-commutative 21.5%

          \[\leadsto -0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{\color{blue}{\left(h \cdot {M}^{2}\right) \cdot w0}}{\ell}\right) \]

      11. associate-/l* 21.3%

          \[\leadsto -0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \color{blue}{\frac{h \cdot {M}^{2}}{\frac{\ell}{w0}}}\right) \]

      12. unpow2 21.3%

          \[\leadsto -0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\frac{\ell}{w0}}\right) \]

      13. associate-*r* 21.5%

          \[\leadsto -0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{\color{blue}{\left(h \cdot M\right) \cdot M}}{\frac{\ell}{w0}}\right) \]

      14. *-commutative 21.5%

          \[\leadsto -0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{\color{blue}{M \cdot \left(h \cdot M\right)}}{\frac{\ell}{w0}}\right) \]

      15. *-commutative 21.5%

          \[\leadsto -0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{M \cdot \color{blue}{\left(M \cdot h\right)}}{\frac{\ell}{w0}}\right) \]

   10. Simplified 21.5%

       \[\leadsto \color{blue}{-0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{\ell}{w0}}\right)} \]
   11. Step-by-step derivation

       1. unpow2 21.5%

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

   12. Applied egg-rr 21.5%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 0.000136:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{\ell}{w0}}\right)\\ \end{array} \]

Alternative 12: 67.5% accurate, 216.0× speedup

Math

\[\begin{array}{l} D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \end{array} \]

FPCore

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

C

D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Fortran

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 81.1%

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

2. Taylor expanded in M around 0 67.6%

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

3. Final simplification 67.6%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023279 
(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))))))