Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.2% → 90.8%

Time: 20.0s

Alternatives: 19

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.8% accurate, 0.2× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) (- INFINITY))
   (*
    w0
    (pow
     (exp 0.5)
     (fma
      2.0
      (log D)
      (fma -2.0 (log d) (fma 2.0 (log M) (log (* (/ h l) -0.25)))))))
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* 0.5 (/ (* M D) d)) 2.0)) l))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -((double) INFINITY)) {
		tmp = w0 * pow(exp(0.5), fma(2.0, log(D), fma(-2.0, log(d), fma(2.0, log(M), log(((h / l) * -0.25))))));
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow((0.5 * ((M * D) / d)), 2.0)) / l)));
	}
	return tmp;
}

Julia

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

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(w0 * N[Power[N[Exp[0.5], $MachinePrecision], N[(2.0 * N[Log[D], $MachinePrecision] + N[(-2.0 * N[Log[d], $MachinePrecision] + N[(2.0 * N[Log[M], $MachinePrecision] + N[Log[N[(N[(h / l), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(0.5 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -inf.0

   1. Initial program 53.3%

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

      1. *-commutative 53.3%

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

      2. times-frac 53.3%

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

   3. Simplified 53.3%

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

   5. Applied egg-rr 53.3%

      \[\leadsto w0 \cdot \color{blue}{e^{\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-\frac{h}{\ell}\right)\right) \cdot 0.5}} \]

   6. Taylor expanded in d around 0 25.1%

      \[\leadsto w0 \cdot e^{\color{blue}{\left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right) + -2 \cdot \log d\right)} \cdot 0.5} \]
   7. Step-by-step derivation

      1. associate-/l* 23.7%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell}{h \cdot {M}^{2}}}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

      2. *-commutative 23.7%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{{D}^{2}}{\frac{\ell}{\color{blue}{{M}^{2} \cdot h}}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

      3. unpow2 23.7%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{{D}^{2}}{\frac{\ell}{\color{blue}{\left(M \cdot M\right)} \cdot h}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

      4. associate-*r* 26.5%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{{D}^{2}}{\frac{\ell}{\color{blue}{M \cdot \left(M \cdot h\right)}}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

      5. unpow2 26.5%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{\ell}{M \cdot \left(M \cdot h\right)}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

   8. Simplified 26.5%

      \[\leadsto w0 \cdot e^{\color{blue}{\left(\log \left(-0.25 \cdot \frac{D \cdot D}{\frac{\ell}{M \cdot \left(M \cdot h\right)}}\right) + -2 \cdot \log d\right)} \cdot 0.5} \]

   9. Taylor expanded in D around 0 6.6%

      \[\leadsto w0 \cdot e^{\left(\color{blue}{\left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell}\right) + 2 \cdot \log D\right)} + -2 \cdot \log d\right) \cdot 0.5} \]
   10. Step-by-step derivation

       1. associate-/l* 5.2%

          \[\leadsto w0 \cdot e^{\left(\left(\log \left(-0.25 \cdot \color{blue}{\frac{h}{\frac{\ell}{{M}^{2}}}}\right) + 2 \cdot \log D\right) + -2 \cdot \log d\right) \cdot 0.5} \]

       2. unpow2 5.2%

          \[\leadsto w0 \cdot e^{\left(\left(\log \left(-0.25 \cdot \frac{h}{\frac{\ell}{\color{blue}{M \cdot M}}}\right) + 2 \cdot \log D\right) + -2 \cdot \log d\right) \cdot 0.5} \]

   11. Simplified 5.2%

       \[\leadsto w0 \cdot e^{\left(\color{blue}{\left(\log \left(-0.25 \cdot \frac{h}{\frac{\ell}{M \cdot M}}\right) + 2 \cdot \log D\right)} + -2 \cdot \log d\right) \cdot 0.5} \]

   12. Taylor expanded in w0 around 0 6.7%

       \[\leadsto \color{blue}{e^{0.5 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell}\right) + \left(2 \cdot \log D + -2 \cdot \log d\right)\right)} \cdot w0} \]
   13. Step-by-step derivation

       1. *-commutative 6.7%

          \[\leadsto \color{blue}{w0 \cdot e^{0.5 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell}\right) + \left(2 \cdot \log D + -2 \cdot \log d\right)\right)}} \]

       2. exp-prod 6.6%

          \[\leadsto w0 \cdot \color{blue}{{\left(e^{0.5}\right)}^{\left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell}\right) + \left(2 \cdot \log D + -2 \cdot \log d\right)\right)}} \]

       3. +-commutative 6.6%

          \[\leadsto w0 \cdot {\left(e^{0.5}\right)}^{\color{blue}{\left(\left(2 \cdot \log D + -2 \cdot \log d\right) + \log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right)}} \]

       4. *-commutative 6.6%

          \[\leadsto w0 \cdot {\left(e^{0.5}\right)}^{\left(\left(2 \cdot \log D + -2 \cdot \log d\right) + \log \left(-0.25 \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right)} \]

       5. associate-+r+ 6.6%

          \[\leadsto w0 \cdot {\left(e^{0.5}\right)}^{\color{blue}{\left(2 \cdot \log D + \left(-2 \cdot \log d + \log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right)\right)}} \]

       6. fma-def 6.6%

          \[\leadsto w0 \cdot {\left(e^{0.5}\right)}^{\color{blue}{\left(\mathsf{fma}\left(2, \log D, -2 \cdot \log d + \log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right)\right)}} \]

       7. fma-def 6.6%

          \[\leadsto w0 \cdot {\left(e^{0.5}\right)}^{\left(\mathsf{fma}\left(2, \log D, \color{blue}{\mathsf{fma}\left(-2, \log d, \log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right)}\right)\right)} \]

       8. log-prod 0.0%

          \[\leadsto w0 \cdot {\left(e^{0.5}\right)}^{\left(\mathsf{fma}\left(2, \log D, \mathsf{fma}\left(-2, \log d, \color{blue}{\log -0.25 + \log \left(\frac{{M}^{2} \cdot h}{\ell}\right)}\right)\right)\right)} \]

       9. *-commutative 0.0%

          \[\leadsto w0 \cdot {\left(e^{0.5}\right)}^{\left(\mathsf{fma}\left(2, \log D, \mathsf{fma}\left(-2, \log d, \log -0.25 + \log \left(\frac{\color{blue}{h \cdot {M}^{2}}}{\ell}\right)\right)\right)\right)} \]

       10. associate-*l/ 0.0%

           \[\leadsto w0 \cdot {\left(e^{0.5}\right)}^{\left(\mathsf{fma}\left(2, \log D, \mathsf{fma}\left(-2, \log d, \log -0.25 + \log \color{blue}{\left(\frac{h}{\ell} \cdot {M}^{2}\right)}\right)\right)\right)} \]

       11. unpow2 0.0%

           \[\leadsto w0 \cdot {\left(e^{0.5}\right)}^{\left(\mathsf{fma}\left(2, \log D, \mathsf{fma}\left(-2, \log d, \log -0.25 + \log \left(\frac{h}{\ell} \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)\right)\right)} \]

   14. Simplified 4.7%

       \[\leadsto \color{blue}{w0 \cdot {\left(e^{0.5}\right)}^{\left(\mathsf{fma}\left(2, \log D, \mathsf{fma}\left(-2, \log d, \mathsf{fma}\left(2, \log M, \log \left(-0.25 \cdot \frac{h}{\ell}\right)\right)\right)\right)\right)}} \]

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 92.2%

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

      1. *-commutative 92.2%

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

      2. times-frac 92.3%

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

   3. Simplified 92.3%

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

      1. unpow2 92.3%

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

      2. associate-*l* 92.9%

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

      3. div-inv 92.8%

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

      4. associate-*l* 92.4%

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

      5. times-frac 92.4%

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

      6. *-un-lft-identity 92.4%

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

      7. *-commutative 92.4%

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

      8. clear-num 92.4%

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

      9. div-inv 92.4%

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

      10. div-inv 92.3%

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

      11. associate-*l* 93.4%

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

      12. times-frac 93.4%

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

      13. *-un-lft-identity 93.4%

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

      14. *-commutative 93.4%

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

      15. clear-num 93.4%

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

      16. div-inv 93.4%

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

      17. associate-*l* 92.8%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in M around 0 97.5%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot {\left(e^{0.5}\right)}^{\left(\mathsf{fma}\left(2, \log D, \mathsf{fma}\left(-2, \log d, \mathsf{fma}\left(2, \log M, \log \left(\frac{h}{\ell} \cdot -0.25\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 2: 90.8% accurate, 0.3× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) (- INFINITY))
   (*
    w0
    (exp
     (*
      0.5
      (+
       (log (* (/ h l) -0.25))
       (+ (* 2.0 (log M)) (+ (* 2.0 (log D)) (* -2.0 (log d))))))))
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* 0.5 (/ (* M D) d)) 2.0)) l))))))

C

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

Java

M = Math.abs(M);
D = Math.abs(D);
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 ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.exp((0.5 * (Math.log(((h / l) * -0.25)) + ((2.0 * Math.log(M)) + ((2.0 * Math.log(D)) + (-2.0 * Math.log(d)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((0.5 * ((M * D) / d)), 2.0)) / l)));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(w0 * N[Exp[N[(0.5 * N[(N[Log[N[(N[(h / l), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] + N[(N[(2.0 * N[Log[M], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[Log[D], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[Log[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(0.5 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -inf.0

   1. Initial program 53.3%

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

      1. *-commutative 53.3%

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

      2. times-frac 53.3%

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

   3. Simplified 53.3%

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

   5. Applied egg-rr 53.3%

      \[\leadsto w0 \cdot \color{blue}{e^{\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-\frac{h}{\ell}\right)\right) \cdot 0.5}} \]

   6. Taylor expanded in d around 0 25.1%

      \[\leadsto w0 \cdot e^{\color{blue}{\left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right) + -2 \cdot \log d\right)} \cdot 0.5} \]
   7. Step-by-step derivation

      1. associate-/l* 23.7%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell}{h \cdot {M}^{2}}}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

      2. *-commutative 23.7%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{{D}^{2}}{\frac{\ell}{\color{blue}{{M}^{2} \cdot h}}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

      3. unpow2 23.7%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{{D}^{2}}{\frac{\ell}{\color{blue}{\left(M \cdot M\right)} \cdot h}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

      4. associate-*r* 26.5%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{{D}^{2}}{\frac{\ell}{\color{blue}{M \cdot \left(M \cdot h\right)}}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

      5. unpow2 26.5%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{\ell}{M \cdot \left(M \cdot h\right)}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

   8. Simplified 26.5%

      \[\leadsto w0 \cdot e^{\color{blue}{\left(\log \left(-0.25 \cdot \frac{D \cdot D}{\frac{\ell}{M \cdot \left(M \cdot h\right)}}\right) + -2 \cdot \log d\right)} \cdot 0.5} \]

   9. Taylor expanded in D around 0 6.6%

      \[\leadsto w0 \cdot e^{\left(\color{blue}{\left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell}\right) + 2 \cdot \log D\right)} + -2 \cdot \log d\right) \cdot 0.5} \]
   10. Step-by-step derivation

       1. associate-/l* 5.2%

          \[\leadsto w0 \cdot e^{\left(\left(\log \left(-0.25 \cdot \color{blue}{\frac{h}{\frac{\ell}{{M}^{2}}}}\right) + 2 \cdot \log D\right) + -2 \cdot \log d\right) \cdot 0.5} \]

       2. unpow2 5.2%

          \[\leadsto w0 \cdot e^{\left(\left(\log \left(-0.25 \cdot \frac{h}{\frac{\ell}{\color{blue}{M \cdot M}}}\right) + 2 \cdot \log D\right) + -2 \cdot \log d\right) \cdot 0.5} \]

   11. Simplified 5.2%

       \[\leadsto w0 \cdot e^{\left(\color{blue}{\left(\log \left(-0.25 \cdot \frac{h}{\frac{\ell}{M \cdot M}}\right) + 2 \cdot \log D\right)} + -2 \cdot \log d\right) \cdot 0.5} \]

   12. Taylor expanded in M around 0 4.7%

       \[\leadsto \color{blue}{e^{0.5 \cdot \left(\log \left(-0.25 \cdot \frac{h}{\ell}\right) + \left(2 \cdot \log M + \left(2 \cdot \log D + -2 \cdot \log d\right)\right)\right)} \cdot w0} \]

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 92.2%

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

      1. *-commutative 92.2%

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

      2. times-frac 92.3%

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

   3. Simplified 92.3%

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

      1. unpow2 92.3%

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

      2. associate-*l* 92.9%

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

      3. div-inv 92.8%

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

      4. associate-*l* 92.4%

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

      5. times-frac 92.4%

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

      6. *-un-lft-identity 92.4%

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

      7. *-commutative 92.4%

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

      8. clear-num 92.4%

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

      9. div-inv 92.4%

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

      10. div-inv 92.3%

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

      11. associate-*l* 93.4%

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

      12. times-frac 93.4%

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

      13. *-un-lft-identity 93.4%

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

      14. *-commutative 93.4%

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

      15. clear-num 93.4%

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

      16. div-inv 93.4%

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

      17. associate-*l* 92.8%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in M around 0 97.5%

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

4. Final simplification 75.4%

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

Alternative 3: 88.7% accurate, 0.3× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) (- INFINITY))
   (*
    w0
    (exp
     (*
      0.5
      (+
       (* -2.0 (log d))
       (fma 2.0 (log D) (log (* -0.25 (/ (* M h) (/ l M)))))))))
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* 0.5 (/ (* M D) d)) 2.0)) l))))))

C

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

Julia

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

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(w0 * N[Exp[N[(0.5 * N[(N[(-2.0 * N[Log[d], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Log[D], $MachinePrecision] + N[Log[N[(-0.25 * N[(N[(M * h), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(0.5 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -inf.0

   1. Initial program 53.3%

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

      1. *-commutative 53.3%

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

      2. times-frac 53.3%

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

   3. Simplified 53.3%

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

   5. Applied egg-rr 53.3%

      \[\leadsto w0 \cdot \color{blue}{e^{\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-\frac{h}{\ell}\right)\right) \cdot 0.5}} \]

   6. Taylor expanded in d around 0 25.1%

      \[\leadsto w0 \cdot e^{\color{blue}{\left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right) + -2 \cdot \log d\right)} \cdot 0.5} \]
   7. Step-by-step derivation

      1. associate-/l* 23.7%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell}{h \cdot {M}^{2}}}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

      2. *-commutative 23.7%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{{D}^{2}}{\frac{\ell}{\color{blue}{{M}^{2} \cdot h}}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

      3. unpow2 23.7%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{{D}^{2}}{\frac{\ell}{\color{blue}{\left(M \cdot M\right)} \cdot h}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

      4. associate-*r* 26.5%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{{D}^{2}}{\frac{\ell}{\color{blue}{M \cdot \left(M \cdot h\right)}}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

      5. unpow2 26.5%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{\ell}{M \cdot \left(M \cdot h\right)}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

   8. Simplified 26.5%

      \[\leadsto w0 \cdot e^{\color{blue}{\left(\log \left(-0.25 \cdot \frac{D \cdot D}{\frac{\ell}{M \cdot \left(M \cdot h\right)}}\right) + -2 \cdot \log d\right)} \cdot 0.5} \]

   9. Taylor expanded in D around 0 6.6%

      \[\leadsto w0 \cdot e^{\left(\color{blue}{\left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell}\right) + 2 \cdot \log D\right)} + -2 \cdot \log d\right) \cdot 0.5} \]
   10. Step-by-step derivation

       1. +-commutative 6.6%

          \[\leadsto w0 \cdot e^{\left(\color{blue}{\left(2 \cdot \log D + \log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right)} + -2 \cdot \log d\right) \cdot 0.5} \]

       2. fma-def 6.6%

          \[\leadsto w0 \cdot e^{\left(\color{blue}{\mathsf{fma}\left(2, \log D, \log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell}\right)\right)} + -2 \cdot \log d\right) \cdot 0.5} \]

       3. *-commutative 6.6%

          \[\leadsto w0 \cdot e^{\left(\mathsf{fma}\left(2, \log D, \log \left(-0.25 \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{\ell}\right)\right) + -2 \cdot \log d\right) \cdot 0.5} \]

       4. unpow2 6.6%

          \[\leadsto w0 \cdot e^{\left(\mathsf{fma}\left(2, \log D, \log \left(-0.25 \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right) + -2 \cdot \log d\right) \cdot 0.5} \]

       5. associate-*r* 8.1%

          \[\leadsto w0 \cdot e^{\left(\mathsf{fma}\left(2, \log D, \log \left(-0.25 \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{\ell}\right)\right) + -2 \cdot \log d\right) \cdot 0.5} \]

       6. *-commutative 8.1%

          \[\leadsto w0 \cdot e^{\left(\mathsf{fma}\left(2, \log D, \log \left(-0.25 \cdot \frac{\color{blue}{\left(M \cdot h\right) \cdot M}}{\ell}\right)\right) + -2 \cdot \log d\right) \cdot 0.5} \]

       7. associate-/l* 8.1%

          \[\leadsto w0 \cdot e^{\left(\mathsf{fma}\left(2, \log D, \log \left(-0.25 \cdot \color{blue}{\frac{M \cdot h}{\frac{\ell}{M}}}\right)\right) + -2 \cdot \log d\right) \cdot 0.5} \]

       8. *-commutative 8.1%

          \[\leadsto w0 \cdot e^{\left(\mathsf{fma}\left(2, \log D, \log \left(-0.25 \cdot \frac{\color{blue}{h \cdot M}}{\frac{\ell}{M}}\right)\right) + -2 \cdot \log d\right) \cdot 0.5} \]

   11. Simplified 8.1%

       \[\leadsto w0 \cdot e^{\left(\color{blue}{\mathsf{fma}\left(2, \log D, \log \left(-0.25 \cdot \frac{h \cdot M}{\frac{\ell}{M}}\right)\right)} + -2 \cdot \log d\right) \cdot 0.5} \]

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 92.2%

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

      1. *-commutative 92.2%

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

      2. times-frac 92.3%

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

   3. Simplified 92.3%

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

      1. unpow2 92.3%

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

      2. associate-*l* 92.9%

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

      3. div-inv 92.8%

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

      4. associate-*l* 92.4%

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

      5. times-frac 92.4%

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

      6. *-un-lft-identity 92.4%

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

      7. *-commutative 92.4%

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

      8. clear-num 92.4%

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

      9. div-inv 92.4%

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

      10. div-inv 92.3%

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

      11. associate-*l* 93.4%

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

      12. times-frac 93.4%

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

      13. *-un-lft-identity 93.4%

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

      14. *-commutative 93.4%

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

      15. clear-num 93.4%

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

      16. div-inv 93.4%

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

      17. associate-*l* 92.8%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in M around 0 97.5%

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

4. Final simplification 76.2%

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

Alternative 4: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{+230}:\\ \;\;\;\;w0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \frac{h}{d}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 5e+230)
     (* w0 (sqrt t_0))
     (*
      w0
      (sqrt (- 1.0 (/ (* 0.25 (* (/ (* (* M D) (* M D)) d) (/ h d))) l)))))))

C

M = abs(M);
D = abs(D);
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 <= 5e+230) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 <= 5d+230) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * ((((m * d) * (m * d)) / d_1) * (h / d_1))) / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 5e+230) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
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 <= 5e+230:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)))
	return tmp

Julia

M = abs(M)
D = abs(D)
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 <= 5e+230)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) / d) * Float64(h / d))) / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
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 <= 5e+230)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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, 5e+230], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $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))) < 5.0000000000000003e230

   1. Initial program 99.9%

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

   if 5.0000000000000003e230 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 44.2%

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

      1. *-commutative 44.2%

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

      2. times-frac 46.8%

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

   3. Simplified 46.8%

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

      1. unpow2 46.8%

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

      2. associate-*l* 50.7%

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

      3. div-inv 50.7%

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

      4. associate-*l* 49.5%

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

      5. times-frac 49.5%

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

      6. *-un-lft-identity 49.5%

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

      7. *-commutative 49.5%

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

      8. clear-num 49.5%

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

      9. div-inv 49.5%

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

      10. div-inv 49.5%

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

      11. associate-*l* 50.9%

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

      12. times-frac 50.9%

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

      13. *-un-lft-identity 50.9%

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

      14. *-commutative 50.9%

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

      15. clear-num 50.9%

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

      16. div-inv 50.9%

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

      17. associate-*l* 46.8%

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

   5. Applied egg-rr 63.5%

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

   6. Taylor expanded in M around 0 61.1%

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

   7. Taylor expanded in D around 0 50.1%

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

      1. associate-*r* 50.2%

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

      2. unpow2 50.2%

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

      3. times-frac 54.5%

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

      4. unpow2 54.5%

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

      5. unpow2 54.5%

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

      6. swap-sqr 62.0%

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

   9. Simplified 62.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+230}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \frac{h}{d}\right)}{\ell}}\\ \end{array} \]

Alternative 5: 87.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{+157}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot e^{0.5 \cdot \left(-2 \cdot \log d + \log \left(-0.25 \cdot \left(\left(M \cdot h\right) \cdot \frac{M \cdot D}{\frac{\ell}{D}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 D) (* 2.0 d)) 1e+157)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* 0.5 (/ (* M D) d)) 2.0)) l))))
   (*
    w0
    (exp
     (*
      0.5
      (+ (* -2.0 (log d)) (log (* -0.25 (* (* M h) (/ (* M D) (/ l D)))))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 1e+157) {
		tmp = w0 * sqrt((1.0 - ((h * pow((0.5 * ((M * D) / d)), 2.0)) / l)));
	} else {
		tmp = w0 * exp((0.5 * ((-2.0 * log(d)) + log((-0.25 * ((M * h) * ((M * D) / (l / D))))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 * d) / (2.0d0 * d_1)) <= 1d+157) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((0.5d0 * ((m * d) / d_1)) ** 2.0d0)) / l)))
    else
        tmp = w0 * exp((0.5d0 * (((-2.0d0) * log(d_1)) + log(((-0.25d0) * ((m * h) * ((m * d) / (l / d))))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 * D) / (2.0 * d)) <= 1e+157) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((0.5 * ((M * D) / d)), 2.0)) / l)));
	} else {
		tmp = w0 * Math.exp((0.5 * ((-2.0 * Math.log(d)) + Math.log((-0.25 * ((M * h) * ((M * D) / (l / D))))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if ((M * D) / (2.0 * d)) <= 1e+157:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((0.5 * ((M * D) / d)), 2.0)) / l)))
	else:
		tmp = w0 * math.exp((0.5 * ((-2.0 * math.log(d)) + math.log((-0.25 * ((M * h) * ((M * D) / (l / D))))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(Float64(M * D) / Float64(2.0 * d)) <= 1e+157)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(0.5 * Float64(Float64(M * D) / d)) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * exp(Float64(0.5 * Float64(Float64(-2.0 * log(d)) + log(Float64(-0.25 * Float64(Float64(M * h) * Float64(Float64(M * D) / Float64(l / D)))))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((M * D) / (2.0 * d)) <= 1e+157)
		tmp = w0 * sqrt((1.0 - ((h * ((0.5 * ((M * D) / d)) ^ 2.0)) / l)));
	else
		tmp = w0 * exp((0.5 * ((-2.0 * log(d)) + log((-0.25 * ((M * h) * ((M * D) / (l / D))))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 1e+157], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(0.5 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Exp[N[(0.5 * N[(N[(-2.0 * N[Log[d], $MachinePrecision]), $MachinePrecision] + N[Log[N[(-0.25 * N[(N[(M * h), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 9.99999999999999983e156

   1. Initial program 88.7%

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

      1. *-commutative 88.7%

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

      2. times-frac 88.4%

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

   3. Simplified 88.4%

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

      1. unpow2 88.4%

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

      2. associate-*l* 89.2%

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

      3. div-inv 89.2%

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

      4. associate-*l* 88.8%

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

      5. times-frac 88.8%

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

      6. *-un-lft-identity 88.8%

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

      7. *-commutative 88.8%

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

      8. clear-num 88.8%

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

      9. div-inv 88.8%

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

      10. div-inv 88.8%

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

      11. associate-*l* 89.7%

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

      12. times-frac 89.7%

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

      13. *-un-lft-identity 89.7%

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

      14. *-commutative 89.7%

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

      15. clear-num 89.7%

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

      16. div-inv 89.7%

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

      17. associate-*l* 88.8%

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

   5. Applied egg-rr 94.4%

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

   6. Taylor expanded in M around 0 94.5%

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

   if 9.99999999999999983e156 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 35.5%

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

      1. *-commutative 35.5%

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

      2. times-frac 39.1%

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

   3. Simplified 39.1%

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

   5. Applied egg-rr 39.1%

      \[\leadsto w0 \cdot \color{blue}{e^{\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-\frac{h}{\ell}\right)\right) \cdot 0.5}} \]

   6. Taylor expanded in d around 0 29.2%

      \[\leadsto w0 \cdot e^{\color{blue}{\left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right) + -2 \cdot \log d\right)} \cdot 0.5} \]
   7. Step-by-step derivation

      1. associate-/l* 29.1%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell}{h \cdot {M}^{2}}}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

      2. *-commutative 29.1%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{{D}^{2}}{\frac{\ell}{\color{blue}{{M}^{2} \cdot h}}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

      3. unpow2 29.1%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{{D}^{2}}{\frac{\ell}{\color{blue}{\left(M \cdot M\right)} \cdot h}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

      4. associate-*r* 29.1%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{{D}^{2}}{\frac{\ell}{\color{blue}{M \cdot \left(M \cdot h\right)}}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

      5. unpow2 29.1%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{\ell}{M \cdot \left(M \cdot h\right)}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

   8. Simplified 29.1%

      \[\leadsto w0 \cdot e^{\color{blue}{\left(\log \left(-0.25 \cdot \frac{D \cdot D}{\frac{\ell}{M \cdot \left(M \cdot h\right)}}\right) + -2 \cdot \log d\right)} \cdot 0.5} \]

   9. Taylor expanded in l around 0 18.1%

      \[\leadsto w0 \cdot e^{\left(\color{blue}{\left(-1 \cdot \log \ell + \log \left(-0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right)\right)} + -2 \cdot \log d\right) \cdot 0.5} \]
   10. Step-by-step derivation

       1. +-commutative 18.1%

          \[\leadsto w0 \cdot e^{\left(\color{blue}{\left(\log \left(-0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right) + -1 \cdot \log \ell\right)} + -2 \cdot \log d\right) \cdot 0.5} \]

       2. *-commutative 18.1%

          \[\leadsto w0 \cdot e^{\left(\left(\log \left(-0.25 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)\right) + -1 \cdot \log \ell\right) + -2 \cdot \log d\right) \cdot 0.5} \]

       3. mul-1-neg 18.1%

          \[\leadsto w0 \cdot e^{\left(\left(\log \left(-0.25 \cdot \left({D}^{2} \cdot \left(h \cdot {M}^{2}\right)\right)\right) + \color{blue}{\left(-\log \ell\right)}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

       4. unsub-neg 18.1%

          \[\leadsto w0 \cdot e^{\left(\color{blue}{\left(\log \left(-0.25 \cdot \left({D}^{2} \cdot \left(h \cdot {M}^{2}\right)\right)\right) - \log \ell\right)} + -2 \cdot \log d\right) \cdot 0.5} \]

       5. *-commutative 18.1%

          \[\leadsto w0 \cdot e^{\left(\left(\log \left(-0.25 \cdot \left({D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}\right)\right) - \log \ell\right) + -2 \cdot \log d\right) \cdot 0.5} \]

       6. log-div 29.2%

          \[\leadsto w0 \cdot e^{\left(\color{blue}{\log \left(\frac{-0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell}\right)} + -2 \cdot \log d\right) \cdot 0.5} \]

       7. associate-*r/ 29.2%

          \[\leadsto w0 \cdot e^{\left(\log \color{blue}{\left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)} + -2 \cdot \log d\right) \cdot 0.5} \]

       8. associate-/l* 29.1%

          \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{\ell}{{M}^{2} \cdot h}}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

       9. unpow2 29.1%

          \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{{D}^{2}}{\frac{\ell}{\color{blue}{\left(M \cdot M\right)} \cdot h}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

       10. associate-*r* 29.1%

           \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{{D}^{2}}{\frac{\ell}{\color{blue}{M \cdot \left(M \cdot h\right)}}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

       11. associate-/r* 29.1%

           \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{{D}^{2}}{\color{blue}{\frac{\frac{\ell}{M}}{M \cdot h}}}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

       12. associate-/r/ 26.2%

           \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{\frac{\ell}{M}} \cdot \left(M \cdot h\right)\right)}\right) + -2 \cdot \log d\right) \cdot 0.5} \]

   11. Simplified 25.9%

       \[\leadsto w0 \cdot e^{\left(\color{blue}{\log \left(-0.25 \cdot \left(\left(h \cdot M\right) \cdot \frac{D \cdot M}{\frac{\ell}{D}}\right)\right)} + -2 \cdot \log d\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{+157}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot e^{0.5 \cdot \left(-2 \cdot \log d + \log \left(-0.25 \cdot \left(\left(M \cdot h\right) \cdot \frac{M \cdot D}{\frac{\ell}{D}}\right)\right)\right)}\\ \end{array} \]

Alternative 6: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.8 \cdot 10^{-117}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \frac{h}{d}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 2.8e-117)
   (* w0 (sqrt (- 1.0 (/ (* 0.25 (* (/ (* (* M D) (* M D)) d) (/ h d))) l))))
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.8e-117) {
		tmp = w0 * sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 <= 2.8d-117) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * ((((m * d) * (m * d)) / d_1) * (h / d_1))) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / d_1) * (d / 2.0d0)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 2.8e-117) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.8e-117:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.8e-117)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) / d) * Float64(h / d))) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.8e-117)
		tmp = w0 * sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)));
	else
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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, 2.8e-117], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.8e-117

   1. Initial program 84.3%

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

      1. *-commutative 84.3%

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

      2. times-frac 84.3%

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

   3. Simplified 84.3%

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

      1. unpow2 84.3%

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

      2. associate-*l* 85.4%

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

      3. div-inv 85.4%

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

      4. associate-*l* 84.9%

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

      5. times-frac 84.9%

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

      6. *-un-lft-identity 84.9%

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

      7. *-commutative 84.9%

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

      8. clear-num 84.9%

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

      9. div-inv 84.9%

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

      10. div-inv 84.9%

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

      11. associate-*l* 86.0%

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

      12. times-frac 86.0%

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

      13. *-un-lft-identity 86.0%

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

      14. *-commutative 86.0%

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

      15. clear-num 86.0%

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

      16. div-inv 86.0%

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

      17. associate-*l* 84.3%

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

   5. Applied egg-rr 89.8%

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

   6. Taylor expanded in M around 0 89.8%

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

   7. Taylor expanded in D around 0 59.4%

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

      1. associate-*r* 59.4%

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

      2. unpow2 59.4%

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

      3. times-frac 66.8%

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

      4. unpow2 66.8%

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

      5. unpow2 66.8%

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

      6. swap-sqr 84.8%

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

   9. Simplified 84.8%

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

   if 2.8e-117 < M

   1. Initial program 79.8%

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

      1. *-commutative 79.8%

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

      2. times-frac 79.8%

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

   3. Simplified 79.8%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.8 \cdot 10^{-117}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \frac{h}{d}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \end{array} \]

Alternative 7: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \sqrt{1 - \frac{h \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell}} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 (sqrt (- 1.0 (/ (* h (pow (* 0.5 (/ (* M D) d)) 2.0)) l)))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - ((h * pow((0.5 * ((M * D) / d)), 2.0)) / l)));
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 * sqrt((1.0d0 - ((h * ((0.5d0 * ((m * d) / d_1)) ** 2.0d0)) / l)))
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 * Math.sqrt((1.0 - ((h * Math.pow((0.5 * ((M * D) / d)), 2.0)) / l)));
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - ((h * math.pow((0.5 * ((M * D) / d)), 2.0)) / l)))

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(0.5 * Float64(Float64(M * D) / d)) ^ 2.0)) / l))))
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((h * ((0.5 * ((M * D) / d)) ^ 2.0)) / l)));
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(0.5 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

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

   1. *-commutative 82.9%

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

   2. times-frac 83.0%

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

3. Simplified 83.0%

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

   1. unpow2 83.0%

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

   2. associate-*l* 84.2%

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

   3. div-inv 84.2%

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

   4. associate-*l* 83.4%

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

   5. times-frac 83.4%

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

   6. *-un-lft-identity 83.4%

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

   7. *-commutative 83.4%

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

   8. clear-num 83.4%

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

   9. div-inv 83.4%

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

   10. div-inv 83.4%

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

   11. associate-*l* 84.6%

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

   12. times-frac 84.6%

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

   13. *-un-lft-identity 84.6%

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

   14. *-commutative 84.6%

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

   15. clear-num 84.6%

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

   16. div-inv 84.6%

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

   17. associate-*l* 83.3%

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

5. Applied egg-rr 88.4%

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

6. Taylor expanded in M around 0 88.1%

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

7. Final simplification 88.1%

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

Alternative 8: 83.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 3.1 \cdot 10^{-153}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right) \cdot \frac{D \cdot D}{\ell}\right)}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+106}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D \cdot \left(h \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)}{\ell} \cdot \frac{\frac{0.25}{d}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\frac{M \cdot h}{\frac{\ell}{M}} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 3.1e-153)
   (* w0 (sqrt (+ 1.0 (* -0.25 (* (* (/ M d) (/ (* M h) d)) (/ (* D D) l))))))
   (if (<= d 7e+106)
     (* w0 (sqrt (- 1.0 (* (/ (* D (* h (* D (* M M)))) l) (/ (/ 0.25 d) d)))))
     (* w0 (+ 1.0 (* (* (/ (* M h) (/ l M)) (* (/ D d) (/ D d))) -0.125))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 3.1e-153) {
		tmp = w0 * sqrt((1.0 + (-0.25 * (((M / d) * ((M * h) / d)) * ((D * D) / l)))));
	} else if (d <= 7e+106) {
		tmp = w0 * sqrt((1.0 - (((D * (h * (D * (M * M)))) / l) * ((0.25 / d) / d))));
	} else {
		tmp = w0 * (1.0 + ((((M * h) / (l / M)) * ((D / d) * (D / d))) * -0.125));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 <= 3.1d-153) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (((m / d_1) * ((m * h) / d_1)) * ((d * d) / l)))))
    else if (d_1 <= 7d+106) then
        tmp = w0 * sqrt((1.0d0 - (((d * (h * (d * (m * m)))) / l) * ((0.25d0 / d_1) / d_1))))
    else
        tmp = w0 * (1.0d0 + ((((m * h) / (l / m)) * ((d / d_1) * (d / d_1))) * (-0.125d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 3.1e-153) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (((M / d) * ((M * h) / d)) * ((D * D) / l)))));
	} else if (d <= 7e+106) {
		tmp = w0 * Math.sqrt((1.0 - (((D * (h * (D * (M * M)))) / l) * ((0.25 / d) / d))));
	} else {
		tmp = w0 * (1.0 + ((((M * h) / (l / M)) * ((D / d) * (D / d))) * -0.125));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 3.1e-153:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (((M / d) * ((M * h) / d)) * ((D * D) / l)))))
	elif d <= 7e+106:
		tmp = w0 * math.sqrt((1.0 - (((D * (h * (D * (M * M)))) / l) * ((0.25 / d) / d))))
	else:
		tmp = w0 * (1.0 + ((((M * h) / (l / M)) * ((D / d) * (D / d))) * -0.125))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 3.1e-153)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(Float64(M / d) * Float64(Float64(M * h) / d)) * Float64(Float64(D * D) / l))))));
	elseif (d <= 7e+106)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D * Float64(h * Float64(D * Float64(M * M)))) / l) * Float64(Float64(0.25 / d) / d)))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(Float64(M * h) / Float64(l / M)) * Float64(Float64(D / d) * Float64(D / d))) * -0.125)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 3.1e-153)
		tmp = w0 * sqrt((1.0 + (-0.25 * (((M / d) * ((M * h) / d)) * ((D * D) / l)))));
	elseif (d <= 7e+106)
		tmp = w0 * sqrt((1.0 - (((D * (h * (D * (M * M)))) / l) * ((0.25 / d) / d))));
	else
		tmp = w0 * (1.0 + ((((M * h) / (l / M)) * ((D / d) * (D / d))) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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, 3.1e-153], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(N[(M / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7e+106], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D * N[(h * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(0.25 / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(N[(N[(M * h), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < 3.09999999999999995e-153

   1. Initial program 81.7%

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

      1. *-commutative 81.7%

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

      2. times-frac 81.2%

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

   3. Simplified 81.2%

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

   5. Applied egg-rr 81.4%

      \[\leadsto w0 \cdot \color{blue}{e^{\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-\frac{h}{\ell}\right)\right) \cdot 0.5}} \]

   6. Taylor expanded in w0 around 0 50.4%

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

      1. associate-*r/ 50.4%

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

      2. *-commutative 50.4%

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

      3. associate-*r/ 50.4%

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

      4. times-frac 51.8%

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

      5. unpow2 51.8%

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

      6. associate-*r* 54.1%

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

      7. unpow2 54.1%

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

      8. unpow2 54.1%

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

      9. associate-/l* 57.9%

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

      10. *-commutative 57.9%

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

      11. times-frac 71.7%

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

      12. associate-/l* 67.3%

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

   8. Simplified 67.3%

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

   if 3.09999999999999995e-153 < d < 6.99999999999999962e106

   1. Initial program 82.7%

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

      1. *-commutative 82.7%

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

      2. times-frac 82.7%

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

   3. Simplified 82.7%

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

   4. Taylor expanded in M around 0 64.6%

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

      1. associate-*r/ 64.6%

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

      2. *-commutative 64.6%

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

      3. associate-/r* 74.6%

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

      4. *-commutative 74.6%

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

      5. *-commutative 74.6%

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

      6. associate-*l* 74.5%

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

      7. unpow2 74.5%

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

      8. unpow2 74.5%

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

      9. unpow2 74.5%

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

   6. Simplified 74.5%

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

      1. pow1/2 74.5%

         \[\leadsto w0 \cdot \color{blue}{{\left(1 - \frac{\frac{0.25 \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)}{\ell}}{d \cdot d}\right)}^{0.5}} \]

      2. associate-/l/ 64.6%

         \[\leadsto w0 \cdot {\left(1 - \color{blue}{\frac{0.25 \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}}\right)}^{0.5} \]

      3. associate-*r* 64.6%

         \[\leadsto w0 \cdot {\left(1 - \frac{0.25 \cdot \color{blue}{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)\right)}}{\left(d \cdot d\right) \cdot \ell}\right)}^{0.5} \]

      4. *-commutative 64.6%

         \[\leadsto w0 \cdot {\left(1 - \frac{0.25 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}}{\left(d \cdot d\right) \cdot \ell}\right)}^{0.5} \]

      5. *-commutative 64.6%

         \[\leadsto w0 \cdot {\left(1 - \frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot 0.25}}{\left(d \cdot d\right) \cdot \ell}\right)}^{0.5} \]

      6. associate-*l* 69.0%

         \[\leadsto w0 \cdot {\left(1 - \frac{\color{blue}{\left(D \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right)} \cdot 0.25}{\left(d \cdot d\right) \cdot \ell}\right)}^{0.5} \]

      7. associate-*l* 69.1%

         \[\leadsto w0 \cdot {\left(1 - \frac{\left(D \cdot \left(D \cdot \color{blue}{\left(M \cdot \left(M \cdot h\right)\right)}\right)\right) \cdot 0.25}{\left(d \cdot d\right) \cdot \ell}\right)}^{0.5} \]

      8. *-commutative 69.1%

         \[\leadsto w0 \cdot {\left(1 - \frac{\left(D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \cdot 0.25}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)}^{0.5} \]

   8. Applied egg-rr 69.1%

      \[\leadsto w0 \cdot \color{blue}{{\left(1 - \frac{\left(D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \cdot 0.25}{\ell \cdot \left(d \cdot d\right)}\right)}^{0.5}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 69.1%

         \[\leadsto w0 \cdot \color{blue}{\left(1 \cdot {\left(1 - \frac{\left(D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \cdot 0.25}{\ell \cdot \left(d \cdot d\right)}\right)}^{0.5}\right)} \]

      2. unpow1/2 69.0%

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

      3. times-frac 80.9%

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

      4. associate-*r* 86.6%

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

   10. Applied egg-rr 86.6%

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

       1. *-lft-identity 86.6%

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

       2. *-commutative 86.6%

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

       3. *-commutative 86.6%

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

       4. associate-*l* 80.9%

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

       5. associate-*r* 79.0%

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

       6. unpow2 79.0%

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

       7. associate-*r* 79.0%

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

       8. unpow2 79.0%

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

       9. associate-/r* 79.0%

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

   12. Simplified 79.0%

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

   if 6.99999999999999962e106 < d

   1. Initial program 87.3%

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

      1. *-commutative 87.3%

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

      2. times-frac 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in M around 0 55.4%

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

      1. *-commutative 55.4%

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

      2. unpow2 55.4%

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

      3. unpow2 55.4%

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

      4. unpow2 55.4%

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

   6. Simplified 55.4%

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

   7. Taylor expanded in D around 0 55.4%

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

      1. times-frac 51.1%

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

      2. unpow2 51.1%

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

      3. associate-*r* 55.5%

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

      4. unpow2 55.5%

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

      5. unpow2 55.5%

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

      6. associate-/l* 59.7%

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

      7. *-commutative 59.7%

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

      8. times-frac 64.3%

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

      9. associate-/l* 60.0%

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

   9. Simplified 60.0%

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

   10. Taylor expanded in M around 0 55.4%

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

       1. *-commutative 55.4%

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

       2. *-commutative 55.4%

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

       3. times-frac 53.3%

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

       4. unpow2 53.3%

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

       5. associate-*r* 57.5%

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

       6. *-commutative 57.5%

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

       7. associate-/l* 57.5%

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

       8. *-commutative 57.5%

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

       9. unpow2 57.5%

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

       10. unpow2 57.5%

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

       11. times-frac 70.6%

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

   12. Simplified 70.6%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 3.1 \cdot 10^{-153}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right) \cdot \frac{D \cdot D}{\ell}\right)}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+106}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D \cdot \left(h \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)}{\ell} \cdot \frac{\frac{0.25}{d}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\frac{M \cdot h}{\frac{\ell}{M}} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 9: 83.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 9 \cdot 10^{-58}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \frac{h}{d}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left(1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{M}{\frac{\ell}{h}}\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 9e-58)
   (* w0 (sqrt (- 1.0 (/ (* 0.25 (* (/ (* (* M D) (* M D)) d) (/ h d))) l))))
   (*
    w0
    (pow (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (* M (/ M (/ l h)))))) 0.5))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 9e-58) {
		tmp = w0 * sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)));
	} else {
		tmp = w0 * pow((1.0 - (0.25 * (((D / d) * (D / d)) * (M * (M / (l / h)))))), 0.5);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 <= 9d-58) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * ((((m * d) * (m * d)) / d_1) * (h / d_1))) / l)))
    else
        tmp = w0 * ((1.0d0 - (0.25d0 * (((d / d_1) * (d / d_1)) * (m * (m / (l / h)))))) ** 0.5d0)
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 9e-58) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)));
	} else {
		tmp = w0 * Math.pow((1.0 - (0.25 * (((D / d) * (D / d)) * (M * (M / (l / h)))))), 0.5);
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 9e-58:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)))
	else:
		tmp = w0 * math.pow((1.0 - (0.25 * (((D / d) * (D / d)) * (M * (M / (l / h)))))), 0.5)
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 9e-58)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) / d) * Float64(h / d))) / l))));
	else
		tmp = Float64(w0 * (Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(M * Float64(M / Float64(l / h)))))) ^ 0.5));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 9e-58)
		tmp = w0 * sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)));
	else
		tmp = w0 * ((1.0 - (0.25 * (((D / d) * (D / d)) * (M * (M / (l / h)))))) ^ 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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, 9e-58], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Power[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(M * N[(M / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 9.0000000000000006e-58

   1. Initial program 83.1%

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

      1. *-commutative 83.1%

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

      2. times-frac 83.2%

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

   3. Simplified 83.2%

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

      1. unpow2 83.2%

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

      2. associate-*l* 84.7%

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

      3. div-inv 84.7%

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

      4. associate-*l* 83.7%

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

      5. times-frac 83.7%

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

      6. *-un-lft-identity 83.7%

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

      7. *-commutative 83.7%

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

      8. clear-num 83.7%

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

      9. div-inv 83.7%

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

      10. div-inv 83.7%

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

      11. associate-*l* 84.8%

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

      12. times-frac 84.8%

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

      13. *-un-lft-identity 84.8%

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

      14. *-commutative 84.8%

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

      15. clear-num 84.7%

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

      16. div-inv 84.7%

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

      17. associate-*l* 83.2%

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

   5. Applied egg-rr 89.2%

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

   6. Taylor expanded in M around 0 89.2%

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

   7. Taylor expanded in D around 0 61.0%

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

      1. associate-*r* 61.0%

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

      2. unpow2 61.0%

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

      3. times-frac 67.8%

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

      4. unpow2 67.8%

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

      5. unpow2 67.8%

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

      6. swap-sqr 84.2%

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

   9. Simplified 84.2%

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

   if 9.0000000000000006e-58 < M

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. times-frac 82.2%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in M around 0 49.9%

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

      1. associate-*r/ 49.9%

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

      2. *-commutative 49.9%

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

      3. associate-/r* 51.7%

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

      4. *-commutative 51.7%

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

      5. *-commutative 51.7%

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

      6. associate-*l* 58.2%

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

      7. unpow2 58.2%

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

      8. unpow2 58.2%

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

      9. unpow2 58.2%

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

   6. Simplified 58.2%

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

      1. pow1/2 58.4%

         \[\leadsto w0 \cdot \color{blue}{{\left(1 - \frac{\frac{0.25 \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)}{\ell}}{d \cdot d}\right)}^{0.5}} \]

      2. associate-/l/ 56.5%

         \[\leadsto w0 \cdot {\left(1 - \color{blue}{\frac{0.25 \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}}\right)}^{0.5} \]

      3. associate-*r* 49.9%

         \[\leadsto w0 \cdot {\left(1 - \frac{0.25 \cdot \color{blue}{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)\right)}}{\left(d \cdot d\right) \cdot \ell}\right)}^{0.5} \]

      4. *-commutative 49.9%

         \[\leadsto w0 \cdot {\left(1 - \frac{0.25 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}}{\left(d \cdot d\right) \cdot \ell}\right)}^{0.5} \]

      5. *-commutative 49.9%

         \[\leadsto w0 \cdot {\left(1 - \frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot 0.25}}{\left(d \cdot d\right) \cdot \ell}\right)}^{0.5} \]

      6. associate-*l* 55.4%

         \[\leadsto w0 \cdot {\left(1 - \frac{\color{blue}{\left(D \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right)} \cdot 0.25}{\left(d \cdot d\right) \cdot \ell}\right)}^{0.5} \]

      7. associate-*l* 58.8%

         \[\leadsto w0 \cdot {\left(1 - \frac{\left(D \cdot \left(D \cdot \color{blue}{\left(M \cdot \left(M \cdot h\right)\right)}\right)\right) \cdot 0.25}{\left(d \cdot d\right) \cdot \ell}\right)}^{0.5} \]

      8. *-commutative 58.8%

         \[\leadsto w0 \cdot {\left(1 - \frac{\left(D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \cdot 0.25}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)}^{0.5} \]

   8. Applied egg-rr 58.8%

      \[\leadsto w0 \cdot \color{blue}{{\left(1 - \frac{\left(D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \cdot 0.25}{\ell \cdot \left(d \cdot d\right)}\right)}^{0.5}} \]

   9. Taylor expanded in D around 0 49.9%

      \[\leadsto w0 \cdot {\left(1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right)}^{0.5} \]
   10. Step-by-step derivation

       1. *-commutative 49.9%

          \[\leadsto w0 \cdot {\left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.25}\right)}^{0.5} \]

       2. *-commutative 49.9%

          \[\leadsto w0 \cdot {\left(1 - \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}}{\ell \cdot {d}^{2}} \cdot 0.25\right)}^{0.5} \]

       3. times-frac 50.0%

          \[\leadsto w0 \cdot {\left(1 - \color{blue}{\left(\frac{{M}^{2} \cdot h}{\ell} \cdot \frac{{D}^{2}}{{d}^{2}}\right)} \cdot 0.25\right)}^{0.5} \]

       4. unpow2 50.0%

          \[\leadsto w0 \cdot {\left(1 - \left(\frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot 0.25\right)}^{0.5} \]

       5. associate-*r* 53.3%

          \[\leadsto w0 \cdot {\left(1 - \left(\frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{\ell} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot 0.25\right)}^{0.5} \]

       6. *-commutative 53.3%

          \[\leadsto w0 \cdot {\left(1 - \left(\frac{\color{blue}{\left(M \cdot h\right) \cdot M}}{\ell} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot 0.25\right)}^{0.5} \]

       7. associate-/l* 58.4%

          \[\leadsto w0 \cdot {\left(1 - \left(\color{blue}{\frac{M \cdot h}{\frac{\ell}{M}}} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot 0.25\right)}^{0.5} \]

       8. *-commutative 58.4%

          \[\leadsto w0 \cdot {\left(1 - \left(\frac{\color{blue}{h \cdot M}}{\frac{\ell}{M}} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot 0.25\right)}^{0.5} \]

       9. unpow2 58.4%

          \[\leadsto w0 \cdot {\left(1 - \left(\frac{h \cdot M}{\frac{\ell}{M}} \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot 0.25\right)}^{0.5} \]

       10. unpow2 58.4%

           \[\leadsto w0 \cdot {\left(1 - \left(\frac{h \cdot M}{\frac{\ell}{M}} \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot 0.25\right)}^{0.5} \]

       11. times-frac 70.4%

           \[\leadsto w0 \cdot {\left(1 - \left(\frac{h \cdot M}{\frac{\ell}{M}} \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot 0.25\right)}^{0.5} \]

   11. Simplified 70.4%

       \[\leadsto w0 \cdot {\left(1 - \color{blue}{\left(\frac{h \cdot M}{\frac{\ell}{M}} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot 0.25}\right)}^{0.5} \]
   12. Step-by-step derivation

       1. associate-/r/ 72.1%

          \[\leadsto w0 \cdot {\left(1 - \left(\color{blue}{\left(\frac{h \cdot M}{\ell} \cdot M\right)} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot 0.25\right)}^{0.5} \]

       2. *-commutative 72.1%

          \[\leadsto w0 \cdot {\left(1 - \left(\left(\frac{\color{blue}{M \cdot h}}{\ell} \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot 0.25\right)}^{0.5} \]

   13. Applied egg-rr 72.1%

       \[\leadsto w0 \cdot {\left(1 - \left(\color{blue}{\left(\frac{M \cdot h}{\ell} \cdot M\right)} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot 0.25\right)}^{0.5} \]
   14. Step-by-step derivation

       1. *-commutative 72.1%

          \[\leadsto w0 \cdot {\left(1 - \left(\color{blue}{\left(M \cdot \frac{M \cdot h}{\ell}\right)} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot 0.25\right)}^{0.5} \]

       2. associate-/l* 72.3%

          \[\leadsto w0 \cdot {\left(1 - \left(\left(M \cdot \color{blue}{\frac{M}{\frac{\ell}{h}}}\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot 0.25\right)}^{0.5} \]

   15. Simplified 72.3%

       \[\leadsto w0 \cdot {\left(1 - \left(\color{blue}{\left(M \cdot \frac{M}{\frac{\ell}{h}}\right)} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot 0.25\right)}^{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9 \cdot 10^{-58}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \frac{h}{d}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left(1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{M}{\frac{\ell}{h}}\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 10: 83.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4 \cdot 10^{+119}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \frac{h}{d}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left(1 - 0.25 \cdot \left(\frac{M \cdot h}{\frac{\ell}{M}} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 4e+119)
   (* w0 (sqrt (- 1.0 (/ (* 0.25 (* (/ (* (* M D) (* M D)) d) (/ h d))) l))))
   (*
    w0
    (pow (- 1.0 (* 0.25 (* (/ (* M h) (/ l M)) (* (/ D d) (/ D d))))) 0.5))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 4e+119) {
		tmp = w0 * sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)));
	} else {
		tmp = w0 * pow((1.0 - (0.25 * (((M * h) / (l / M)) * ((D / d) * (D / d))))), 0.5);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 <= 4d+119) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * ((((m * d) * (m * d)) / d_1) * (h / d_1))) / l)))
    else
        tmp = w0 * ((1.0d0 - (0.25d0 * (((m * h) / (l / m)) * ((d / d_1) * (d / d_1))))) ** 0.5d0)
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 4e+119) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)));
	} else {
		tmp = w0 * Math.pow((1.0 - (0.25 * (((M * h) / (l / M)) * ((D / d) * (D / d))))), 0.5);
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 4e+119:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)))
	else:
		tmp = w0 * math.pow((1.0 - (0.25 * (((M * h) / (l / M)) * ((D / d) * (D / d))))), 0.5)
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 4e+119)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) / d) * Float64(h / d))) / l))));
	else
		tmp = Float64(w0 * (Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(M * h) / Float64(l / M)) * Float64(Float64(D / d) * Float64(D / d))))) ^ 0.5));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 4e+119)
		tmp = w0 * sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)));
	else
		tmp = w0 * ((1.0 - (0.25 * (((M * h) / (l / M)) * ((D / d) * (D / d))))) ^ 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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, 4e+119], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Power[N[(1.0 - N[(0.25 * N[(N[(N[(M * h), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 3.99999999999999978e119

   1. Initial program 82.5%

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

      1. *-commutative 82.5%

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

      2. times-frac 82.1%

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

   3. Simplified 82.1%

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

      1. unpow2 82.1%

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

      2. associate-*l* 83.5%

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

      3. div-inv 83.5%

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

      4. associate-*l* 82.6%

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

      5. times-frac 82.6%

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

      6. *-un-lft-identity 82.6%

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

      7. *-commutative 82.6%

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

      8. clear-num 82.6%

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

      9. div-inv 82.6%

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

      10. div-inv 82.6%

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

      11. associate-*l* 84.0%

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

      12. times-frac 84.0%

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

      13. *-un-lft-identity 84.0%

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

      14. *-commutative 84.0%

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

      15. clear-num 84.0%

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

      16. div-inv 84.0%

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

      17. associate-*l* 82.5%

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

   5. Applied egg-rr 86.8%

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

   6. Taylor expanded in M around 0 86.8%

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

   7. Taylor expanded in D around 0 58.6%

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

      1. associate-*r* 60.0%

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

      2. unpow2 60.0%

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

      3. times-frac 66.2%

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

      4. unpow2 66.2%

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

      5. unpow2 66.2%

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

      6. swap-sqr 82.5%

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

   9. Simplified 82.5%

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

   if 3.99999999999999978e119 < d

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

      2. times-frac 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in M around 0 60.0%

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

      1. associate-*r/ 60.0%

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

      2. *-commutative 60.0%

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

      3. associate-/r* 60.0%

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

      4. *-commutative 60.0%

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

      5. *-commutative 60.0%

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

      6. associate-*l* 52.5%

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

      7. unpow2 52.5%

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

      8. unpow2 52.5%

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

      9. unpow2 52.5%

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

   6. Simplified 52.5%

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

      1. pow1/2 52.6%

         \[\leadsto w0 \cdot \color{blue}{{\left(1 - \frac{\frac{0.25 \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)}{\ell}}{d \cdot d}\right)}^{0.5}} \]

      2. associate-/l/ 52.5%

         \[\leadsto w0 \cdot {\left(1 - \color{blue}{\frac{0.25 \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \left(D \cdot D\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell}}\right)}^{0.5} \]

      3. associate-*r* 60.0%

         \[\leadsto w0 \cdot {\left(1 - \frac{0.25 \cdot \color{blue}{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)\right)}}{\left(d \cdot d\right) \cdot \ell}\right)}^{0.5} \]

      4. *-commutative 60.0%

         \[\leadsto w0 \cdot {\left(1 - \frac{0.25 \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}}{\left(d \cdot d\right) \cdot \ell}\right)}^{0.5} \]

      5. *-commutative 60.0%

         \[\leadsto w0 \cdot {\left(1 - \frac{\color{blue}{\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot 0.25}}{\left(d \cdot d\right) \cdot \ell}\right)}^{0.5} \]

      6. associate-*l* 67.5%

         \[\leadsto w0 \cdot {\left(1 - \frac{\color{blue}{\left(D \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right)} \cdot 0.25}{\left(d \cdot d\right) \cdot \ell}\right)}^{0.5} \]

      7. associate-*l* 70.2%

         \[\leadsto w0 \cdot {\left(1 - \frac{\left(D \cdot \left(D \cdot \color{blue}{\left(M \cdot \left(M \cdot h\right)\right)}\right)\right) \cdot 0.25}{\left(d \cdot d\right) \cdot \ell}\right)}^{0.5} \]

      8. *-commutative 70.2%

         \[\leadsto w0 \cdot {\left(1 - \frac{\left(D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \cdot 0.25}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)}^{0.5} \]

   8. Applied egg-rr 70.2%

      \[\leadsto w0 \cdot \color{blue}{{\left(1 - \frac{\left(D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right) \cdot 0.25}{\ell \cdot \left(d \cdot d\right)}\right)}^{0.5}} \]

   9. Taylor expanded in D around 0 60.0%

      \[\leadsto w0 \cdot {\left(1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right)}^{0.5} \]
   10. Step-by-step derivation

       1. *-commutative 60.0%

          \[\leadsto w0 \cdot {\left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}} \cdot 0.25}\right)}^{0.5} \]

       2. *-commutative 60.0%

          \[\leadsto w0 \cdot {\left(1 - \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}}{\ell \cdot {d}^{2}} \cdot 0.25\right)}^{0.5} \]

       3. times-frac 57.5%

          \[\leadsto w0 \cdot {\left(1 - \color{blue}{\left(\frac{{M}^{2} \cdot h}{\ell} \cdot \frac{{D}^{2}}{{d}^{2}}\right)} \cdot 0.25\right)}^{0.5} \]

       4. unpow2 57.5%

          \[\leadsto w0 \cdot {\left(1 - \left(\frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot 0.25\right)}^{0.5} \]

       5. associate-*r* 60.0%

          \[\leadsto w0 \cdot {\left(1 - \left(\frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{\ell} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot 0.25\right)}^{0.5} \]

       6. *-commutative 60.0%

          \[\leadsto w0 \cdot {\left(1 - \left(\frac{\color{blue}{\left(M \cdot h\right) \cdot M}}{\ell} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot 0.25\right)}^{0.5} \]

       7. associate-/l* 60.0%

          \[\leadsto w0 \cdot {\left(1 - \left(\color{blue}{\frac{M \cdot h}{\frac{\ell}{M}}} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot 0.25\right)}^{0.5} \]

       8. *-commutative 60.0%

          \[\leadsto w0 \cdot {\left(1 - \left(\frac{\color{blue}{h \cdot M}}{\frac{\ell}{M}} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot 0.25\right)}^{0.5} \]

       9. unpow2 60.0%

          \[\leadsto w0 \cdot {\left(1 - \left(\frac{h \cdot M}{\frac{\ell}{M}} \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot 0.25\right)}^{0.5} \]

       10. unpow2 60.0%

           \[\leadsto w0 \cdot {\left(1 - \left(\frac{h \cdot M}{\frac{\ell}{M}} \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot 0.25\right)}^{0.5} \]

       11. times-frac 77.6%

           \[\leadsto w0 \cdot {\left(1 - \left(\frac{h \cdot M}{\frac{\ell}{M}} \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot 0.25\right)}^{0.5} \]

   11. Simplified 77.6%

       \[\leadsto w0 \cdot {\left(1 - \color{blue}{\left(\frac{h \cdot M}{\frac{\ell}{M}} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot 0.25}\right)}^{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4 \cdot 10^{+119}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \frac{h}{d}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left(1 - 0.25 \cdot \left(\frac{M \cdot h}{\frac{\ell}{M}} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 11: 81.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.15 \cdot 10^{-35}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \frac{h}{d}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\frac{M \cdot h}{\frac{\ell}{M}} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 2.15e-35)
   (* w0 (sqrt (- 1.0 (/ (* 0.25 (* (/ (* (* M D) (* M D)) d) (/ h d))) l))))
   (* w0 (+ 1.0 (* (* (/ (* M h) (/ l M)) (* (/ D d) (/ D d))) -0.125)))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.15e-35) {
		tmp = w0 * sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)));
	} else {
		tmp = w0 * (1.0 + ((((M * h) / (l / M)) * ((D / d) * (D / d))) * -0.125));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 <= 2.15d-35) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * ((((m * d) * (m * d)) / d_1) * (h / d_1))) / l)))
    else
        tmp = w0 * (1.0d0 + ((((m * h) / (l / m)) * ((d / d_1) * (d / d_1))) * (-0.125d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 2.15e-35) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)));
	} else {
		tmp = w0 * (1.0 + ((((M * h) / (l / M)) * ((D / d) * (D / d))) * -0.125));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.15e-35:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)))
	else:
		tmp = w0 * (1.0 + ((((M * h) / (l / M)) * ((D / d) * (D / d))) * -0.125))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.15e-35)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) / d) * Float64(h / d))) / l))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(Float64(M * h) / Float64(l / M)) * Float64(Float64(D / d) * Float64(D / d))) * -0.125)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.15e-35)
		tmp = w0 * sqrt((1.0 - ((0.25 * ((((M * D) * (M * D)) / d) * (h / d))) / l)));
	else
		tmp = w0 * (1.0 + ((((M * h) / (l / M)) * ((D / d) * (D / d))) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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, 2.15e-35], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(N[(N[(M * h), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.1500000000000001e-35

   1. Initial program 83.8%

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

      1. *-commutative 83.8%

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

      2. times-frac 83.9%

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

   3. Simplified 83.9%

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

      1. unpow2 83.9%

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

      2. associate-*l* 85.3%

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

      3. div-inv 85.3%

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

      4. associate-*l* 84.4%

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

      5. times-frac 84.4%

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

      6. *-un-lft-identity 84.4%

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

      7. *-commutative 84.4%

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

      8. clear-num 84.3%

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

      9. div-inv 84.3%

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

      10. div-inv 84.3%

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

      11. associate-*l* 85.3%

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

      12. times-frac 85.3%

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

      13. *-un-lft-identity 85.3%

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

      14. *-commutative 85.3%

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

      15. clear-num 85.3%

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

      16. div-inv 85.3%

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

      17. associate-*l* 83.8%

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

   5. Applied egg-rr 89.6%

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

   6. Taylor expanded in M around 0 89.6%

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

   7. Taylor expanded in D around 0 62.5%

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

      1. associate-*r* 62.5%

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

      2. unpow2 62.5%

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

      3. times-frac 69.0%

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

      4. unpow2 69.0%

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

      5. unpow2 69.0%

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

      6. swap-sqr 84.8%

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

   9. Simplified 84.8%

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

   if 2.1500000000000001e-35 < M

   1. Initial program 79.4%

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

      1. *-commutative 79.4%

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

      2. times-frac 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in M around 0 44.0%

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

      1. *-commutative 44.0%

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

      2. unpow2 44.0%

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

      3. unpow2 44.0%

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

      4. unpow2 44.0%

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

   6. Simplified 44.0%

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

   7. Taylor expanded in D around 0 44.0%

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

      1. times-frac 42.1%

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

      2. unpow2 42.1%

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

      3. associate-*r* 46.0%

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

      4. unpow2 46.0%

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

      5. unpow2 46.0%

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

      6. associate-/l* 52.0%

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

      7. *-commutative 52.0%

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

      8. times-frac 64.2%

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

      9. associate-/l* 58.1%

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

   9. Simplified 58.1%

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

   10. Taylor expanded in M around 0 44.0%

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

       1. *-commutative 44.0%

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

       2. *-commutative 44.0%

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

       3. times-frac 42.1%

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

       4. unpow2 42.1%

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

       5. associate-*r* 46.0%

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

       6. *-commutative 46.0%

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

       7. associate-/l* 53.8%

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

       8. *-commutative 53.8%

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

       9. unpow2 53.8%

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

       10. unpow2 53.8%

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

       11. times-frac 66.0%

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

   12. Simplified 66.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.15 \cdot 10^{-35}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot \frac{h}{d}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\frac{M \cdot h}{\frac{\ell}{M}} \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 12: 76.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \mathsf{fma}\left(-0.125, D \cdot \frac{\left(h \cdot \frac{M}{d}\right) \cdot \frac{M \cdot w0}{d}}{\frac{\ell}{D}}, w0\right) \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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
 (fma -0.125 (* D (/ (* (* h (/ M d)) (/ (* M w0) d)) (/ l D))) w0))

C

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

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return fma(-0.125, Float64(D * Float64(Float64(Float64(h * Float64(M / d)) * Float64(Float64(M * w0) / d)) / Float64(l / D))), w0)
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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_] := N[(-0.125 * N[(D * N[(N[(N[(h * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * w0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]

Derivation

1. Initial program 82.9%

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

   1. *-commutative 82.9%

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

   2. times-frac 83.0%

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

3. Simplified 83.0%

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

   1. unpow2 83.0%

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

   2. associate-*l* 84.2%

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

   3. div-inv 84.2%

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

   4. associate-*l* 83.4%

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

   5. times-frac 83.4%

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

   6. *-un-lft-identity 83.4%

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

   7. *-commutative 83.4%

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

   8. clear-num 83.4%

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

   9. div-inv 83.4%

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

   10. div-inv 83.4%

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

   11. associate-*l* 84.6%

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

   12. times-frac 84.6%

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

   13. *-un-lft-identity 84.6%

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

   14. *-commutative 84.6%

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

   15. clear-num 84.6%

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

   16. div-inv 84.6%

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

   17. associate-*l* 83.3%

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

5. Applied egg-rr 88.4%

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

6. Taylor expanded in M around 0 88.1%

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

7. Taylor expanded in D around 0 52.2%

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

   1. fma-def 52.2%

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

9. Simplified 72.0%

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

10. Final simplification 72.0%

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

Alternative 13: 76.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.6 \cdot 10^{+46}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right) \cdot \left(D \cdot \frac{D}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 1.6e+46)
   (* w0 (+ 1.0 (* -0.125 (* (* (/ M d) (/ (* M h) d)) (* D (/ D l))))))
   w0))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.6e+46) {
		tmp = w0 * (1.0 + (-0.125 * (((M / d) * ((M * h) / d)) * (D * (D / l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 <= 1.6d+46) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((m / d_1) * ((m * h) / d_1)) * (d * (d / l)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 1.6e+46) {
		tmp = w0 * (1.0 + (-0.125 * (((M / d) * ((M * h) / d)) * (D * (D / l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 1.6e+46:
		tmp = w0 * (1.0 + (-0.125 * (((M / d) * ((M * h) / d)) * (D * (D / l)))))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 1.6e+46)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(M / d) * Float64(Float64(M * h) / d)) * Float64(D * Float64(D / l))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 1.6e+46)
		tmp = w0 * (1.0 + (-0.125 * (((M / d) * ((M * h) / d)) * (D * (D / l)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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, 1.6e+46], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(M / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 1.5999999999999999e46

   1. Initial program 81.3%

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

      1. *-commutative 81.3%

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

      2. times-frac 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in M around 0 51.0%

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

      1. *-commutative 51.0%

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

      2. unpow2 51.0%

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

      3. unpow2 51.0%

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

      4. unpow2 51.0%

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

   6. Simplified 51.0%

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

   7. Taylor expanded in D around 0 51.0%

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

      1. times-frac 53.4%

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

      2. unpow2 53.4%

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

      3. associate-*r* 55.0%

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

      4. unpow2 55.0%

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

      5. unpow2 55.0%

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

      6. associate-/l* 58.6%

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

      7. *-commutative 58.6%

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

      8. times-frac 66.7%

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

      9. associate-/l* 63.1%

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

   9. Simplified 63.1%

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

       1. associate-/l* 66.7%

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

       2. associate-/r/ 66.7%

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

   11. Applied egg-rr 66.7%

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

   if 1.5999999999999999e46 < d

   1. Initial program 88.2%

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

      1. *-commutative 88.2%

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

      2. times-frac 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in M around 0 87.2%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.6 \cdot 10^{+46}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right) \cdot \left(D \cdot \frac{D}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 14: 71.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{+49}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \left(\frac{D}{d \cdot d} \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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+49)
   w0
   (* -0.125 (* D (* (/ D (* d d)) (/ (* w0 (* h (* M M))) l))))))

C

M = abs(M);
D = abs(D);
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+49) {
		tmp = w0;
	} else {
		tmp = -0.125 * (D * ((D / (d * d)) * ((w0 * (h * (M * M))) / l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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+49) then
        tmp = w0
    else
        tmp = (-0.125d0) * (d * ((d / (d_1 * d_1)) * ((w0 * (h * (m * m))) / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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+49) {
		tmp = w0;
	} else {
		tmp = -0.125 * (D * ((D / (d * d)) * ((w0 * (h * (M * M))) / l)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2e+49:
		tmp = w0
	else:
		tmp = -0.125 * (D * ((D / (d * d)) * ((w0 * (h * (M * M))) / l)))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2e+49)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(D * Float64(Float64(D / Float64(d * d)) * Float64(Float64(w0 * Float64(h * Float64(M * M))) / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
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+49)
		tmp = w0;
	else
		tmp = -0.125 * (D * ((D / (d * d)) * ((w0 * (h * (M * M))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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+49], w0, N[(-0.125 * N[(D * N[(N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.99999999999999989e49

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. times-frac 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in M around 0 69.7%

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

   if 1.99999999999999989e49 < M

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. times-frac 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in M around 0 34.2%

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

      1. *-commutative 34.2%

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

      2. unpow2 34.2%

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

      3. unpow2 34.2%

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

      4. unpow2 34.2%

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

   6. Simplified 34.2%

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

   7. Taylor expanded in D around 0 34.2%

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

      1. times-frac 31.2%

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

      2. unpow2 31.2%

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

      3. associate-*r* 37.3%

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

      4. unpow2 37.3%

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

      5. unpow2 37.3%

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

      6. associate-/l* 43.6%

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

      7. *-commutative 43.6%

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

      8. times-frac 62.4%

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

      9. associate-/l* 56.1%

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

   9. Simplified 56.1%

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

   10. Taylor expanded in M around inf 19.7%

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

       1. *-commutative 19.7%

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

       2. times-frac 19.6%

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

       3. unpow2 19.6%

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

       4. associate-/l* 23.0%

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

       5. associate-*l/ 23.2%

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

       6. associate-*r* 23.2%

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

       7. unpow2 23.2%

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

       8. times-frac 23.6%

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

       9. *-commutative 23.6%

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

       10. unpow2 23.6%

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

       11. associate-*r* 23.8%

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

       12. associate-/r/ 23.7%

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

   12. Simplified 24.5%

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

   13. Taylor expanded in h around 0 23.3%

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

       1. *-commutative 23.3%

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

       2. times-frac 23.4%

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

       3. unpow2 23.4%

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

       4. *-commutative 23.4%

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

       5. *-commutative 23.4%

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

       6. unpow2 23.4%

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

   15. Simplified 23.4%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{+49}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \left(\frac{D}{d \cdot d} \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell}\right)\right)\\ \end{array} \]

Alternative 15: 71.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 5.4 \cdot 10^{+48}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \left(\frac{D \cdot w0}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 5.4e+48)
   w0
   (* -0.125 (* D (* (/ (* D w0) l) (/ (* h (* M M)) (* d d)))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 5.4e+48) {
		tmp = w0;
	} else {
		tmp = -0.125 * (D * (((D * w0) / l) * ((h * (M * M)) / (d * d))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 <= 5.4d+48) then
        tmp = w0
    else
        tmp = (-0.125d0) * (d * (((d * w0) / l) * ((h * (m * m)) / (d_1 * d_1))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 5.4e+48) {
		tmp = w0;
	} else {
		tmp = -0.125 * (D * (((D * w0) / l) * ((h * (M * M)) / (d * d))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 5.4e+48:
		tmp = w0
	else:
		tmp = -0.125 * (D * (((D * w0) / l) * ((h * (M * M)) / (d * d))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 5.4e+48)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(D * Float64(Float64(Float64(D * w0) / l) * Float64(Float64(h * Float64(M * M)) / Float64(d * d)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 5.4e+48)
		tmp = w0;
	else
		tmp = -0.125 * (D * (((D * w0) / l) * ((h * (M * M)) / (d * d))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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, 5.4e+48], w0, N[(-0.125 * N[(D * N[(N[(N[(D * w0), $MachinePrecision] / l), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.40000000000000007e48

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. times-frac 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in M around 0 69.7%

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

   if 5.40000000000000007e48 < M

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. times-frac 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in M around 0 34.2%

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

      1. *-commutative 34.2%

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

      2. unpow2 34.2%

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

      3. unpow2 34.2%

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

      4. unpow2 34.2%

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

   6. Simplified 34.2%

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

   7. Taylor expanded in D around 0 34.2%

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

      1. times-frac 31.2%

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

      2. unpow2 31.2%

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

      3. associate-*r* 37.3%

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

      4. unpow2 37.3%

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

      5. unpow2 37.3%

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

      6. associate-/l* 43.6%

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

      7. *-commutative 43.6%

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

      8. times-frac 62.4%

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

      9. associate-/l* 56.1%

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

   9. Simplified 56.1%

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

   10. Taylor expanded in M around inf 19.7%

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

       1. *-commutative 19.7%

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

       2. times-frac 19.6%

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

       3. unpow2 19.6%

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

       4. associate-/l* 23.0%

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

       5. associate-*l/ 23.2%

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

       6. associate-*r* 23.2%

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

       7. unpow2 23.2%

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

       8. times-frac 23.6%

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

       9. *-commutative 23.6%

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

       10. unpow2 23.6%

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

       11. associate-*r* 23.8%

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

       12. associate-/r/ 23.7%

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

   12. Simplified 24.5%

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

   13. Taylor expanded in h around 0 23.3%

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

       1. associate-*r* 23.2%

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

       2. *-commutative 23.2%

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

       3. times-frac 23.2%

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

       4. unpow2 23.2%

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

       5. unpow2 23.2%

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

   15. Simplified 23.2%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5.4 \cdot 10^{+48}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \left(\frac{D \cdot w0}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)\right)\\ \end{array} \]

Alternative 16: 68.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 5.2 \cdot 10^{-184}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \frac{\left(D \cdot h\right) \cdot \left(\frac{M}{d} \cdot \left(w0 \cdot \frac{M}{d}\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 5.2e-184)
   (* -0.125 (* D (/ (* (* D h) (* (/ M d) (* w0 (/ M d)))) l)))
   w0))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 5.2e-184) {
		tmp = -0.125 * (D * (((D * h) * ((M / d) * (w0 * (M / d)))) / l));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 <= 5.2d-184) then
        tmp = (-0.125d0) * (d * (((d * h) * ((m / d_1) * (w0 * (m / d_1)))) / l))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 5.2e-184) {
		tmp = -0.125 * (D * (((D * h) * ((M / d) * (w0 * (M / d)))) / l));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 5.2e-184:
		tmp = -0.125 * (D * (((D * h) * ((M / d) * (w0 * (M / d)))) / l))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 5.2e-184)
		tmp = Float64(-0.125 * Float64(D * Float64(Float64(Float64(D * h) * Float64(Float64(M / d) * Float64(w0 * Float64(M / d)))) / l)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 5.2e-184)
		tmp = -0.125 * (D * (((D * h) * ((M / d) * (w0 * (M / d)))) / l));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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, 5.2e-184], N[(-0.125 * N[(D * N[(N[(N[(D * h), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(w0 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 5.19999999999999957e-184

   1. Initial program 81.5%

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

      1. *-commutative 81.5%

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

      2. times-frac 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in M around 0 51.1%

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

      1. *-commutative 51.1%

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

      2. unpow2 51.1%

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

      3. unpow2 51.1%

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

      4. unpow2 51.1%

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

   6. Simplified 51.1%

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

   7. Taylor expanded in D around 0 51.1%

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

      1. times-frac 53.0%

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

      2. unpow2 53.0%

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

      3. associate-*r* 55.0%

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

      4. unpow2 55.0%

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

      5. unpow2 55.0%

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

      6. associate-/l* 59.1%

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

      7. *-commutative 59.1%

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

      8. times-frac 68.6%

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

      9. associate-/l* 64.5%

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

   9. Simplified 64.5%

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

   10. Taylor expanded in M around inf 16.5%

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

       1. *-commutative 16.5%

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

       2. times-frac 15.2%

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

       3. unpow2 15.2%

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

       4. associate-/l* 16.0%

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

       5. associate-*l/ 17.9%

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

       6. associate-*r* 17.8%

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

       7. unpow2 17.8%

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

       8. times-frac 18.9%

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

       9. *-commutative 18.9%

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

       10. unpow2 18.9%

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

       11. associate-*r* 19.7%

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

       12. associate-/r/ 19.4%

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

   12. Simplified 19.9%

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

       1. *-un-lft-identity 19.9%

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

       2. associate-/r/ 21.3%

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

       3. associate-*l* 21.3%

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

       4. associate-/l* 21.3%

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

   14. Applied egg-rr 21.3%

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

       1. *-lft-identity 21.3%

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

       2. associate-*l/ 21.4%

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

       3. *-commutative 21.4%

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

       4. associate-*r* 19.8%

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

       5. associate-/r/ 19.8%

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

   16. Simplified 19.8%

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

   if 5.19999999999999957e-184 < d

   1. Initial program 84.8%

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

      1. *-commutative 84.8%

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

      2. times-frac 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in M around 0 73.1%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5.2 \cdot 10^{-184}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \frac{\left(D \cdot h\right) \cdot \left(\frac{M}{d} \cdot \left(w0 \cdot \frac{M}{d}\right)\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 17: 68.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.15 \cdot 10^{-184}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \frac{\left(h \cdot \frac{M}{d}\right) \cdot \frac{w0}{\frac{d}{M}}}{\frac{\ell}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 2.15e-184)
   (* -0.125 (* D (/ (* (* h (/ M d)) (/ w0 (/ d M))) (/ l D))))
   w0))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 2.15e-184) {
		tmp = -0.125 * (D * (((h * (M / d)) * (w0 / (d / M))) / (l / D)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 <= 2.15d-184) then
        tmp = (-0.125d0) * (d * (((h * (m / d_1)) * (w0 / (d_1 / m))) / (l / d)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 2.15e-184) {
		tmp = -0.125 * (D * (((h * (M / d)) * (w0 / (d / M))) / (l / D)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 2.15e-184:
		tmp = -0.125 * (D * (((h * (M / d)) * (w0 / (d / M))) / (l / D)))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 2.15e-184)
		tmp = Float64(-0.125 * Float64(D * Float64(Float64(Float64(h * Float64(M / d)) * Float64(w0 / Float64(d / M))) / Float64(l / D))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 2.15e-184)
		tmp = -0.125 * (D * (((h * (M / d)) * (w0 / (d / M))) / (l / D)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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, 2.15e-184], N[(-0.125 * N[(D * N[(N[(N[(h * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(w0 / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 2.15000000000000003e-184

   1. Initial program 81.5%

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

      1. *-commutative 81.5%

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

      2. times-frac 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in M around 0 51.1%

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

      1. *-commutative 51.1%

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

      2. unpow2 51.1%

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

      3. unpow2 51.1%

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

      4. unpow2 51.1%

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

   6. Simplified 51.1%

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

   7. Taylor expanded in D around 0 51.1%

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

      1. times-frac 53.0%

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

      2. unpow2 53.0%

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

      3. associate-*r* 55.0%

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

      4. unpow2 55.0%

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

      5. unpow2 55.0%

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

      6. associate-/l* 59.1%

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

      7. *-commutative 59.1%

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

      8. times-frac 68.6%

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

      9. associate-/l* 64.5%

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

   9. Simplified 64.5%

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

   10. Taylor expanded in M around inf 16.5%

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

       1. *-commutative 16.5%

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

       2. times-frac 15.2%

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

       3. unpow2 15.2%

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

       4. associate-/l* 16.0%

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

       5. associate-*l/ 17.9%

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

       6. associate-*r* 17.8%

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

       7. unpow2 17.8%

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

       8. times-frac 18.9%

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

       9. *-commutative 18.9%

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

       10. unpow2 18.9%

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

       11. associate-*r* 19.7%

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

       12. associate-/r/ 19.4%

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

   12. Simplified 19.9%

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

   13. Taylor expanded in M around 0 19.9%

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

       1. associate-/l* 20.0%

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

   15. Simplified 20.0%

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

   if 2.15000000000000003e-184 < d

   1. Initial program 84.8%

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

      1. *-commutative 84.8%

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

      2. times-frac 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in M around 0 73.1%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.15 \cdot 10^{-184}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \frac{\left(h \cdot \frac{M}{d}\right) \cdot \frac{w0}{\frac{d}{M}}}{\frac{\ell}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 18: 68.4% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.7 \cdot 10^{-184}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \frac{\left(h \cdot \frac{M}{d}\right) \cdot \frac{M \cdot w0}{d}}{\frac{\ell}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 2.7e-184)
   (* -0.125 (* D (/ (* (* h (/ M d)) (/ (* M w0) d)) (/ l D))))
   w0))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 2.7e-184) {
		tmp = -0.125 * (D * (((h * (M / d)) * ((M * w0) / d)) / (l / D)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 <= 2.7d-184) then
        tmp = (-0.125d0) * (d * (((h * (m / d_1)) * ((m * w0) / d_1)) / (l / d)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 2.7e-184) {
		tmp = -0.125 * (D * (((h * (M / d)) * ((M * w0) / d)) / (l / D)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 2.7e-184:
		tmp = -0.125 * (D * (((h * (M / d)) * ((M * w0) / d)) / (l / D)))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 2.7e-184)
		tmp = Float64(-0.125 * Float64(D * Float64(Float64(Float64(h * Float64(M / d)) * Float64(Float64(M * w0) / d)) / Float64(l / D))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 2.7e-184)
		tmp = -0.125 * (D * (((h * (M / d)) * ((M * w0) / d)) / (l / D)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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, 2.7e-184], N[(-0.125 * N[(D * N[(N[(N[(h * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * w0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 2.7000000000000001e-184

   1. Initial program 81.5%

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

      1. *-commutative 81.5%

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

      2. times-frac 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in M around 0 51.1%

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

      1. *-commutative 51.1%

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

      2. unpow2 51.1%

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

      3. unpow2 51.1%

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

      4. unpow2 51.1%

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

   6. Simplified 51.1%

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

   7. Taylor expanded in D around 0 51.1%

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

      1. times-frac 53.0%

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

      2. unpow2 53.0%

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

      3. associate-*r* 55.0%

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

      4. unpow2 55.0%

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

      5. unpow2 55.0%

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

      6. associate-/l* 59.1%

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

      7. *-commutative 59.1%

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

      8. times-frac 68.6%

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

      9. associate-/l* 64.5%

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

   9. Simplified 64.5%

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

   10. Taylor expanded in M around inf 16.5%

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

       1. *-commutative 16.5%

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

       2. times-frac 15.2%

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

       3. unpow2 15.2%

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

       4. associate-/l* 16.0%

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

       5. associate-*l/ 17.9%

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

       6. associate-*r* 17.8%

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

       7. unpow2 17.8%

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

       8. times-frac 18.9%

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

       9. *-commutative 18.9%

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

       10. unpow2 18.9%

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

       11. associate-*r* 19.7%

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

       12. associate-/r/ 19.4%

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

   12. Simplified 19.9%

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

   if 2.7000000000000001e-184 < d

   1. Initial program 84.8%

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

      1. *-commutative 84.8%

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

      2. times-frac 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in M around 0 73.1%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.7 \cdot 10^{-184}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \frac{\left(h \cdot \frac{M}{d}\right) \cdot \frac{M \cdot w0}{d}}{\frac{\ell}{D}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 19: 68.0% accurate, 216.0× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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

M = Math.abs(M);
D = Math.abs(D);
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

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

Julia

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

MATLAB

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

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
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 82.9%

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

   1. *-commutative 82.9%

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

   2. times-frac 83.0%

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

3. Simplified 83.0%

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

4. Taylor expanded in M around 0 68.4%

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

5. Final simplification 68.4%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023217 
(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))))))