Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.7% → 92.6%

Time: 17.2s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+44}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(-2 \cdot \log \left(\frac{1}{M}\right) + \left(\log \left(\frac{h}{\ell} \cdot -0.25\right) + \left(2 \cdot \log D + -2 \cdot \log d\right)\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

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)) -1e+44)
   (pow
    (*
     (cbrt w0)
     (exp
      (*
       0.16666666666666666
       (+
        (* -2.0 (log (/ 1.0 M)))
        (+ (log (* (/ h l) -0.25)) (+ (* 2.0 (log D)) (* -2.0 (log d))))))))
    3.0)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* M (* D (/ 0.5 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)) <= -1e+44) {
		tmp = pow((cbrt(w0) * exp((0.16666666666666666 * ((-2.0 * log((1.0 / M))) + (log(((h / l) * -0.25)) + ((2.0 * log(D)) + (-2.0 * log(d)))))))), 3.0);
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow((M * (D * (0.5 / 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)) <= -1e+44) {
		tmp = Math.pow((Math.cbrt(w0) * Math.exp((0.16666666666666666 * ((-2.0 * Math.log((1.0 / M))) + (Math.log(((h / l) * -0.25)) + ((2.0 * Math.log(D)) + (-2.0 * Math.log(d)))))))), 3.0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M * (D * (0.5 / 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)) <= -1e+44)
		tmp = Float64(cbrt(w0) * exp(Float64(0.16666666666666666 * Float64(Float64(-2.0 * log(Float64(1.0 / M))) + Float64(log(Float64(Float64(h / l) * -0.25)) + Float64(Float64(2.0 * log(D)) + Float64(-2.0 * log(d)))))))) ^ 3.0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M * Float64(D * Float64(0.5 / 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], -1e+44], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[(N[(-2.0 * N[Log[N[(1.0 / M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Log[N[(N[(h / l), $MachinePrecision] * -0.25), $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], 3.0], $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $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)) < -1.0000000000000001e44

   1. Initial program 56.8%

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

      1. times-frac 57.3%

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

   3. Simplified 57.3%

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

      1. add-cube-cbrt 57.3%

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

      2. pow3 57.2%

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

   5. Applied egg-rr 57.3%

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

   6. Taylor expanded in D around inf 14.1%

      \[\leadsto {\color{blue}{\left({\left(1 \cdot w0\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 26.0%

         \[\leadsto {\left(\color{blue}{\sqrt[3]{1 \cdot w0}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      2. *-lft-identity 26.0%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{w0}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      3. exp-prod 25.6%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot \color{blue}{{\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}}\right)}^{3} \]

      4. distribute-lft-neg-in 25.6%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \color{blue}{\left(\left(-0.25\right) \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}\right)} + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      5. metadata-eval 25.6%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(\color{blue}{-0.25} \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      6. associate-/r* 24.6%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \color{blue}{\frac{\frac{h \cdot {M}^{2}}{\ell}}{{d}^{2}}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      7. unpow2 24.6%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}{{d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      8. associate-/l* 24.7%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{\color{blue}{\frac{h}{\frac{\ell}{M \cdot M}}}}{{d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      9. unpow2 24.7%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{\frac{h}{\frac{\ell}{M \cdot M}}}{\color{blue}{d \cdot d}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      10. log-rec 24.7%

          \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{\frac{h}{\frac{\ell}{M \cdot M}}}{d \cdot d}\right) + -2 \cdot \color{blue}{\left(-\log D\right)}\right)}\right)}^{3} \]

   8. Simplified 24.7%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{\frac{h}{\frac{\ell}{M \cdot M}}}{d \cdot d}\right) + -2 \cdot \left(-\log D\right)\right)}\right)}}^{3} \]

   9. Taylor expanded in d around 0 14.3%

      \[\leadsto {\left(\sqrt[3]{w0} \cdot \color{blue}{e^{0.16666666666666666 \cdot \left(2 \cdot \log D + \left(-2 \cdot \log d + \log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right)\right)}}\right)}^{3} \]

   10. Taylor expanded in M around inf 8.6%

       \[\leadsto {\left(\sqrt[3]{w0} \cdot \color{blue}{e^{0.16666666666666666 \cdot \left(-2 \cdot \log \left(\frac{1}{M}\right) + \left(\log \left(-0.25 \cdot \frac{h}{\ell}\right) + \left(2 \cdot \log D + -2 \cdot \log d\right)\right)\right)}}\right)}^{3} \]

   if -1.0000000000000001e44 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 91.5%

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

      1. times-frac 91.5%

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

   3. Simplified 91.5%

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

      1. associate-*r/ 97.3%

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

      2. frac-times 97.3%

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

      3. div-inv 97.3%

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

      4. associate-*l* 97.3%

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

      5. associate-/r* 97.3%

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

      6. metadata-eval 97.3%

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

   5. Applied egg-rr 97.3%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+44}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(-2 \cdot \log \left(\frac{1}{M}\right) + \left(\log \left(\frac{h}{\ell} \cdot -0.25\right) + \left(2 \cdot \log D + -2 \cdot \log d\right)\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 2: 92.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+44}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\log \left(\frac{h}{\ell} \cdot -0.25\right) + \left(\left(2 \cdot \log D + -2 \cdot \log d\right) + 2 \cdot \log M\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

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)) -1e+44)
   (pow
    (*
     (cbrt w0)
     (exp
      (*
       0.16666666666666666
       (+
        (log (* (/ h l) -0.25))
        (+ (+ (* 2.0 (log D)) (* -2.0 (log d))) (* 2.0 (log M)))))))
    3.0)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* M (* D (/ 0.5 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)) <= -1e+44) {
		tmp = pow((cbrt(w0) * exp((0.16666666666666666 * (log(((h / l) * -0.25)) + (((2.0 * log(D)) + (-2.0 * log(d))) + (2.0 * log(M))))))), 3.0);
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow((M * (D * (0.5 / 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)) <= -1e+44) {
		tmp = Math.pow((Math.cbrt(w0) * Math.exp((0.16666666666666666 * (Math.log(((h / l) * -0.25)) + (((2.0 * Math.log(D)) + (-2.0 * Math.log(d))) + (2.0 * Math.log(M))))))), 3.0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M * (D * (0.5 / 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)) <= -1e+44)
		tmp = Float64(cbrt(w0) * exp(Float64(0.16666666666666666 * Float64(log(Float64(Float64(h / l) * -0.25)) + Float64(Float64(Float64(2.0 * log(D)) + Float64(-2.0 * log(d))) + Float64(2.0 * log(M))))))) ^ 3.0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M * Float64(D * Float64(0.5 / 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], -1e+44], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[(N[Log[N[(N[(h / l), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(2.0 * N[Log[D], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[Log[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Log[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $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)) < -1.0000000000000001e44

   1. Initial program 56.8%

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

      1. times-frac 57.3%

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

   3. Simplified 57.3%

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

      1. add-cube-cbrt 57.3%

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

      2. pow3 57.2%

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

   5. Applied egg-rr 57.3%

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

   6. Taylor expanded in D around inf 14.1%

      \[\leadsto {\color{blue}{\left({\left(1 \cdot w0\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 26.0%

         \[\leadsto {\left(\color{blue}{\sqrt[3]{1 \cdot w0}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      2. *-lft-identity 26.0%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{w0}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      3. exp-prod 25.6%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot \color{blue}{{\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}}\right)}^{3} \]

      4. distribute-lft-neg-in 25.6%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \color{blue}{\left(\left(-0.25\right) \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}\right)} + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      5. metadata-eval 25.6%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(\color{blue}{-0.25} \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      6. associate-/r* 24.6%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \color{blue}{\frac{\frac{h \cdot {M}^{2}}{\ell}}{{d}^{2}}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      7. unpow2 24.6%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}{{d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      8. associate-/l* 24.7%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{\color{blue}{\frac{h}{\frac{\ell}{M \cdot M}}}}{{d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      9. unpow2 24.7%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{\frac{h}{\frac{\ell}{M \cdot M}}}{\color{blue}{d \cdot d}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      10. log-rec 24.7%

          \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{\frac{h}{\frac{\ell}{M \cdot M}}}{d \cdot d}\right) + -2 \cdot \color{blue}{\left(-\log D\right)}\right)}\right)}^{3} \]

   8. Simplified 24.7%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{\frac{h}{\frac{\ell}{M \cdot M}}}{d \cdot d}\right) + -2 \cdot \left(-\log D\right)\right)}\right)}}^{3} \]

   9. Taylor expanded in d around 0 14.3%

      \[\leadsto {\left(\sqrt[3]{w0} \cdot \color{blue}{e^{0.16666666666666666 \cdot \left(2 \cdot \log D + \left(-2 \cdot \log d + \log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right)\right)}}\right)}^{3} \]

   10. Taylor expanded in M around 0 8.6%

       \[\leadsto {\left(\sqrt[3]{w0} \cdot \color{blue}{e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h}{\ell}\right) + \left(2 \cdot \log M + \left(2 \cdot \log D + -2 \cdot \log d\right)\right)\right)}}\right)}^{3} \]

   if -1.0000000000000001e44 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 91.5%

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

      1. times-frac 91.5%

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

   3. Simplified 91.5%

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

      1. associate-*r/ 97.3%

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

      2. frac-times 97.3%

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

      3. div-inv 97.3%

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

      4. associate-*l* 97.3%

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

      5. associate-/r* 97.3%

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

      6. metadata-eval 97.3%

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

   5. Applied egg-rr 97.3%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+44}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\log \left(\frac{h}{\ell} \cdot -0.25\right) + \left(\left(2 \cdot \log D + -2 \cdot \log d\right) + 2 \cdot \log M\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 3: 88.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+268}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(2 \cdot \log D + \left(-2 \cdot \log d + \log \left(-0.25 \cdot \left(M \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right)\right)\right)}\right)}^{3}\\ \end{array} \end{array} \]

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) 5e+268)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* M (* D (/ 0.5 d))) 2.0)) l))))
   (pow
    (*
     (cbrt w0)
     (exp
      (*
       0.16666666666666666
       (+
        (* 2.0 (log D))
        (+ (* -2.0 (log d)) (log (* -0.25 (* M (* M (/ h l))))))))))
    3.0)))

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 (pow(((M * D) / (2.0 * d)), 2.0) <= 5e+268) {
		tmp = w0 * sqrt((1.0 - ((h * pow((M * (D * (0.5 / d))), 2.0)) / l)));
	} else {
		tmp = pow((cbrt(w0) * exp((0.16666666666666666 * ((2.0 * log(D)) + ((-2.0 * log(d)) + log((-0.25 * (M * (M * (h / l)))))))))), 3.0);
	}
	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) <= 5e+268) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M * (D * (0.5 / d))), 2.0)) / l)));
	} else {
		tmp = Math.pow((Math.cbrt(w0) * Math.exp((0.16666666666666666 * ((2.0 * Math.log(D)) + ((-2.0 * Math.log(d)) + Math.log((-0.25 * (M * (M * (h / l)))))))))), 3.0);
	}
	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 ((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) <= 5e+268)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(cbrt(w0) * exp(Float64(0.16666666666666666 * Float64(Float64(2.0 * log(D)) + Float64(Float64(-2.0 * log(d)) + log(Float64(-0.25 * Float64(M * Float64(M * Float64(h / l)))))))))) ^ 3.0;
	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[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 5e+268], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[(N[(2.0 * N[Log[D], $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[Log[d], $MachinePrecision]), $MachinePrecision] + N[Log[N[(-0.25 * N[(M * N[(M * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 5.0000000000000002e268

   1. Initial program 90.2%

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

      1. times-frac 89.9%

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

   3. Simplified 89.9%

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

      1. associate-*r/ 95.5%

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

      2. frac-times 95.8%

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

      3. div-inv 95.8%

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

      4. associate-*l* 95.5%

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

      5. associate-/r* 95.5%

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

      6. metadata-eval 95.5%

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

   5. Applied egg-rr 95.5%

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

   if 5.0000000000000002e268 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 52.1%

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

      1. times-frac 53.7%

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

   3. Simplified 53.7%

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

      1. add-cube-cbrt 53.7%

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

      2. pow3 53.7%

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

   5. Applied egg-rr 53.7%

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

   6. Taylor expanded in D around inf 17.5%

      \[\leadsto {\color{blue}{\left({\left(1 \cdot w0\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 29.4%

         \[\leadsto {\left(\color{blue}{\sqrt[3]{1 \cdot w0}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      2. *-lft-identity 29.4%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{w0}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      3. exp-prod 29.0%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot \color{blue}{{\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}}\right)}^{3} \]

      4. distribute-lft-neg-in 29.0%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \color{blue}{\left(\left(-0.25\right) \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}\right)} + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      5. metadata-eval 29.0%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(\color{blue}{-0.25} \cdot \frac{h \cdot {M}^{2}}{\ell \cdot {d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      6. associate-/r* 28.9%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \color{blue}{\frac{\frac{h \cdot {M}^{2}}{\ell}}{{d}^{2}}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      7. unpow2 28.9%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{\frac{h \cdot \color{blue}{\left(M \cdot M\right)}}{\ell}}{{d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      8. associate-/l* 29.0%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{\color{blue}{\frac{h}{\frac{\ell}{M \cdot M}}}}{{d}^{2}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      9. unpow2 29.0%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{\frac{h}{\frac{\ell}{M \cdot M}}}{\color{blue}{d \cdot d}}\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3} \]

      10. log-rec 29.0%

          \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{\frac{h}{\frac{\ell}{M \cdot M}}}{d \cdot d}\right) + -2 \cdot \color{blue}{\left(-\log D\right)}\right)}\right)}^{3} \]

   8. Simplified 29.0%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{\frac{h}{\frac{\ell}{M \cdot M}}}{d \cdot d}\right) + -2 \cdot \left(-\log D\right)\right)}\right)}}^{3} \]

   9. Taylor expanded in d around 0 17.5%

      \[\leadsto {\left(\sqrt[3]{w0} \cdot \color{blue}{e^{0.16666666666666666 \cdot \left(2 \cdot \log D + \left(-2 \cdot \log d + \log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{\ell}\right)\right)\right)}}\right)}^{3} \]
   10. Step-by-step derivation

       1. pow2 17.5%

          \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(2 \cdot \log D + \left(-2 \cdot \log d + \log \left(-0.25 \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{\ell}\right)\right)\right)}\right)}^{3} \]

       2. associate-*r/ 16.2%

          \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(2 \cdot \log D + \left(-2 \cdot \log d + \log \left(-0.25 \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot \frac{h}{\ell}\right)}\right)\right)\right)}\right)}^{3} \]

       3. associate-*l* 17.5%

          \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(2 \cdot \log D + \left(-2 \cdot \log d + \log \left(-0.25 \cdot \color{blue}{\left(M \cdot \left(M \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right)}\right)}^{3} \]

   11. Applied egg-rr 17.5%

       \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(2 \cdot \log D + \left(-2 \cdot \log d + \log \left(-0.25 \cdot \color{blue}{\left(M \cdot \left(M \cdot \frac{h}{\ell}\right)\right)}\right)\right)\right)}\right)}^{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+268}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(2 \cdot \log D + \left(-2 \cdot \log d + \log \left(-0.25 \cdot \left(M \cdot \left(M \cdot \frac{h}{\ell}\right)\right)\right)\right)\right)}\right)}^{3}\\ \end{array} \]

Alternative 4: 87.5% accurate, 0.5× speedup

Math

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

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
 (let* ((t_0
         (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))))
   (if (<= t_0 INFINITY) t_0 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 t_0 = w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = w0;
	}
	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 t_0 = w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} 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):
	t_0 = w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0
	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)
	t_0 = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	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)
	t_0 = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0;
	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_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, Infinity], t$95$0, w0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < +inf.0

   1. Initial program 86.0%

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

   if +inf.0 < (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))))

   1. Initial program 0.0%

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

      1. times-frac 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in M around 0 62.6%

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

4. Final simplification 84.6%

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

Alternative 5: 83.5% accurate, 1.0× speedup

Math

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

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 (<= (/ h l) 2e+230)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M 2.0) (/ D d)) 2.0)))))
   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 ((h / l) <= 2e+230) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / 2.0) * (D / d)), 2.0))));
	} 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 ((h / l) <= 2d+230) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / 2.0d0) * (d / d_1)) ** 2.0d0))))
    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 ((h / l) <= 2e+230) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / 2.0) * (D / d)), 2.0))));
	} 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 (h / l) <= 2e+230:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / 2.0) * (D / d)), 2.0))))
	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 (Float64(h / l) <= 2e+230)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0)))));
	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 ((h / l) <= 2e+230)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / 2.0) * (D / d)) ^ 2.0))));
	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[N[(h / l), $MachinePrecision], 2e+230], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < 2.0000000000000002e230

   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. times-frac 83.3%

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

   3. Simplified 83.3%

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

   if 2.0000000000000002e230 < (/.f64 h l)

   1. Initial program 22.2%

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

      1. times-frac 22.2%

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

   3. Simplified 22.2%

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

   4. Taylor expanded in M around 0 100.0%

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

4. Final simplification 83.9%

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

Alternative 6: 86.2% accurate, 1.0× speedup

Math

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

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 (* M (* D (/ 0.5 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((M * (D * (0.5 / 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 * ((m * (d * (0.5d0 / 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((M * (D * (0.5 / 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((M * (D * (0.5 / 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(M * Float64(D * Float64(0.5 / 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 * ((M * (D * (0.5 / 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[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

   1. times-frac 81.1%

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

3. Simplified 81.1%

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

   1. associate-*r/ 85.9%

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

   2. frac-times 85.7%

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

   3. div-inv 85.7%

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

   4. associate-*l* 85.9%

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

   5. associate-/r* 85.9%

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

   6. metadata-eval 85.9%

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

5. Applied egg-rr 85.9%

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

6. Final simplification 85.9%

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

Alternative 7: 79.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{-151}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 155000000:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D \cdot \left(M \cdot M\right)}{\ell} \cdot \frac{D}{\frac{d \cdot d}{h}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot \ell}{\left(M \cdot M\right) \cdot \frac{h}{d}}}}\\ \end{array} \end{array} \]

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-151)
   w0
   (if (<= M 155000000.0)
     (* w0 (sqrt (- 1.0 (* 0.25 (* (/ (* D (* M M)) l) (/ D (/ (* d d) h)))))))
     (*
      w0
      (sqrt (- 1.0 (* 0.25 (/ (* D D) (/ (* d l) (* (* M M) (/ h 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 <= 2e-151) {
		tmp = w0;
	} else if (M <= 155000000.0) {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D * (M * M)) / l) * (D / ((d * d) / h))))));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * ((D * D) / ((d * l) / ((M * M) * (h / 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 <= 2d-151) then
        tmp = w0
    else if (m <= 155000000.0d0) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d * (m * m)) / l) * (d / ((d_1 * d_1) / h))))))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * ((d * d) / ((d_1 * l) / ((m * m) * (h / 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 <= 2e-151) {
		tmp = w0;
	} else if (M <= 155000000.0) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D * (M * M)) / l) * (D / ((d * d) / h))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * ((D * D) / ((d * l) / ((M * M) * (h / 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 <= 2e-151:
		tmp = w0
	elif M <= 155000000.0:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D * (M * M)) / l) * (D / ((d * d) / h))))))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * ((D * D) / ((d * l) / ((M * M) * (h / 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 <= 2e-151)
		tmp = w0;
	elseif (M <= 155000000.0)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D * Float64(M * M)) / l) * Float64(D / Float64(Float64(d * d) / h)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d * l) / Float64(Float64(M * M) * Float64(h / 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 <= 2e-151)
		tmp = w0;
	elseif (M <= 155000000.0)
		tmp = w0 * sqrt((1.0 - (0.25 * (((D * (M * M)) / l) * (D / ((d * d) / h))))));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * ((D * D) / ((d * l) / ((M * M) * (h / 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, 2e-151], w0, If[LessEqual[M, 155000000.0], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(D / N[(N[(d * d), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(D * D), $MachinePrecision] / N[(N[(d * l), $MachinePrecision] / N[(N[(M * M), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < 1.9999999999999999e-151

   1. Initial program 81.1%

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

      1. times-frac 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in M around 0 71.0%

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

   if 1.9999999999999999e-151 < M < 1.55e8

   1. Initial program 91.4%

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

      1. times-frac 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in M around 0 85.2%

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

      1. *-commutative 85.2%

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

      2. associate-/l* 85.2%

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

      3. unpow2 85.2%

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

      4. unpow2 85.2%

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

      5. *-commutative 85.2%

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

      6. unpow2 85.2%

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

   6. Simplified 85.2%

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

   7. Taylor expanded in l around 0 85.2%

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

      1. unpow2 79.8%

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

      2. times-frac 77.0%

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

      3. unpow2 77.0%

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

      4. associate-/l* 77.0%

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

   9. Simplified 82.5%

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

       1. *-un-lft-identity 82.5%

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

       2. times-frac 85.7%

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

       3. associate-/r/ 85.7%

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

   11. Applied egg-rr 85.7%

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

       1. *-lft-identity 85.7%

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

       2. unpow2 85.7%

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

       3. associate-/r/ 88.6%

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

       4. unpow2 88.6%

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

       5. associate-*l/ 88.6%

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

   13. Simplified 88.6%

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

       1. associate-*l/ 88.7%

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

   15. Applied egg-rr 88.7%

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

   if 1.55e8 < M

   1. Initial program 73.3%

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

      1. times-frac 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in M around 0 49.2%

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

      1. *-commutative 49.2%

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

      2. associate-/l* 51.2%

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

      3. unpow2 51.2%

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

      4. unpow2 51.2%

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

      5. *-commutative 51.2%

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

      6. unpow2 51.2%

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

   6. Simplified 51.2%

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

   7. Taylor expanded in l around 0 51.2%

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

      1. unpow2 45.8%

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

      2. times-frac 39.7%

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

      3. unpow2 39.7%

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

      4. associate-/l* 40.0%

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

   9. Simplified 43.6%

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

       1. frac-times 48.1%

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

   11. Applied egg-rr 55.3%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{-151}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 155000000:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D \cdot \left(M \cdot M\right)}{\ell} \cdot \frac{D}{\frac{d \cdot d}{h}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot \ell}{\left(M \cdot M\right) \cdot \frac{h}{d}}}}\\ \end{array} \]

Alternative 8: 78.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.9 \cdot 10^{-151}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{h}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{\ell}\right)\right)}\\ \end{array} \end{array} \]

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 2.9e-151)
   w0
   (* w0 (sqrt (- 1.0 (* 0.25 (* (/ D (/ (* d d) h)) (* (* M M) (/ 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 tmp;
	if (M <= 2.9e-151) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * ((D / ((d * d) / h)) * ((M * M) * (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) :: tmp
    if (m <= 2.9d-151) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * ((d / ((d_1 * d_1) / h)) * ((m * m) * (d / 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 <= 2.9e-151) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * ((D / ((d * d) / h)) * ((M * M) * (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):
	tmp = 0
	if M <= 2.9e-151:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * ((D / ((d * d) / h)) * ((M * M) * (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)
	tmp = 0.0
	if (M <= 2.9e-151)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(D / Float64(Float64(d * d) / h)) * Float64(Float64(M * M) * Float64(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)
	tmp = 0.0;
	if (M <= 2.9e-151)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - (0.25 * ((D / ((d * d) / h)) * ((M * M) * (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_] := If[LessEqual[M, 2.9e-151], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(D / N[(N[(d * d), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.90000000000000013e-151

   1. Initial program 81.1%

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

      1. times-frac 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in M around 0 71.0%

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

   if 2.90000000000000013e-151 < M

   1. Initial program 80.6%

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

      1. times-frac 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in M around 0 63.8%

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

      1. *-commutative 63.8%

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

      2. associate-/l* 65.0%

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

      3. unpow2 65.0%

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

      4. unpow2 65.0%

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

      5. *-commutative 65.0%

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

      6. unpow2 65.0%

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

   6. Simplified 65.0%

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

   7. Taylor expanded in l around 0 65.0%

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

      1. unpow2 59.6%

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

      2. times-frac 54.8%

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

      3. unpow2 54.8%

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

      4. associate-/l* 55.0%

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

   9. Simplified 59.3%

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

       1. *-un-lft-identity 59.3%

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

       2. times-frac 60.5%

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

       3. associate-/r/ 60.5%

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

   11. Applied egg-rr 60.5%

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

       1. *-lft-identity 60.5%

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

       2. unpow2 60.5%

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

       3. associate-/r/ 64.0%

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

       4. unpow2 64.0%

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

       5. associate-*l/ 63.9%

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

   13. Simplified 63.9%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.9 \cdot 10^{-151}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{h}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{\ell}\right)\right)}\\ \end{array} \]

Alternative 9: 80.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 8.8 \cdot 10^{-150}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{\frac{\ell}{M \cdot M}} \cdot \frac{D}{d \cdot \frac{d}{h}}\right)}\\ \end{array} \end{array} \]

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 8.8e-150)
   w0
   (* w0 (sqrt (- 1.0 (* 0.25 (* (/ D (/ l (* M M))) (/ D (* d (/ d h))))))))))

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 <= 8.8e-150) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * ((D / (l / (M * M))) * (D / (d * (d / h)))))));
	}
	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 <= 8.8d-150) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * ((d / (l / (m * m))) * (d / (d_1 * (d_1 / h)))))))
    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 <= 8.8e-150) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * ((D / (l / (M * M))) * (D / (d * (d / h)))))));
	}
	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 <= 8.8e-150:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * ((D / (l / (M * M))) * (D / (d * (d / h)))))))
	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 <= 8.8e-150)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(D / Float64(l / Float64(M * M))) * Float64(D / Float64(d * Float64(d / h))))))));
	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 <= 8.8e-150)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - (0.25 * ((D / (l / (M * M))) * (D / (d * (d / h)))))));
	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, 8.8e-150], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(D / N[(l / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d * N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 8.7999999999999997e-150

   1. Initial program 81.1%

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

      1. times-frac 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in M around 0 71.0%

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

   if 8.7999999999999997e-150 < M

   1. Initial program 80.6%

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

      1. times-frac 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in M around 0 63.8%

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

      1. *-commutative 63.8%

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

      2. associate-/l* 65.0%

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

      3. unpow2 65.0%

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

      4. unpow2 65.0%

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

      5. *-commutative 65.0%

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

      6. unpow2 65.0%

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

   6. Simplified 65.0%

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

   7. Taylor expanded in l around 0 65.0%

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

      1. unpow2 59.6%

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

      2. times-frac 54.8%

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

      3. unpow2 54.8%

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

      4. associate-/l* 55.0%

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

   9. Simplified 59.3%

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

       1. times-frac 60.5%

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

       2. associate-/r/ 60.5%

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

   11. Applied egg-rr 60.5%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 8.8 \cdot 10^{-150}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D}{\frac{\ell}{M \cdot M}} \cdot \frac{D}{d \cdot \frac{d}{h}}\right)}\\ \end{array} \]

Alternative 10: 78.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.25 \cdot 10^{-151}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D \cdot \left(M \cdot M\right)}{\ell} \cdot \frac{D}{\frac{d \cdot d}{h}}\right)}\\ \end{array} \end{array} \]

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 1.25e-151)
   w0
   (* w0 (sqrt (- 1.0 (* 0.25 (* (/ (* D (* M M)) l) (/ D (/ (* d d) h)))))))))

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 <= 1.25e-151) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D * (M * M)) / l) * (D / ((d * d) / h))))));
	}
	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 <= 1.25d-151) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d * (m * m)) / l) * (d / ((d_1 * d_1) / h))))))
    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 <= 1.25e-151) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D * (M * M)) / l) * (D / ((d * d) / h))))));
	}
	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 <= 1.25e-151:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D * (M * M)) / l) * (D / ((d * d) / h))))))
	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 <= 1.25e-151)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D * Float64(M * M)) / l) * Float64(D / Float64(Float64(d * d) / h)))))));
	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 <= 1.25e-151)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (((D * (M * M)) / l) * (D / ((d * d) / h))))));
	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, 1.25e-151], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(D / N[(N[(d * d), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.25000000000000001e-151

   1. Initial program 81.1%

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

      1. times-frac 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in M around 0 71.0%

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

   if 1.25000000000000001e-151 < M

   1. Initial program 80.6%

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

      1. times-frac 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in M around 0 63.8%

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

      1. *-commutative 63.8%

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

      2. associate-/l* 65.0%

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

      3. unpow2 65.0%

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

      4. unpow2 65.0%

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

      5. *-commutative 65.0%

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

      6. unpow2 65.0%

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

   6. Simplified 65.0%

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

   7. Taylor expanded in l around 0 65.0%

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

      1. unpow2 59.6%

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

      2. times-frac 54.8%

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

      3. unpow2 54.8%

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

      4. associate-/l* 55.0%

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

   9. Simplified 59.3%

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

       1. *-un-lft-identity 59.3%

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

       2. times-frac 60.5%

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

       3. associate-/r/ 60.5%

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

   11. Applied egg-rr 60.5%

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

       1. *-lft-identity 60.5%

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

       2. unpow2 60.5%

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

       3. associate-/r/ 64.0%

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

       4. unpow2 64.0%

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

       5. associate-*l/ 63.9%

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

   13. Simplified 63.9%

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

       1. associate-*l/ 63.9%

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

   15. Applied egg-rr 63.9%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.25 \cdot 10^{-151}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D \cdot \left(M \cdot M\right)}{\ell} \cdot \frac{D}{\frac{d \cdot d}{h}}\right)}\\ \end{array} \]

Alternative 11: 76.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.1 \cdot 10^{-137}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{\ell}{w0}}, w0\right)\\ \end{array} \end{array} \]

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 1.1e-137)
   w0
   (fma -0.125 (* (* (/ D d) (/ D d)) (/ (* M (* M h)) (/ l w0))) 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 (M <= 1.1e-137) {
		tmp = w0;
	} else {
		tmp = fma(-0.125, (((D / d) * (D / d)) * ((M * (M * h)) / (l / w0))), 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 (M <= 1.1e-137)
		tmp = w0;
	else
		tmp = fma(-0.125, Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * Float64(M * h)) / Float64(l / w0))), w0);
	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[M, 1.1e-137], w0, N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(l / w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.1000000000000001e-137

   1. Initial program 81.1%

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

      1. times-frac 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in M around 0 71.0%

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

   if 1.1000000000000001e-137 < M

   1. Initial program 80.6%

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

      1. times-frac 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in M around 0 53.5%

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

      1. fma-def 53.5%

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

      2. *-commutative 53.5%

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

      3. times-frac 54.6%

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

      4. unpow2 54.6%

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

      5. unpow2 54.6%

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

      6. times-frac 57.3%

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

      7. unpow2 57.3%

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

      8. *-commutative 57.3%

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

      9. associate-/l* 56.0%

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

      10. *-commutative 56.0%

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

      11. unpow2 56.0%

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

      12. associate-*l* 59.6%

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

   6. Simplified 59.6%

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

      1. unpow2 59.6%

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

   8. Applied egg-rr 59.6%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.1 \cdot 10^{-137}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\frac{\ell}{w0}}, w0\right)\\ \end{array} \]

Alternative 12: 73.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.7 \cdot 10^{-100}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot D}{\frac{\ell}{h} \cdot \frac{d \cdot d}{M \cdot M}}\right)\\ \end{array} \end{array} \]

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 1.7e-100)
   w0
   (* w0 (+ 1.0 (* -0.125 (/ (* D D) (* (/ l h) (/ (* d d) (* M M)))))))))

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 <= 1.7e-100) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / ((l / h) * ((d * d) / (M * M))))));
	}
	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 <= 1.7d-100) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * d) / ((l / h) * ((d_1 * d_1) / (m * m))))))
    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 <= 1.7e-100) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / ((l / h) * ((d * d) / (M * M))))));
	}
	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 <= 1.7e-100:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / ((l / h) * ((d * d) / (M * M))))))
	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 <= 1.7e-100)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * D) / Float64(Float64(l / h) * Float64(Float64(d * d) / Float64(M * M)))))));
	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 <= 1.7e-100)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / ((l / h) * ((d * d) / (M * M))))));
	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, 1.7e-100], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * D), $MachinePrecision] / N[(N[(l / h), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.69999999999999988e-100

   1. Initial program 81.1%

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

      1. times-frac 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in M around 0 71.0%

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

   if 1.69999999999999988e-100 < M

   1. Initial program 80.6%

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

      1. times-frac 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in M around 0 55.5%

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

      1. *-commutative 55.5%

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

      2. *-commutative 55.5%

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

      3. associate-/l* 56.8%

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

      4. unpow2 56.8%

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

      5. unpow2 56.8%

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

      6. *-commutative 56.8%

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

      7. unpow2 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in l around 0 56.8%

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

      1. unpow2 56.8%

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

      2. times-frac 51.3%

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

      3. unpow2 51.3%

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

      4. associate-/l* 51.5%

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

   9. Simplified 51.5%

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

   10. Taylor expanded in l around 0 56.8%

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

       1. times-frac 58.3%

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

       2. unpow2 58.3%

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

       3. unpow2 58.3%

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

   12. Simplified 58.3%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.7 \cdot 10^{-100}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot D}{\frac{\ell}{h} \cdot \frac{d \cdot d}{M \cdot M}}\right)\\ \end{array} \]

Alternative 13: 74.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.35 \cdot 10^{-100}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot D}{\frac{d \cdot \ell}{\left(M \cdot M\right) \cdot \frac{h}{d}}} \cdot -0.125\right)\\ \end{array} \end{array} \]

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 1.35e-100)
   w0
   (* w0 (+ 1.0 (* (/ (* D D) (/ (* d l) (* (* M M) (/ h 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 <= 1.35e-100) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (((D * D) / ((d * l) / ((M * M) * (h / 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 <= 1.35d-100) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + (((d * d) / ((d_1 * l) / ((m * m) * (h / 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 <= 1.35e-100) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (((D * D) / ((d * l) / ((M * M) * (h / 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 <= 1.35e-100:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (((D * D) / ((d * l) / ((M * M) * (h / 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 <= 1.35e-100)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D * D) / Float64(Float64(d * l) / Float64(Float64(M * M) * Float64(h / 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 <= 1.35e-100)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (((D * D) / ((d * l) / ((M * M) * (h / 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, 1.35e-100], w0, N[(w0 * N[(1.0 + N[(N[(N[(D * D), $MachinePrecision] / N[(N[(d * l), $MachinePrecision] / N[(N[(M * M), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.35000000000000008e-100

   1. Initial program 81.1%

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

      1. times-frac 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in M around 0 71.0%

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

   if 1.35000000000000008e-100 < M

   1. Initial program 80.6%

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

      1. times-frac 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in M around 0 55.5%

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

      1. *-commutative 55.5%

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

      2. *-commutative 55.5%

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

      3. associate-/l* 56.8%

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

      4. unpow2 56.8%

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

      5. unpow2 56.8%

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

      6. *-commutative 56.8%

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

      7. unpow2 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in l around 0 56.8%

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

      1. unpow2 56.8%

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

      2. times-frac 51.3%

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

      3. unpow2 51.3%

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

      4. associate-/l* 51.5%

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

   9. Simplified 51.5%

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

       1. frac-times 59.7%

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

   11. Applied egg-rr 59.7%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.35 \cdot 10^{-100}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot D}{\frac{d \cdot \ell}{\left(M \cdot M\right) \cdot \frac{h}{d}}} \cdot -0.125\right)\\ \end{array} \]

Alternative 14: 76.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 3.2 \cdot 10^{-151}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + w0 \cdot \left(\left(\frac{D}{\frac{d \cdot d}{h}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{\ell}\right)\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

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 3.2e-151)
   w0
   (+ w0 (* w0 (* (* (/ D (/ (* d d) h)) (* (* M M) (/ D l))) -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 <= 3.2e-151) {
		tmp = w0;
	} else {
		tmp = w0 + (w0 * (((D / ((d * d) / h)) * ((M * M) * (D / l))) * -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 <= 3.2d-151) then
        tmp = w0
    else
        tmp = w0 + (w0 * (((d / ((d_1 * d_1) / h)) * ((m * m) * (d / l))) * (-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 <= 3.2e-151) {
		tmp = w0;
	} else {
		tmp = w0 + (w0 * (((D / ((d * d) / h)) * ((M * M) * (D / l))) * -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 <= 3.2e-151:
		tmp = w0
	else:
		tmp = w0 + (w0 * (((D / ((d * d) / h)) * ((M * M) * (D / l))) * -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 <= 3.2e-151)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(w0 * Float64(Float64(Float64(D / Float64(Float64(d * d) / h)) * Float64(Float64(M * M) * Float64(D / l))) * -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 <= 3.2e-151)
		tmp = w0;
	else
		tmp = w0 + (w0 * (((D / ((d * d) / h)) * ((M * M) * (D / l))) * -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, 3.2e-151], w0, N[(w0 + N[(w0 * N[(N[(N[(D / N[(N[(d * d), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.20000000000000021e-151

   1. Initial program 81.1%

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

      1. times-frac 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in M around 0 71.0%

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

   if 3.20000000000000021e-151 < M

   1. Initial program 80.6%

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

      1. times-frac 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in M around 0 58.4%

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

      1. *-commutative 58.4%

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

      2. *-commutative 58.4%

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

      3. associate-/l* 59.6%

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

      4. unpow2 59.6%

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

      5. unpow2 59.6%

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

      6. *-commutative 59.6%

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

      7. unpow2 59.6%

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

   6. Simplified 59.6%

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

   7. Taylor expanded in l around 0 59.6%

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

      1. unpow2 59.6%

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

      2. times-frac 54.8%

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

      3. unpow2 54.8%

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

      4. associate-/l* 55.0%

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

   9. Simplified 55.0%

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

       1. distribute-rgt-in 55.0%

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

       2. *-un-lft-identity 55.0%

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

       3. times-frac 55.1%

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

       4. associate-/r/ 55.1%

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

   11. Applied egg-rr 55.1%

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

       1. *-commutative 55.1%

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

       2. *-commutative 55.1%

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

       3. unpow2 55.1%

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

       4. associate-/r/ 58.6%

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

       5. unpow2 58.6%

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

       6. associate-*l/ 58.5%

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

   13. Simplified 58.5%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.2 \cdot 10^{-151}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + w0 \cdot \left(\left(\frac{D}{\frac{d \cdot d}{h}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D}{\ell}\right)\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 15: 77.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{-151}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + w0 \cdot \left(\left(\frac{D}{\frac{\ell}{M \cdot M}} \cdot \frac{D}{d \cdot \frac{d}{h}}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

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 4e-151)
   w0
   (+ w0 (* w0 (* (* (/ D (/ l (* M M))) (/ D (* d (/ d h)))) -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 <= 4e-151) {
		tmp = w0;
	} else {
		tmp = w0 + (w0 * (((D / (l / (M * M))) * (D / (d * (d / h)))) * -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 <= 4d-151) then
        tmp = w0
    else
        tmp = w0 + (w0 * (((d / (l / (m * m))) * (d / (d_1 * (d_1 / h)))) * (-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 <= 4e-151) {
		tmp = w0;
	} else {
		tmp = w0 + (w0 * (((D / (l / (M * M))) * (D / (d * (d / h)))) * -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 <= 4e-151:
		tmp = w0
	else:
		tmp = w0 + (w0 * (((D / (l / (M * M))) * (D / (d * (d / h)))) * -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 <= 4e-151)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(w0 * Float64(Float64(Float64(D / Float64(l / Float64(M * M))) * Float64(D / Float64(d * Float64(d / h)))) * -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 <= 4e-151)
		tmp = w0;
	else
		tmp = w0 + (w0 * (((D / (l / (M * M))) * (D / (d * (d / h)))) * -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, 4e-151], w0, N[(w0 + N[(w0 * N[(N[(N[(D / N[(l / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d * N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.9999999999999998e-151

   1. Initial program 81.1%

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

      1. times-frac 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in M around 0 71.0%

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

   if 3.9999999999999998e-151 < M

   1. Initial program 80.6%

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

      1. times-frac 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in M around 0 58.4%

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

      1. *-commutative 58.4%

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

      2. *-commutative 58.4%

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

      3. associate-/l* 59.6%

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

      4. unpow2 59.6%

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

      5. unpow2 59.6%

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

      6. *-commutative 59.6%

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

      7. unpow2 59.6%

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

   6. Simplified 59.6%

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

   7. Taylor expanded in l around 0 59.6%

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

      1. unpow2 59.6%

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

      2. times-frac 54.8%

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

      3. unpow2 54.8%

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

      4. associate-/l* 55.0%

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

   9. Simplified 55.0%

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

       1. distribute-rgt-in 55.0%

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

       2. *-un-lft-identity 55.0%

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

       3. times-frac 55.1%

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

       4. associate-/r/ 55.1%

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

   11. Applied egg-rr 55.1%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{-151}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + w0 \cdot \left(\left(\frac{D}{\frac{\ell}{M \cdot M}} \cdot \frac{D}{d \cdot \frac{d}{h}}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 16: 73.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.02 \cdot 10^{+41}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(w0 \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot M\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 1.02e+41)
   w0
   (* -0.125 (* (* (/ D d) (/ D d)) (* w0 (* (/ h l) (* M M)))))))

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 <= 1.02e+41) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (D / d)) * (w0 * ((h / l) * (M * M))));
	}
	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 <= 1.02d+41) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d / d_1) * (d / d_1)) * (w0 * ((h / l) * (m * m))))
    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 <= 1.02e+41) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (D / d)) * (w0 * ((h / l) * (M * M))));
	}
	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 <= 1.02e+41:
		tmp = w0
	else:
		tmp = -0.125 * (((D / d) * (D / d)) * (w0 * ((h / l) * (M * M))))
	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 <= 1.02e+41)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(w0 * Float64(Float64(h / l) * Float64(M * M)))));
	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 <= 1.02e+41)
		tmp = w0;
	else
		tmp = -0.125 * (((D / d) * (D / d)) * (w0 * ((h / l) * (M * M))));
	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, 1.02e+41], w0, N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(w0 * N[(N[(h / l), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.01999999999999992e41

   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. times-frac 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in M around 0 70.2%

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

   if 1.01999999999999992e41 < M

   1. Initial program 76.9%

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

      1. times-frac 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in M around 0 45.4%

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

      1. *-commutative 45.4%

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

      2. *-commutative 45.4%

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

      3. associate-/l* 45.4%

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

      4. unpow2 45.4%

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

      5. unpow2 45.4%

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

      6. *-commutative 45.4%

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

      7. unpow2 45.4%

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

   6. Simplified 45.4%

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

   7. Taylor expanded in l around 0 45.4%

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

      1. unpow2 45.4%

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

      2. times-frac 37.9%

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

      3. unpow2 37.9%

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

      4. associate-/l* 38.3%

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

   9. Simplified 38.3%

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

   10. Taylor expanded in D around inf 26.7%

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

       1. *-commutative 26.7%

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

       2. times-frac 26.4%

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

       3. unpow2 26.4%

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

       4. unpow2 26.4%

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

       5. unpow2 26.4%

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

       6. associate-*r* 26.6%

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

       7. associate-*r/ 26.6%

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

       8. associate-*r* 26.4%

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

       9. unpow2 26.4%

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

       10. associate-*r/ 26.5%

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

       11. unpow2 26.5%

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

   12. Simplified 26.5%

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

       1. times-frac 26.8%

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

   14. Applied egg-rr 26.8%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.02 \cdot 10^{+41}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(w0 \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot M\right)\right)\right)\right)\\ \end{array} \]

Alternative 17: 69.0% 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 81.0%

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

   1. times-frac 81.1%

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

3. Simplified 81.1%

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

4. Taylor expanded in M around 0 66.8%

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

5. Final simplification 66.8%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023238 
(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))))))