Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 86.7%

Time: 21.5s

Alternatives: 6

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: 81.0% 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: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D\_m}{2 \cdot \frac{d\_m}{M\_m}}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left({\left(e^{0.16666666666666666}\right)}^{\left(\log \left(\frac{-0.25 \cdot \left(h \cdot {\left(M\_m \cdot D\_m\right)}^{2}\right)}{\ell}\right) + -2 \cdot \log d\_m\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ (* M_m D_m) (* 2.0 d_m)) 5e+116)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (/ D_m (* 2.0 (/ d_m M_m))) 2.0)) l))))
   (*
    w0
    (pow
     (pow
      (exp 0.16666666666666666)
      (+ (log (/ (* -0.25 (* h (pow (* M_m D_m) 2.0))) l)) (* -2.0 (log d_m))))
     3.0))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+116) {
		tmp = w0 * sqrt((1.0 - ((h * pow((D_m / (2.0 * (d_m / M_m))), 2.0)) / l)));
	} else {
		tmp = w0 * pow(pow(exp(0.16666666666666666), (log(((-0.25 * (h * pow((M_m * D_m), 2.0))) / l)) + (-2.0 * log(d_m)))), 3.0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((m_m * d_m) / (2.0d0 * d_m_1)) <= 5d+116) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((d_m / (2.0d0 * (d_m_1 / m_m))) ** 2.0d0)) / l)))
    else
        tmp = w0 * ((exp(0.16666666666666666d0) ** (log((((-0.25d0) * (h * ((m_m * d_m) ** 2.0d0))) / l)) + ((-2.0d0) * log(d_m_1)))) ** 3.0d0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+116) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((D_m / (2.0 * (d_m / M_m))), 2.0)) / l)));
	} else {
		tmp = w0 * Math.pow(Math.pow(Math.exp(0.16666666666666666), (Math.log(((-0.25 * (h * Math.pow((M_m * D_m), 2.0))) / l)) + (-2.0 * Math.log(d_m)))), 3.0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if ((M_m * D_m) / (2.0 * d_m)) <= 5e+116:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((D_m / (2.0 * (d_m / M_m))), 2.0)) / l)))
	else:
		tmp = w0 * math.pow(math.pow(math.exp(0.16666666666666666), (math.log(((-0.25 * (h * math.pow((M_m * D_m), 2.0))) / l)) + (-2.0 * math.log(d_m)))), 3.0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) <= 5e+116)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(D_m / Float64(2.0 * Float64(d_m / M_m))) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * ((exp(0.16666666666666666) ^ Float64(log(Float64(Float64(-0.25 * Float64(h * (Float64(M_m * D_m) ^ 2.0))) / l)) + Float64(-2.0 * log(d_m)))) ^ 3.0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+116)
		tmp = w0 * sqrt((1.0 - ((h * ((D_m / (2.0 * (d_m / M_m))) ^ 2.0)) / l)));
	else
		tmp = w0 * ((exp(0.16666666666666666) ^ (log(((-0.25 * (h * ((M_m * D_m) ^ 2.0))) / l)) + (-2.0 * log(d_m)))) ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 5e+116], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(D$95$m / N[(2.0 * N[(d$95$m / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Power[N[Power[N[Exp[0.16666666666666666], $MachinePrecision], N[(N[Log[N[(N[(-0.25 * N[(h * N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 5.00000000000000025e116

   1. Initial program 84.5%

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

   3. Simplified 84.1%

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

      1. unpow2 84.1%

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

      2. unpow2 84.1%

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

      3. associate-/r* 84.1%

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

      4. associate-*r/ 84.5%

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

      5. *-commutative 84.5%

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

      6. associate-*r/ 89.5%

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

      7. *-commutative 89.5%

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

      8. associate-*l/ 89.1%

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

      9. associate-/r* 89.1%

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

      10. clear-num 89.1%

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

      11. associate-*l/ 89.5%

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

      12. *-un-lft-identity 89.5%

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

      13. associate-/r/ 89.5%

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

   6. Applied egg-rr 89.5%

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

   if 5.00000000000000025e116 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 48.1%

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

   3. Simplified 50.6%

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

   5. Taylor expanded in D around 0 47.9%

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

      1. associate-/l* 47.9%

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

      2. associate-/l* 50.5%

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

   7. Simplified 50.5%

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

      1. add-cube-cbrt 50.5%

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

      2. pow3 50.5%

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

   9. Applied egg-rr 47.7%

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

   10. Taylor expanded in d around 0 30.4%

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

       1. exp-prod 30.2%

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

       2. associate-*r* 30.2%

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

       3. unpow2 30.2%

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

       4. unpow2 30.2%

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

       5. swap-sqr 30.2%

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

       6. unpow2 30.2%

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

       7. associate-*r/ 30.2%

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

       8. *-commutative 30.2%

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

   12. Simplified 30.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+116}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left({\left(e^{0.16666666666666666}\right)}^{\left(\log \left(\frac{-0.25 \cdot \left(h \cdot {\left(M \cdot D\right)}^{2}\right)}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq 0:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) 0.0)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* D_m (/ (/ M_m 2.0) d_m)) 2.0)))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= 0.0) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((D_m * ((M_m / 2.0) / d_m)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)) <= 0.0d0) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d_m * ((m_m / 2.0d0) / d_m_1)) ** 2.0d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= 0.0) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((D_m * ((M_m / 2.0) / d_m)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= 0.0:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((D_m * ((M_m / 2.0) / d_m)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 0.0], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -0.0

   1. Initial program 84.0%

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

   3. Simplified 84.4%

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

   if -0.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 43.3%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in D around 0 74.7%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 1.0× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (* w0 (sqrt (- 1.0 (/ (* h (pow (/ D_m (* 2.0 (/ d_m M_m))) 2.0)) l)))))

C

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

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0 * sqrt((1.0d0 - ((h * ((d_m / (2.0d0 * (d_m_1 / m_m))) ** 2.0d0)) / l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(D$95$m / N[(2.0 * N[(d$95$m / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

3. Simplified 79.2%

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

   1. unpow2 79.2%

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

   2. unpow2 79.2%

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

   3. associate-/r* 79.2%

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

   4. associate-*r/ 79.3%

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

   5. *-commutative 79.3%

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

   6. associate-*r/ 84.0%

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

   7. *-commutative 84.0%

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

   8. associate-*l/ 84.0%

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

   9. associate-/r* 84.0%

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

   10. clear-num 84.0%

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

   11. associate-*l/ 84.3%

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

   12. *-un-lft-identity 84.3%

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

   13. associate-/r/ 84.3%

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

6. Applied egg-rr 84.3%

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

7. Final simplification 84.3%

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

Alternative 4: 72.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 3.7 \cdot 10^{+143}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(h \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)\right)\right) \cdot \frac{\frac{w0}{{d\_m}^{2}}}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= D_m 3.7e+143)
   w0
   (+
    w0
    (*
     -0.125
     (* (* h (* (* M_m D_m) (* M_m D_m))) (/ (/ w0 (pow d_m 2.0)) l))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 3.7e+143) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((h * ((M_m * D_m) * (M_m * D_m))) * ((w0 / pow(d_m, 2.0)) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (d_m <= 3.7d+143) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((h * ((m_m * d_m) * (m_m * d_m))) * ((w0 / (d_m_1 ** 2.0d0)) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 3.7e+143) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((h * ((M_m * D_m) * (M_m * D_m))) * ((w0 / Math.pow(d_m, 2.0)) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if D_m <= 3.7e+143:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((h * ((M_m * D_m) * (M_m * D_m))) * ((w0 / math.pow(d_m, 2.0)) / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (D_m <= 3.7e+143)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(h * Float64(Float64(M_m * D_m) * Float64(M_m * D_m))) * Float64(Float64(w0 / (d_m ^ 2.0)) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (D_m <= 3.7e+143)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((h * ((M_m * D_m) * (M_m * D_m))) * ((w0 / (d_m ^ 2.0)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[D$95$m, 3.7e+143], w0, N[(w0 + N[(-0.125 * N[(N[(h * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 / N[Power[d$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 3.7000000000000002e143

   1. Initial program 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in D around 0 70.1%

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

   if 3.7000000000000002e143 < D

   1. Initial program 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in D around 0 24.1%

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

      1. associate-/l* 24.1%

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

      2. associate-/l* 24.2%

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

   7. Simplified 24.2%

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

      1. pow1 24.2%

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

      2. associate-*r* 24.2%

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

      3. pow-prod-down 60.9%

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

      4. associate-/l* 60.9%

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

      5. *-un-lft-identity 60.9%

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

      6. times-frac 57.5%

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

      7. pow-flip 57.5%

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

      8. metadata-eval 57.5%

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

   9. Applied egg-rr 57.5%

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

       1. unpow1 57.5%

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

       2. associate-*r* 57.6%

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

   11. Simplified 57.6%

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

       1. unpow2 57.6%

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

   13. Applied egg-rr 57.6%

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

   14. Taylor expanded in d around 0 61.1%

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

       1. associate-/r* 60.9%

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

   16. Simplified 60.9%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 3.7 \cdot 10^{+143}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right) \cdot \frac{\frac{w0}{{d}^{2}}}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 7.5 \cdot 10^{+144}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(h \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)\right)\right) \cdot \left({d\_m}^{-2} \cdot \frac{w0}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= D_m 7.5e+144)
   w0
   (+
    w0
    (*
     -0.125
     (* (* h (* (* M_m D_m) (* M_m D_m))) (* (pow d_m -2.0) (/ w0 l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 7.5e+144) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((h * ((M_m * D_m) * (M_m * D_m))) * (pow(d_m, -2.0) * (w0 / l))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (d_m <= 7.5d+144) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((h * ((m_m * d_m) * (m_m * d_m))) * ((d_m_1 ** (-2.0d0)) * (w0 / l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 7.5e+144) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((h * ((M_m * D_m) * (M_m * D_m))) * (Math.pow(d_m, -2.0) * (w0 / l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if D_m <= 7.5e+144:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((h * ((M_m * D_m) * (M_m * D_m))) * (math.pow(d_m, -2.0) * (w0 / l))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (D_m <= 7.5e+144)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(h * Float64(Float64(M_m * D_m) * Float64(M_m * D_m))) * Float64((d_m ^ -2.0) * Float64(w0 / l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (D_m <= 7.5e+144)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((h * ((M_m * D_m) * (M_m * D_m))) * ((d_m ^ -2.0) * (w0 / l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[D$95$m, 7.5e+144], w0, N[(w0 + N[(-0.125 * N[(N[(h * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[d$95$m, -2.0], $MachinePrecision] * N[(w0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 7.5000000000000006e144

   1. Initial program 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in D around 0 70.1%

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

   if 7.5000000000000006e144 < D

   1. Initial program 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in D around 0 24.1%

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

      1. associate-/l* 24.1%

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

      2. associate-/l* 24.2%

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

   7. Simplified 24.2%

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

      1. pow1 24.2%

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

      2. associate-*r* 24.2%

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

      3. pow-prod-down 60.9%

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

      4. associate-/l* 60.9%

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

      5. *-un-lft-identity 60.9%

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

      6. times-frac 57.5%

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

      7. pow-flip 57.5%

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

      8. metadata-eval 57.5%

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

   9. Applied egg-rr 57.5%

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

       1. unpow1 57.5%

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

       2. associate-*r* 57.6%

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

   11. Simplified 57.6%

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

       1. unpow2 57.6%

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

   13. Applied egg-rr 57.6%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 7.5 \cdot 10^{+144}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right) \cdot \left({d}^{-2} \cdot \frac{w0}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 216.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ w0 \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m) :precision binary64 w0)

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0
end function

Java

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

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	return w0

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := w0

Derivation

1. Initial program 79.3%

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

3. Simplified 79.2%

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

5. Taylor expanded in D around 0 66.5%

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

Reproduce

herbie shell --seed 2024101 
(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))))))