Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.7% → 92.5%

Time: 22.0s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+88}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\left(-2 \cdot \log d\_m + \left(\log \left(\frac{h}{\ell} \cdot -0.25\right) + 2 \cdot \log M\_m\right)\right) + -2 \cdot \log \left(\frac{1}{D\_m}\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(0.5 \cdot \frac{M\_m \cdot D\_m}{d\_m}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

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)) -5e+88)
   (pow
    (*
     (cbrt w0)
     (exp
      (*
       0.16666666666666666
       (+
        (+ (* -2.0 (log d_m)) (+ (log (* (/ h l) -0.25)) (* 2.0 (log M_m))))
        (* -2.0 (log (/ 1.0 D_m)))))))
    3.0)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* 0.5 (/ (* M_m D_m) 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 ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -5e+88) {
		tmp = pow((cbrt(w0) * exp((0.16666666666666666 * (((-2.0 * log(d_m)) + (log(((h / l) * -0.25)) + (2.0 * log(M_m)))) + (-2.0 * log((1.0 / D_m))))))), 3.0);
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow((0.5 * ((M_m * D_m) / d_m)), 2.0)) / l)));
	}
	return tmp;
}

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)) <= -5e+88) {
		tmp = Math.pow((Math.cbrt(w0) * Math.exp((0.16666666666666666 * (((-2.0 * Math.log(d_m)) + (Math.log(((h / l) * -0.25)) + (2.0 * Math.log(M_m)))) + (-2.0 * Math.log((1.0 / D_m))))))), 3.0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((0.5 * ((M_m * D_m) / 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 (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -5e+88)
		tmp = Float64(cbrt(w0) * exp(Float64(0.16666666666666666 * Float64(Float64(Float64(-2.0 * log(d_m)) + Float64(log(Float64(Float64(h / l) * -0.25)) + Float64(2.0 * log(M_m)))) + Float64(-2.0 * log(Float64(1.0 / D_m))))))) ^ 3.0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(0.5 * Float64(Float64(M_m * D_m) / d_m)) ^ 2.0)) / l))));
	end
	return 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], -5e+88], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[(N[(N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision] + N[(N[Log[N[(N[(h / l), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] + N[(2.0 * N[Log[M$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[Log[N[(1.0 / D$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d$95$m), $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 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.99999999999999997e88

   1. Initial program 65.5%

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

   3. Simplified 67.9%

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

      1. add-cube-cbrt 67.6%

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

      2. pow3 67.6%

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

   6. Applied egg-rr 67.6%

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

   7. Taylor expanded in D around inf 23.7%

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

   8. Taylor expanded in d around 0 8.1%

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

      1. +-commutative 8.1%

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

      2. distribute-lft-neg-in 8.1%

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

      3. metadata-eval 8.1%

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

      4. associate-/l* 9.2%

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

   10. Simplified 9.2%

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

   11. Taylor expanded in M around 0 6.9%

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

   if -4.99999999999999997e88 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.5%

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

   3. Simplified 90.9%

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

      1. associate-*r/ 96.7%

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

      2. add-sqr-sqrt 96.7%

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

      3. pow2 96.7%

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

      4. sqrt-pow1 96.7%

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

      5. metadata-eval 96.7%

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

      6. pow1 96.7%

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

      7. *-un-lft-identity 96.7%

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

      8. times-frac 96.7%

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

      9. metadata-eval 96.7%

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in M around 0 97.2%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+88}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\left(-2 \cdot \log d + \left(\log \left(\frac{h}{\ell} \cdot -0.25\right) + 2 \cdot \log M\right)\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+88}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\log \left(\frac{h}{\ell} \cdot -0.25\right) + \left(-2 \cdot \log d\_m + \left(2 \cdot \log M\_m + -2 \cdot \log \left(\frac{1}{D\_m}\right)\right)\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(0.5 \cdot \frac{M\_m \cdot D\_m}{d\_m}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

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)) -5e+88)
   (pow
    (*
     (cbrt w0)
     (exp
      (*
       0.16666666666666666
       (+
        (log (* (/ h l) -0.25))
        (+
         (* -2.0 (log d_m))
         (+ (* 2.0 (log M_m)) (* -2.0 (log (/ 1.0 D_m)))))))))
    3.0)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* 0.5 (/ (* M_m D_m) 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 ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -5e+88) {
		tmp = pow((cbrt(w0) * exp((0.16666666666666666 * (log(((h / l) * -0.25)) + ((-2.0 * log(d_m)) + ((2.0 * log(M_m)) + (-2.0 * log((1.0 / D_m))))))))), 3.0);
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow((0.5 * ((M_m * D_m) / d_m)), 2.0)) / l)));
	}
	return tmp;
}

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)) <= -5e+88) {
		tmp = Math.pow((Math.cbrt(w0) * Math.exp((0.16666666666666666 * (Math.log(((h / l) * -0.25)) + ((-2.0 * Math.log(d_m)) + ((2.0 * Math.log(M_m)) + (-2.0 * Math.log((1.0 / D_m))))))))), 3.0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((0.5 * ((M_m * D_m) / 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 (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -5e+88)
		tmp = Float64(cbrt(w0) * exp(Float64(0.16666666666666666 * Float64(log(Float64(Float64(h / l) * -0.25)) + Float64(Float64(-2.0 * log(d_m)) + Float64(Float64(2.0 * log(M_m)) + Float64(-2.0 * log(Float64(1.0 / D_m))))))))) ^ 3.0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(0.5 * Float64(Float64(M_m * D_m) / d_m)) ^ 2.0)) / l))));
	end
	return 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], -5e+88], 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[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[Log[M$95$m], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[Log[N[(1.0 / D$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d$95$m), $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 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.99999999999999997e88

   1. Initial program 65.5%

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

   3. Simplified 67.9%

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

      1. add-cube-cbrt 67.6%

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

      2. pow3 67.6%

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

   6. Applied egg-rr 67.6%

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

   7. Taylor expanded in D around inf 23.7%

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

   8. Taylor expanded in d around 0 8.1%

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

      1. +-commutative 8.1%

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

      2. distribute-lft-neg-in 8.1%

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

      3. metadata-eval 8.1%

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

      4. associate-/l* 9.2%

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

   10. Simplified 9.2%

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

   11. Taylor expanded in M around 0 6.8%

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

   if -4.99999999999999997e88 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 91.5%

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

   3. Simplified 90.9%

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

      1. associate-*r/ 96.7%

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

      2. add-sqr-sqrt 96.7%

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

      3. pow2 96.7%

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

      4. sqrt-pow1 96.7%

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

      5. metadata-eval 96.7%

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

      6. pow1 96.7%

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

      7. *-un-lft-identity 96.7%

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

      8. times-frac 96.7%

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

      9. metadata-eval 96.7%

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in M around 0 97.2%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+88}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\log \left(\frac{h}{\ell} \cdot -0.25\right) + \left(-2 \cdot \log d + \left(2 \cdot \log M + -2 \cdot \log \left(\frac{1}{D}\right)\right)\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(-2 \cdot \log \left(\frac{1}{D\_m}\right) + \left(-2 \cdot \log d\_m + \log \left(-0.25 \cdot \left(\frac{h}{\ell} \cdot {M\_m}^{2}\right)\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l))))
   (if (<= t_0 (- INFINITY))
     (pow
      (*
       (cbrt w0)
       (exp
        (*
         0.16666666666666666
         (+
          (* -2.0 (log (/ 1.0 D_m)))
          (+ (* -2.0 (log d_m)) (log (* -0.25 (* (/ h l) (pow M_m 2.0)))))))))
      3.0)
     (if (<= t_0 0.02) (* w0 (sqrt (- 1.0 t_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 t_0 = pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = pow((cbrt(w0) * exp((0.16666666666666666 * ((-2.0 * log((1.0 / D_m))) + ((-2.0 * log(d_m)) + log((-0.25 * ((h / l) * pow(M_m, 2.0))))))))), 3.0);
	} else if (t_0 <= 0.02) {
		tmp = w0 * sqrt((1.0 - t_0));
	} else {
		tmp = w0;
	}
	return tmp;
}

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 t_0 = Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.pow((Math.cbrt(w0) * Math.exp((0.16666666666666666 * ((-2.0 * Math.log((1.0 / D_m))) + ((-2.0 * Math.log(d_m)) + Math.log((-0.25 * ((h / l) * Math.pow(M_m, 2.0))))))))), 3.0);
	} else if (t_0 <= 0.02) {
		tmp = w0 * Math.sqrt((1.0 - t_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)
	t_0 = Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(cbrt(w0) * exp(Float64(0.16666666666666666 * Float64(Float64(-2.0 * log(Float64(1.0 / D_m))) + Float64(Float64(-2.0 * log(d_m)) + log(Float64(-0.25 * Float64(Float64(h / l) * (M_m ^ 2.0))))))))) ^ 3.0;
	elseif (t_0 <= 0.02)
		tmp = Float64(w0 * sqrt(Float64(1.0 - t_0)));
	else
		tmp = w0;
	end
	return 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_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, (-Infinity)], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[(N[(-2.0 * N[Log[N[(1.0 / D$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision] + N[Log[N[(-0.25 * N[(N[(h / l), $MachinePrecision] * N[Power[M$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[t$95$0, 0.02], N[(w0 * N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -inf.0

   1. Initial program 55.7%

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

   3. Simplified 58.8%

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

      1. add-cube-cbrt 58.7%

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

      2. pow3 58.7%

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

   6. Applied egg-rr 58.7%

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

   7. Taylor expanded in D around inf 25.9%

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

   8. Taylor expanded in d around 0 8.8%

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

      1. +-commutative 8.8%

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

      2. distribute-lft-neg-in 8.8%

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

      3. metadata-eval 8.8%

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

      4. associate-/l* 8.8%

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

   10. Simplified 8.8%

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

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 0.0200000000000000004

   1. Initial program 99.9%

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

   if 0.0200000000000000004 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in M around 0 75.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(-2 \cdot \log \left(\frac{1}{D}\right) + \left(-2 \cdot \log d + \log \left(-0.25 \cdot \left(\frac{h}{\ell} \cdot {M}^{2}\right)\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.02:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq 10^{+304}:\\ \;\;\;\;w0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(\left(M\_m \cdot 0.5\right) \cdot \frac{D\_m}{d\_m}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)))))
   (if (<= t_0 1e+304)
     (* w0 (sqrt t_0))
     (* w0 (sqrt (- 1.0 (* h (/ (pow (* (* M_m 0.5) (/ D_m 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 t_0 = 1.0 - (pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 1e+304) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - (h * (pow(((M_m * 0.5) * (D_m / 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l))
    if (t_0 <= 1d+304) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - (h * ((((m_m * 0.5d0) * (d_m / 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 t_0 = 1.0 - (Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 1e+304) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow(((M_m * 0.5) * (D_m / 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):
	t_0 = 1.0 - (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 1e+304:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - (h * (math.pow(((M_m * 0.5) * (D_m / 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)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 1e+304)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(Float64(M_m * 0.5) * Float64(D_m / 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)
	t_0 = 1.0 - ((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 1e+304)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - (h * ((((M_m * 0.5) * (D_m / 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_] := Block[{t$95$0 = N[(1.0 - 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]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+304], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 9.9999999999999994e303

   1. Initial program 99.9%

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

   if 9.9999999999999994e303 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 44.8%

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

   3. Simplified 47.3%

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

      1. associate-*r/ 61.7%

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

      2. add-sqr-sqrt 61.7%

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

      3. pow2 61.7%

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

      4. sqrt-pow1 61.7%

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

      5. metadata-eval 61.7%

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

      6. pow1 61.7%

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

      7. *-un-lft-identity 61.7%

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

      8. times-frac 61.7%

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

      9. metadata-eval 61.7%

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

   6. Applied egg-rr 61.7%

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

   8. Applied egg-rr 47.3%

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

      1. *-lft-identity 47.3%

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

      2. *-commutative 47.3%

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

      3. associate-*l/ 61.7%

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

      4. associate-*r/ 61.7%

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

   10. Simplified 61.7%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{+304}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+294}:\\ \;\;\;\;w0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d\_m}\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)))))
   (if (<= t_0 5e+294)
     (* w0 (sqrt t_0))
     (* w0 (sqrt (- 1.0 (/ (* h (pow (* M_m (* 0.5 (/ D_m 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 t_0 = 1.0 - (pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+294) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow((M_m * (0.5 * (D_m / 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l))
    if (t_0 <= 5d+294) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - ((h * ((m_m * (0.5d0 * (d_m / 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 t_0 = 1.0 - (Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+294) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M_m * (0.5 * (D_m / 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):
	t_0 = 1.0 - (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 5e+294:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((M_m * (0.5 * (D_m / 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)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 5e+294)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / 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)
	t_0 = 1.0 - ((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 5e+294)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - ((h * ((M_m * (0.5 * (D_m / 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_] := Block[{t$95$0 = N[(1.0 - 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]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+294], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 4.9999999999999999e294

   1. Initial program 99.9%

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

   if 4.9999999999999999e294 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 46.2%

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

   3. Simplified 48.6%

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

      1. associate-*r/ 62.7%

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

      2. add-sqr-sqrt 62.7%

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

      3. pow2 62.7%

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

      4. sqrt-pow1 62.7%

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

      5. metadata-eval 62.7%

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

      6. pow1 62.7%

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

      7. *-un-lft-identity 62.7%

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

      8. times-frac 62.7%

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

      9. metadata-eval 62.7%

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

   6. Applied egg-rr 62.7%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+294}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \left(M\_m \cdot 0.5\right) \cdot \frac{D\_m}{d\_m}\\ t_1 := t\_0 \cdot \sqrt{h}\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{t\_0}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_1 \cdot \frac{t\_1}{\ell}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (* M_m 0.5) (/ D_m d_m))) (t_1 (* t_0 (sqrt h))))
   (if (<= h -5e-310)
     (* w0 (sqrt (- 1.0 (* h (/ (pow t_0 2.0) l)))))
     (* w0 (sqrt (- 1.0 (* t_1 (/ t_1 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 t_0 = (M_m * 0.5) * (D_m / d_m);
	double t_1 = t_0 * sqrt(h);
	double tmp;
	if (h <= -5e-310) {
		tmp = w0 * sqrt((1.0 - (h * (pow(t_0, 2.0) / l))));
	} else {
		tmp = w0 * sqrt((1.0 - (t_1 * (t_1 / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (m_m * 0.5d0) * (d_m / d_m_1)
    t_1 = t_0 * sqrt(h)
    if (h <= (-5d-310)) then
        tmp = w0 * sqrt((1.0d0 - (h * ((t_0 ** 2.0d0) / l))))
    else
        tmp = w0 * sqrt((1.0d0 - (t_1 * (t_1 / 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 t_0 = (M_m * 0.5) * (D_m / d_m);
	double t_1 = t_0 * Math.sqrt(h);
	double tmp;
	if (h <= -5e-310) {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow(t_0, 2.0) / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_1 * (t_1 / 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):
	t_0 = (M_m * 0.5) * (D_m / d_m)
	t_1 = t_0 * math.sqrt(h)
	tmp = 0
	if h <= -5e-310:
		tmp = w0 * math.sqrt((1.0 - (h * (math.pow(t_0, 2.0) / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - (t_1 * (t_1 / 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)
	t_0 = Float64(Float64(M_m * 0.5) * Float64(D_m / d_m))
	t_1 = Float64(t_0 * sqrt(h))
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((t_0 ^ 2.0) / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_1 * Float64(t_1 / 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)
	t_0 = (M_m * 0.5) * (D_m / d_m);
	t_1 = t_0 * sqrt(h);
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = w0 * sqrt((1.0 - (h * ((t_0 ^ 2.0) / l))));
	else
		tmp = w0 * sqrt((1.0 - (t_1 * (t_1 / 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_] := Block[{t$95$0 = N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$1 * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 87.4%

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

   3. Simplified 87.4%

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

      1. associate-*r/ 89.0%

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

      2. add-sqr-sqrt 89.0%

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

      3. pow2 89.0%

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

      4. sqrt-pow1 89.0%

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

      5. metadata-eval 89.0%

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

      6. pow1 89.0%

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

      7. *-un-lft-identity 89.0%

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

      8. times-frac 89.0%

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

      9. metadata-eval 89.0%

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

   6. Applied egg-rr 89.0%

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

   8. Applied egg-rr 87.4%

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

      1. *-lft-identity 87.4%

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

      2. *-commutative 87.4%

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

      3. associate-*l/ 89.0%

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

      4. associate-*r/ 90.5%

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

   10. Simplified 90.5%

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

   if -4.999999999999985e-310 < h

   1. Initial program 79.2%

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

   3. Simplified 79.9%

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

      1. associate-*r/ 83.2%

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

      2. add-sqr-sqrt 83.2%

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

      3. pow2 83.2%

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

      4. sqrt-pow1 83.2%

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

      5. metadata-eval 83.2%

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

      6. pow1 83.2%

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

      7. *-un-lft-identity 83.2%

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

      8. times-frac 83.2%

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

      9. metadata-eval 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. add-sqr-sqrt 83.2%

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

      2. *-un-lft-identity 83.2%

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

      3. times-frac 83.2%

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

      4. sqrt-prod 83.2%

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

      5. sqrt-pow1 73.6%

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

      6. metadata-eval 73.6%

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

      7. pow1 73.6%

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

      8. associate-*r* 73.6%

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

      9. sqrt-prod 74.3%

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

      10. sqrt-pow1 88.5%

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

      11. metadata-eval 88.5%

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

      12. pow1 88.5%

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

      13. associate-*r* 88.5%

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

   8. Applied egg-rr 88.5%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{h}\right) \cdot \frac{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.2% accurate, 1.0× speedup

Math

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

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 (* (* M_m 0.5) (/ D_m 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) {
	return w0 * sqrt((1.0 - (h * (pow(((M_m * 0.5) * (D_m / d_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 * ((((m_m * 0.5d0) * (d_m / d_m_1)) ** 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(((M_m * 0.5) * (D_m / d_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(((M_m * 0.5) * (D_m / d_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(h * Float64((Float64(Float64(M_m * 0.5) * Float64(D_m / d_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 * ((((M_m * 0.5) * (D_m / d_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[(h * N[(N[Power[N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.4%

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

3. Simplified 83.7%

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

   1. associate-*r/ 86.2%

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

   2. add-sqr-sqrt 86.2%

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

   3. pow2 86.2%

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

   4. sqrt-pow1 86.2%

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

   5. metadata-eval 86.2%

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

   6. pow1 86.2%

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

   7. *-un-lft-identity 86.2%

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

   8. times-frac 86.2%

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

   9. metadata-eval 86.2%

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

6. Applied egg-rr 86.2%

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

8. Applied egg-rr 83.7%

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

   1. *-lft-identity 83.7%

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

   2. *-commutative 83.7%

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

   3. associate-*l/ 86.2%

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

   4. associate-*r/ 87.3%

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

10. Simplified 87.3%

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

11. Final simplification 87.3%

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

Alternative 8: 68.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 7.2 \cdot 10^{-42}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{w0}\right)\\ \end{array} \end{array} \]

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 (<= M_m 7.2e-42) w0 (log (exp 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 (M_m <= 7.2e-42) {
		tmp = w0;
	} else {
		tmp = log(exp(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 <= 7.2d-42) then
        tmp = w0
    else
        tmp = log(exp(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 (M_m <= 7.2e-42) {
		tmp = w0;
	} else {
		tmp = Math.log(Math.exp(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 M_m <= 7.2e-42:
		tmp = w0
	else:
		tmp = math.log(math.exp(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 (M_m <= 7.2e-42)
		tmp = w0;
	else
		tmp = log(exp(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 <= 7.2e-42)
		tmp = w0;
	else
		tmp = log(exp(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[M$95$m, 7.2e-42], w0, N[Log[N[Exp[w0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.2000000000000004e-42

   1. Initial program 85.5%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in M around 0 73.1%

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

   if 7.2000000000000004e-42 < M

   1. Initial program 76.6%

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

   3. Simplified 76.6%

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

      1. add-sqr-sqrt 39.1%

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

      2. sqrt-unprod 31.0%

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

      3. *-commutative 31.0%

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

      4. *-commutative 31.0%

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

      5. swap-sqr 29.2%

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

   6. Applied egg-rr 29.2%

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

      1. *-commutative 29.2%

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

      2. associate-*r* 29.2%

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

      3. associate-*r/ 29.2%

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

      4. *-commutative 29.2%

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

      5. associate-*r/ 27.6%

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

      6. associate-*r/ 27.5%

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

   8. Simplified 27.5%

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

   9. Taylor expanded in h around 0 25.1%

      \[\leadsto \sqrt{\color{blue}{{w0}^{2}}} \]
   10. Step-by-step derivation

       1. sqrt-pow1 50.8%

          \[\leadsto \color{blue}{{w0}^{\left(\frac{2}{2}\right)}} \]

       2. metadata-eval 50.8%

          \[\leadsto {w0}^{\color{blue}{1}} \]

       3. pow1 50.8%

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

       4. add-log-exp 27.4%

          \[\leadsto \color{blue}{\log \left(e^{w0}\right)} \]

   11. Applied egg-rr 27.4%

       \[\leadsto \color{blue}{\log \left(e^{w0}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.2 \cdot 10^{-42}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{w0}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 10^{-63}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(w0\right)\right)\\ \end{array} \end{array} \]

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 (<= M_m 1e-63) w0 (log1p (expm1 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 (M_m <= 1e-63) {
		tmp = w0;
	} else {
		tmp = log1p(expm1(w0));
	}
	return tmp;
}

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 <= 1e-63) {
		tmp = w0;
	} else {
		tmp = Math.log1p(Math.expm1(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 M_m <= 1e-63:
		tmp = w0
	else:
		tmp = math.log1p(math.expm1(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 (M_m <= 1e-63)
		tmp = w0;
	else
		tmp = log1p(expm1(w0));
	end
	return 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[M$95$m, 1e-63], w0, N[Log[1 + N[(Exp[w0] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.00000000000000007e-63

   1. Initial program 85.0%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in M around 0 72.7%

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

   if 1.00000000000000007e-63 < M

   1. Initial program 78.7%

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

   3. Simplified 78.7%

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

      1. add-sqr-sqrt 40.1%

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

      2. sqrt-unprod 28.7%

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

      3. *-commutative 28.7%

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

      4. *-commutative 28.7%

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

      5. swap-sqr 27.1%

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

   6. Applied egg-rr 27.1%

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

      1. *-commutative 27.1%

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

      2. associate-*r* 27.1%

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

      3. associate-*r/ 27.1%

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

      4. *-commutative 27.1%

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

      5. associate-*r/ 25.5%

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

      6. associate-*r/ 25.5%

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

   8. Simplified 25.5%

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

   9. Taylor expanded in h around 0 23.3%

      \[\leadsto \sqrt{\color{blue}{{w0}^{2}}} \]
   10. Step-by-step derivation

       1. sqrt-pow1 53.8%

          \[\leadsto \color{blue}{{w0}^{\left(\frac{2}{2}\right)}} \]

       2. metadata-eval 53.8%

          \[\leadsto {w0}^{\color{blue}{1}} \]

       3. pow1 53.8%

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

       4. add-exp-log 21.4%

          \[\leadsto \color{blue}{e^{\log w0}} \]

   11. Applied egg-rr 21.4%

       \[\leadsto \color{blue}{e^{\log w0}} \]
   12. Step-by-step derivation

       1. rem-exp-log 53.8%

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

       2. log1p-expm1-u 54.2%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(w0\right)\right)} \]

   13. Applied egg-rr 54.2%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(w0\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 10^{-63}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(w0\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 7.2 \cdot 10^{-30}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(w0 + 1\right)}^{2} + -1}{2 + w0}\\ \end{array} \end{array} \]

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 (<= M_m 7.2e-30) w0 (/ (+ (pow (+ w0 1.0) 2.0) -1.0) (+ 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 (M_m <= 7.2e-30) {
		tmp = w0;
	} else {
		tmp = (pow((w0 + 1.0), 2.0) + -1.0) / (2.0 + 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 <= 7.2d-30) then
        tmp = w0
    else
        tmp = (((w0 + 1.0d0) ** 2.0d0) + (-1.0d0)) / (2.0d0 + 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 (M_m <= 7.2e-30) {
		tmp = w0;
	} else {
		tmp = (Math.pow((w0 + 1.0), 2.0) + -1.0) / (2.0 + 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 M_m <= 7.2e-30:
		tmp = w0
	else:
		tmp = (math.pow((w0 + 1.0), 2.0) + -1.0) / (2.0 + 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 (M_m <= 7.2e-30)
		tmp = w0;
	else
		tmp = Float64(Float64((Float64(w0 + 1.0) ^ 2.0) + -1.0) / Float64(2.0 + 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 <= 7.2e-30)
		tmp = w0;
	else
		tmp = (((w0 + 1.0) ^ 2.0) + -1.0) / (2.0 + 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[M$95$m, 7.2e-30], w0, N[(N[(N[Power[N[(w0 + 1.0), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision] / N[(2.0 + w0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.2000000000000006e-30

   1. Initial program 85.5%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in M around 0 73.1%

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

   if 7.2000000000000006e-30 < M

   1. Initial program 76.6%

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

   3. Simplified 76.6%

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

      1. add-sqr-sqrt 39.1%

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

      2. sqrt-unprod 31.0%

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

      3. *-commutative 31.0%

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

      4. *-commutative 31.0%

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

      5. swap-sqr 29.2%

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

   6. Applied egg-rr 29.2%

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

      1. *-commutative 29.2%

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

      2. associate-*r* 29.2%

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

      3. associate-*r/ 29.2%

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

      4. *-commutative 29.2%

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

      5. associate-*r/ 27.6%

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

      6. associate-*r/ 27.5%

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

   8. Simplified 27.5%

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

   9. Taylor expanded in h around 0 25.1%

      \[\leadsto \sqrt{\color{blue}{{w0}^{2}}} \]
   10. Step-by-step derivation

       1. sqrt-pow1 50.8%

          \[\leadsto \color{blue}{{w0}^{\left(\frac{2}{2}\right)}} \]

       2. metadata-eval 50.8%

          \[\leadsto {w0}^{\color{blue}{1}} \]

       3. pow1 50.8%

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

       4. add-exp-log 20.6%

          \[\leadsto \color{blue}{e^{\log w0}} \]

   11. Applied egg-rr 20.6%

       \[\leadsto \color{blue}{e^{\log w0}} \]
   12. Step-by-step derivation

       1. rem-exp-log 50.8%

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

       2. expm1-log1p-u 36.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w0\right)\right)} \]

       3. expm1-define 9.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(w0\right)} - 1} \]

       4. flip-- 17.9%

          \[\leadsto \color{blue}{\frac{e^{\mathsf{log1p}\left(w0\right)} \cdot e^{\mathsf{log1p}\left(w0\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(w0\right)} + 1}} \]

       5. pow2 17.9%

          \[\leadsto \frac{\color{blue}{{\left(e^{\mathsf{log1p}\left(w0\right)}\right)}^{2}} - 1 \cdot 1}{e^{\mathsf{log1p}\left(w0\right)} + 1} \]

       6. log1p-undefine 17.9%

          \[\leadsto \frac{{\left(e^{\color{blue}{\log \left(1 + w0\right)}}\right)}^{2} - 1 \cdot 1}{e^{\mathsf{log1p}\left(w0\right)} + 1} \]

       7. rem-exp-log 17.9%

          \[\leadsto \frac{{\color{blue}{\left(1 + w0\right)}}^{2} - 1 \cdot 1}{e^{\mathsf{log1p}\left(w0\right)} + 1} \]

       8. metadata-eval 17.9%

          \[\leadsto \frac{{\left(1 + w0\right)}^{2} - \color{blue}{1}}{e^{\mathsf{log1p}\left(w0\right)} + 1} \]

       9. log1p-undefine 17.9%

          \[\leadsto \frac{{\left(1 + w0\right)}^{2} - 1}{e^{\color{blue}{\log \left(1 + w0\right)}} + 1} \]

       10. rem-exp-log 27.3%

           \[\leadsto \frac{{\left(1 + w0\right)}^{2} - 1}{\color{blue}{\left(1 + w0\right)} + 1} \]

   13. Applied egg-rr 27.3%

       \[\leadsto \color{blue}{\frac{{\left(1 + w0\right)}^{2} - 1}{\left(1 + w0\right) + 1}} \]
   14. Step-by-step derivation

       1. sub-neg 27.3%

          \[\leadsto \frac{\color{blue}{{\left(1 + w0\right)}^{2} + \left(-1\right)}}{\left(1 + w0\right) + 1} \]

       2. metadata-eval 27.3%

          \[\leadsto \frac{{\left(1 + w0\right)}^{2} + \color{blue}{-1}}{\left(1 + w0\right) + 1} \]

       3. +-commutative 27.3%

          \[\leadsto \frac{{\left(1 + w0\right)}^{2} + -1}{\color{blue}{1 + \left(1 + w0\right)}} \]

       4. associate-+r+ 27.3%

          \[\leadsto \frac{{\left(1 + w0\right)}^{2} + -1}{\color{blue}{\left(1 + 1\right) + w0}} \]

       5. metadata-eval 27.3%

          \[\leadsto \frac{{\left(1 + w0\right)}^{2} + -1}{\color{blue}{2} + w0} \]

   15. Simplified 27.3%

       \[\leadsto \color{blue}{\frac{{\left(1 + w0\right)}^{2} + -1}{2 + w0}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.2 \cdot 10^{-30}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(w0 + 1\right)}^{2} + -1}{2 + w0}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.9% 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 83.4%

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

3. Simplified 83.7%

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

5. Taylor expanded in M around 0 67.7%

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

6. Final simplification 67.7%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024082 
(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))))))