Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 91.4%

Time: 17.2s

Alternatives: 9

Speedup: 1.8×

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: 80.8% 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: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ w0_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M_m \cdot D_m}{2 \cdot d_m}\right)}^{2} \leq 4 \cdot 10^{+253}:\\ \;\;\;\;w0_m \cdot {\left(1 - \frac{h \cdot {\left(\left(M_m \cdot D_m\right) \cdot \frac{0.5}{d_m}\right)}^{2}}{\ell}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{w0_m} \cdot e^{0.25 \cdot \left(\left(\left(2 \cdot \log M_m + \log \left(-0.25 \cdot \frac{h}{\ell}\right)\right) + -2 \cdot \log d_m\right) + -2 \cdot \log \left(\frac{1}{D_m}\right)\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0_m = (fabs.f64 w0)
w0_s = (copysign.f64 1 w0)
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) 4e+253)
    (*
     w0_m
     (pow (- 1.0 (/ (* h (pow (* (* M_m D_m) (/ 0.5 d_m)) 2.0)) l)) 0.5))
    (pow
     (*
      (sqrt w0_m)
      (exp
       (*
        0.25
        (+
         (+ (+ (* 2.0 (log M_m)) (log (* -0.25 (/ h l)))) (* -2.0 (log d_m)))
         (* -2.0 (log (/ 1.0 D_m)))))))
     2.0))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0_m = fabs(w0);
w0_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, 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) <= 4e+253) {
		tmp = w0_m * pow((1.0 - ((h * pow(((M_m * D_m) * (0.5 / d_m)), 2.0)) / l)), 0.5);
	} else {
		tmp = pow((sqrt(w0_m) * exp((0.25 * ((((2.0 * log(M_m)) + log((-0.25 * (h / l)))) + (-2.0 * log(d_m))) + (-2.0 * log((1.0 / D_m))))))), 2.0);
	}
	return w0_s * tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0d0, w0)
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if ((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) <= 4d+253) then
        tmp = w0_m * ((1.0d0 - ((h * (((m_m * d_m) * (0.5d0 / d_m_1)) ** 2.0d0)) / l)) ** 0.5d0)
    else
        tmp = (sqrt(w0_m) * exp((0.25d0 * ((((2.0d0 * log(m_m)) + log(((-0.25d0) * (h / l)))) + ((-2.0d0) * log(d_m_1))) + ((-2.0d0) * log((1.0d0 / d_m))))))) ** 2.0d0
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0_m = Math.abs(w0);
w0_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, 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) <= 4e+253) {
		tmp = w0_m * Math.pow((1.0 - ((h * Math.pow(((M_m * D_m) * (0.5 / d_m)), 2.0)) / l)), 0.5);
	} else {
		tmp = Math.pow((Math.sqrt(w0_m) * Math.exp((0.25 * ((((2.0 * Math.log(M_m)) + Math.log((-0.25 * (h / l)))) + (-2.0 * Math.log(d_m))) + (-2.0 * Math.log((1.0 / D_m))))))), 2.0);
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0_m = math.fabs(w0)
w0_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	tmp = 0
	if math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) <= 4e+253:
		tmp = w0_m * math.pow((1.0 - ((h * math.pow(((M_m * D_m) * (0.5 / d_m)), 2.0)) / l)), 0.5)
	else:
		tmp = math.pow((math.sqrt(w0_m) * math.exp((0.25 * ((((2.0 * math.log(M_m)) + math.log((-0.25 * (h / l)))) + (-2.0 * math.log(d_m))) + (-2.0 * math.log((1.0 / D_m))))))), 2.0)
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0, w0)
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if ((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) <= 4e+253)
		tmp = Float64(w0_m * (Float64(1.0 - Float64(Float64(h * (Float64(Float64(M_m * D_m) * Float64(0.5 / d_m)) ^ 2.0)) / l)) ^ 0.5));
	else
		tmp = Float64(sqrt(w0_m) * exp(Float64(0.25 * Float64(Float64(Float64(Float64(2.0 * log(M_m)) + log(Float64(-0.25 * Float64(h / l)))) + Float64(-2.0 * log(d_m))) + Float64(-2.0 * log(Float64(1.0 / D_m))))))) ^ 2.0;
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0_m = abs(w0);
w0_s = sign(w0) * abs(1.0);
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) <= 4e+253)
		tmp = w0_m * ((1.0 - ((h * (((M_m * D_m) * (0.5 / d_m)) ^ 2.0)) / l)) ^ 0.5);
	else
		tmp = (sqrt(w0_m) * exp((0.25 * ((((2.0 * log(M_m)) + log((-0.25 * (h / l)))) + (-2.0 * log(d_m))) + (-2.0 * log((1.0 / D_m))))))) ^ 2.0;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0_m = N[Abs[w0], $MachinePrecision]
w0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 4e+253], N[(w0$95$m * N[Power[N[(1.0 - N[(N[(h * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sqrt[w0$95$m], $MachinePrecision] * N[Exp[N[(0.25 * N[(N[(N[(N[(2.0 * N[Log[M$95$m], $MachinePrecision]), $MachinePrecision] + N[Log[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[Log[N[(1.0 / D$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 3.9999999999999997e253

   1. Initial program 90.7%

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

   3. Simplified 90.9%

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

      1. associate-*r/ 98.0%

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

      2. add-sqr-sqrt 47.2%

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

      3. associate-/r* 47.2%

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

      4. associate-*l/ 47.2%

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

      5. div-inv 47.2%

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

      6. associate-*l* 46.7%

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

      7. associate-/r* 46.7%

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

      8. metadata-eval 46.7%

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

   5. Applied egg-rr 46.7%

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

      1. pow1/2 46.7%

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

      2. associate-/l/ 46.7%

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

      3. *-commutative 46.7%

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

      4. associate-*r* 47.2%

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

      5. add-sqr-sqrt 97.9%

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

   7. Applied egg-rr 97.9%

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

   if 3.9999999999999997e253 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 48.7%

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

   3. Simplified 52.0%

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

   5. Applied egg-rr 21.5%

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

   6. Taylor expanded in D around inf 12.4%

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

   7. Taylor expanded in d around 0 7.1%

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

      1. distribute-lft-neg-in 7.1%

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

      2. metadata-eval 7.1%

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

      3. associate-/l* 8.5%

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

   9. Simplified 8.5%

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

   10. Taylor expanded in M around 0 9.3%

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

       1. +-commutative 9.3%

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

   12. Simplified 9.3%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 4 \cdot 10^{+253}:\\ \;\;\;\;w0 \cdot {\left(1 - \frac{h \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\left(\left(2 \cdot \log M + \log \left(-0.25 \cdot \frac{h}{\ell}\right)\right) + -2 \cdot \log d\right) + -2 \cdot \log \left(\frac{1}{D}\right)\right)}\right)}^{2}\\ \end{array} \]

Alternative 2: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ w0_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M_m \cdot D_m}{2 \cdot d_m}\right)}^{2} \leq 4 \cdot 10^{+253}:\\ \;\;\;\;w0_m \cdot {\left(1 - \frac{h \cdot {\left(\left(M_m \cdot D_m\right) \cdot \frac{0.5}{d_m}\right)}^{2}}{\ell}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{w0_m} \cdot e^{0.25 \cdot \left(-2 \cdot \log \left(\frac{1}{D_m}\right) + \left(-2 \cdot \log d_m + \log \left(-0.25 \cdot \frac{{M_m}^{2}}{\frac{\ell}{h}}\right)\right)\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0_m = (fabs.f64 w0)
w0_s = (copysign.f64 1 w0)
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) 4e+253)
    (*
     w0_m
     (pow (- 1.0 (/ (* h (pow (* (* M_m D_m) (/ 0.5 d_m)) 2.0)) l)) 0.5))
    (pow
     (*
      (sqrt w0_m)
      (exp
       (*
        0.25
        (+
         (* -2.0 (log (/ 1.0 D_m)))
         (+ (* -2.0 (log d_m)) (log (* -0.25 (/ (pow M_m 2.0) (/ l h)))))))))
     2.0))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0_m = fabs(w0);
w0_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, 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) <= 4e+253) {
		tmp = w0_m * pow((1.0 - ((h * pow(((M_m * D_m) * (0.5 / d_m)), 2.0)) / l)), 0.5);
	} else {
		tmp = pow((sqrt(w0_m) * exp((0.25 * ((-2.0 * log((1.0 / D_m))) + ((-2.0 * log(d_m)) + log((-0.25 * (pow(M_m, 2.0) / (l / h))))))))), 2.0);
	}
	return w0_s * tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0d0, w0)
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if ((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) <= 4d+253) then
        tmp = w0_m * ((1.0d0 - ((h * (((m_m * d_m) * (0.5d0 / d_m_1)) ** 2.0d0)) / l)) ** 0.5d0)
    else
        tmp = (sqrt(w0_m) * exp((0.25d0 * (((-2.0d0) * log((1.0d0 / d_m))) + (((-2.0d0) * log(d_m_1)) + log(((-0.25d0) * ((m_m ** 2.0d0) / (l / h))))))))) ** 2.0d0
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0_m = Math.abs(w0);
w0_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, 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) <= 4e+253) {
		tmp = w0_m * Math.pow((1.0 - ((h * Math.pow(((M_m * D_m) * (0.5 / d_m)), 2.0)) / l)), 0.5);
	} else {
		tmp = Math.pow((Math.sqrt(w0_m) * Math.exp((0.25 * ((-2.0 * Math.log((1.0 / D_m))) + ((-2.0 * Math.log(d_m)) + Math.log((-0.25 * (Math.pow(M_m, 2.0) / (l / h))))))))), 2.0);
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0_m = math.fabs(w0)
w0_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	tmp = 0
	if math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) <= 4e+253:
		tmp = w0_m * math.pow((1.0 - ((h * math.pow(((M_m * D_m) * (0.5 / d_m)), 2.0)) / l)), 0.5)
	else:
		tmp = math.pow((math.sqrt(w0_m) * math.exp((0.25 * ((-2.0 * math.log((1.0 / D_m))) + ((-2.0 * math.log(d_m)) + math.log((-0.25 * (math.pow(M_m, 2.0) / (l / h))))))))), 2.0)
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0, w0)
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if ((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) <= 4e+253)
		tmp = Float64(w0_m * (Float64(1.0 - Float64(Float64(h * (Float64(Float64(M_m * D_m) * Float64(0.5 / d_m)) ^ 2.0)) / l)) ^ 0.5));
	else
		tmp = Float64(sqrt(w0_m) * exp(Float64(0.25 * Float64(Float64(-2.0 * log(Float64(1.0 / D_m))) + Float64(Float64(-2.0 * log(d_m)) + log(Float64(-0.25 * Float64((M_m ^ 2.0) / Float64(l / h))))))))) ^ 2.0;
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0_m = abs(w0);
w0_s = sign(w0) * abs(1.0);
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) <= 4e+253)
		tmp = w0_m * ((1.0 - ((h * (((M_m * D_m) * (0.5 / d_m)) ^ 2.0)) / l)) ^ 0.5);
	else
		tmp = (sqrt(w0_m) * exp((0.25 * ((-2.0 * log((1.0 / D_m))) + ((-2.0 * log(d_m)) + log((-0.25 * ((M_m ^ 2.0) / (l / h))))))))) ^ 2.0;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0_m = N[Abs[w0], $MachinePrecision]
w0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 4e+253], N[(w0$95$m * N[Power[N[(1.0 - N[(N[(h * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sqrt[w0$95$m], $MachinePrecision] * N[Exp[N[(0.25 * 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[Power[M$95$m, 2.0], $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 3.9999999999999997e253

   1. Initial program 90.7%

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

   3. Simplified 90.9%

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

      1. associate-*r/ 98.0%

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

      2. add-sqr-sqrt 47.2%

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

      3. associate-/r* 47.2%

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

      4. associate-*l/ 47.2%

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

      5. div-inv 47.2%

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

      6. associate-*l* 46.7%

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

      7. associate-/r* 46.7%

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

      8. metadata-eval 46.7%

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

   5. Applied egg-rr 46.7%

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

      1. pow1/2 46.7%

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

      2. associate-/l/ 46.7%

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

      3. *-commutative 46.7%

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

      4. associate-*r* 47.2%

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

      5. add-sqr-sqrt 97.9%

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

   7. Applied egg-rr 97.9%

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

   if 3.9999999999999997e253 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 48.7%

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

   3. Simplified 52.0%

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

   5. Applied egg-rr 21.5%

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

   6. Taylor expanded in D around inf 12.4%

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

   7. Taylor expanded in d around 0 7.1%

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

      1. distribute-lft-neg-in 7.1%

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

      2. metadata-eval 7.1%

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

      3. associate-/l* 8.5%

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

   9. Simplified 8.5%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 4 \cdot 10^{+253}:\\ \;\;\;\;w0 \cdot {\left(1 - \frac{h \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(-2 \cdot \log \left(\frac{1}{D}\right) + \left(-2 \cdot \log d + \log \left(-0.25 \cdot \frac{{M}^{2}}{\frac{\ell}{h}}\right)\right)\right)}\right)}^{2}\\ \end{array} \]

Alternative 3: 86.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ w0_s \cdot \begin{array}{l} \mathbf{if}\;\frac{M_m \cdot D_m}{2 \cdot d_m} \leq 5 \cdot 10^{+126}:\\ \;\;\;\;w0_m \cdot {\left(1 - \frac{h \cdot {\left(\left(M_m \cdot D_m\right) \cdot \frac{0.5}{d_m}\right)}^{2}}{\ell}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{w0_m} \cdot e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M_m}^{2}}{\ell}\right) + \left(-2 \cdot \log d_m + 2 \cdot \log D_m\right)\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0_m = (fabs.f64 w0)
w0_s = (copysign.f64 1 w0)
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= (/ (* M_m D_m) (* 2.0 d_m)) 5e+126)
    (*
     w0_m
     (pow (- 1.0 (/ (* h (pow (* (* M_m D_m) (/ 0.5 d_m)) 2.0)) l)) 0.5))
    (pow
     (*
      (sqrt w0_m)
      (exp
       (*
        0.25
        (+
         (log (* -0.25 (/ (* h (pow M_m 2.0)) l)))
         (+ (* -2.0 (log d_m)) (* 2.0 (log D_m)))))))
     2.0))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0_m = fabs(w0);
w0_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+126) {
		tmp = w0_m * pow((1.0 - ((h * pow(((M_m * D_m) * (0.5 / d_m)), 2.0)) / l)), 0.5);
	} else {
		tmp = pow((sqrt(w0_m) * exp((0.25 * (log((-0.25 * ((h * pow(M_m, 2.0)) / l))) + ((-2.0 * log(d_m)) + (2.0 * log(D_m))))))), 2.0);
	}
	return w0_s * tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0d0, w0)
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((m_m * d_m) / (2.0d0 * d_m_1)) <= 5d+126) then
        tmp = w0_m * ((1.0d0 - ((h * (((m_m * d_m) * (0.5d0 / d_m_1)) ** 2.0d0)) / l)) ** 0.5d0)
    else
        tmp = (sqrt(w0_m) * exp((0.25d0 * (log(((-0.25d0) * ((h * (m_m ** 2.0d0)) / l))) + (((-2.0d0) * log(d_m_1)) + (2.0d0 * log(d_m))))))) ** 2.0d0
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0_m = Math.abs(w0);
w0_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+126) {
		tmp = w0_m * Math.pow((1.0 - ((h * Math.pow(((M_m * D_m) * (0.5 / d_m)), 2.0)) / l)), 0.5);
	} else {
		tmp = Math.pow((Math.sqrt(w0_m) * Math.exp((0.25 * (Math.log((-0.25 * ((h * Math.pow(M_m, 2.0)) / l))) + ((-2.0 * Math.log(d_m)) + (2.0 * Math.log(D_m))))))), 2.0);
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0_m = math.fabs(w0)
w0_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	tmp = 0
	if ((M_m * D_m) / (2.0 * d_m)) <= 5e+126:
		tmp = w0_m * math.pow((1.0 - ((h * math.pow(((M_m * D_m) * (0.5 / d_m)), 2.0)) / l)), 0.5)
	else:
		tmp = math.pow((math.sqrt(w0_m) * math.exp((0.25 * (math.log((-0.25 * ((h * math.pow(M_m, 2.0)) / l))) + ((-2.0 * math.log(d_m)) + (2.0 * math.log(D_m))))))), 2.0)
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0, w0)
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) <= 5e+126)
		tmp = Float64(w0_m * (Float64(1.0 - Float64(Float64(h * (Float64(Float64(M_m * D_m) * Float64(0.5 / d_m)) ^ 2.0)) / l)) ^ 0.5));
	else
		tmp = Float64(sqrt(w0_m) * exp(Float64(0.25 * Float64(log(Float64(-0.25 * Float64(Float64(h * (M_m ^ 2.0)) / l))) + Float64(Float64(-2.0 * log(d_m)) + Float64(2.0 * log(D_m))))))) ^ 2.0;
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0_m = abs(w0);
w0_s = sign(w0) * abs(1.0);
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+126)
		tmp = w0_m * ((1.0 - ((h * (((M_m * D_m) * (0.5 / d_m)) ^ 2.0)) / l)) ^ 0.5);
	else
		tmp = (sqrt(w0_m) * exp((0.25 * (log((-0.25 * ((h * (M_m ^ 2.0)) / l))) + ((-2.0 * log(d_m)) + (2.0 * log(D_m))))))) ^ 2.0;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0_m = N[Abs[w0], $MachinePrecision]
w0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 5e+126], N[(w0$95$m * N[Power[N[(1.0 - N[(N[(h * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sqrt[w0$95$m], $MachinePrecision] * N[Exp[N[(0.25 * N[(N[Log[N[(-0.25 * N[(N[(h * N[Power[M$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Log[D$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 4.99999999999999977e126

   1. Initial program 84.9%

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

   3. Simplified 85.9%

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

      1. associate-*r/ 92.0%

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

      2. add-sqr-sqrt 43.8%

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

      3. associate-/r* 43.8%

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

      4. associate-*l/ 43.3%

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

      5. div-inv 43.3%

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

      6. associate-*l* 43.4%

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

      7. associate-/r* 43.4%

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

      8. metadata-eval 43.4%

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

   5. Applied egg-rr 43.4%

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

      1. pow1/2 43.4%

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

      2. associate-/l/ 43.3%

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

      3. *-commutative 43.3%

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

      4. associate-*r* 43.3%

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

      5. add-sqr-sqrt 91.0%

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

   7. Applied egg-rr 91.0%

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

   if 4.99999999999999977e126 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 45.2%

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

   3. Simplified 45.2%

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

   5. Applied egg-rr 17.6%

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

   6. Taylor expanded in D around inf 12.8%

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

   7. Taylor expanded in d around 0 8.4%

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

      1. distribute-lft-neg-in 8.4%

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

      2. metadata-eval 8.4%

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

      3. associate-/l* 11.6%

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

   9. Simplified 11.6%

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

   10. Taylor expanded in D around 0 8.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+126}:\\ \;\;\;\;w0 \cdot {\left(1 - \frac{h \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell}\right) + \left(-2 \cdot \log d + 2 \cdot \log D\right)\right)}\right)}^{2}\\ \end{array} \]

Alternative 4: 86.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ \begin{array}{l} t_0 := \sqrt[3]{1 - \left(\left(0.5 \cdot \frac{D_m}{d_m}\right) \cdot \frac{M_m}{\frac{\ell}{h}}\right) \cdot \left(M_m \cdot \frac{D_m}{2 \cdot d_m}\right)}\\ w0_s \cdot \begin{array}{l} \mathbf{if}\;M_m \cdot D_m \leq 4 \cdot 10^{+99}:\\ \;\;\;\;w0_m \cdot {\left(1 - \frac{h \cdot {\left(\left(M_m \cdot D_m\right) \cdot \frac{0.5}{d_m}\right)}^{2}}{\ell}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;w0_m \cdot \left({\left({t_0}^{2}\right)}^{0.5} \cdot {t_0}^{0.5}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0_m = (fabs.f64 w0)
w0_s = (copysign.f64 1 w0)
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0
         (cbrt
          (-
           1.0
           (*
            (* (* 0.5 (/ D_m d_m)) (/ M_m (/ l h)))
            (* M_m (/ D_m (* 2.0 d_m))))))))
   (*
    w0_s
    (if (<= (* M_m D_m) 4e+99)
      (*
       w0_m
       (pow (- 1.0 (/ (* h (pow (* (* M_m D_m) (/ 0.5 d_m)) 2.0)) l)) 0.5))
      (* w0_m (* (pow (pow t_0 2.0) 0.5) (pow t_0 0.5)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0_m = fabs(w0);
w0_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = cbrt((1.0 - (((0.5 * (D_m / d_m)) * (M_m / (l / h))) * (M_m * (D_m / (2.0 * d_m))))));
	double tmp;
	if ((M_m * D_m) <= 4e+99) {
		tmp = w0_m * pow((1.0 - ((h * pow(((M_m * D_m) * (0.5 / d_m)), 2.0)) / l)), 0.5);
	} else {
		tmp = w0_m * (pow(pow(t_0, 2.0), 0.5) * pow(t_0, 0.5));
	}
	return w0_s * tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0_m = Math.abs(w0);
w0_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = Math.cbrt((1.0 - (((0.5 * (D_m / d_m)) * (M_m / (l / h))) * (M_m * (D_m / (2.0 * d_m))))));
	double tmp;
	if ((M_m * D_m) <= 4e+99) {
		tmp = w0_m * Math.pow((1.0 - ((h * Math.pow(((M_m * D_m) * (0.5 / d_m)), 2.0)) / l)), 0.5);
	} else {
		tmp = w0_m * (Math.pow(Math.pow(t_0, 2.0), 0.5) * Math.pow(t_0, 0.5));
	}
	return w0_s * tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0, w0)
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	t_0 = cbrt(Float64(1.0 - Float64(Float64(Float64(0.5 * Float64(D_m / d_m)) * Float64(M_m / Float64(l / h))) * Float64(M_m * Float64(D_m / Float64(2.0 * d_m))))))
	tmp = 0.0
	if (Float64(M_m * D_m) <= 4e+99)
		tmp = Float64(w0_m * (Float64(1.0 - Float64(Float64(h * (Float64(Float64(M_m * D_m) * Float64(0.5 / d_m)) ^ 2.0)) / l)) ^ 0.5));
	else
		tmp = Float64(w0_m * Float64(((t_0 ^ 2.0) ^ 0.5) * (t_0 ^ 0.5)));
	end
	return Float64(w0_s * tmp)
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0_m = N[Abs[w0], $MachinePrecision]
w0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[Power[N[(1.0 - N[(N[(N[(0.5 * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * N[(D$95$m / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(w0$95$s * If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 4e+99], N[(w0$95$m * N[Power[N[(1.0 - N[(N[(h * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[(N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 0.5], $MachinePrecision] * N[Power[t$95$0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 3.9999999999999999e99

   1. Initial program 85.3%

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

   3. Simplified 85.9%

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

      1. associate-*r/ 92.6%

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

      2. add-sqr-sqrt 46.3%

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

      3. associate-/r* 46.3%

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

      4. associate-*l/ 46.3%

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

      5. div-inv 46.3%

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

      6. associate-*l* 45.9%

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

      7. associate-/r* 45.9%

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

      8. metadata-eval 45.9%

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

   5. Applied egg-rr 45.9%

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

      1. pow1/2 45.9%

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

      2. associate-/l/ 45.9%

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

      3. *-commutative 45.9%

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

      4. associate-*r* 46.3%

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

      5. add-sqr-sqrt 92.1%

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

   7. Applied egg-rr 92.1%

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

   if 3.9999999999999999e99 < (*.f64 M D)

   1. Initial program 53.8%

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

   3. Simplified 56.3%

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

      1. associate-*r/ 54.0%

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

      2. associate-*l/ 51.4%

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

      3. div-inv 51.4%

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

      4. associate-*l* 54.0%

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

      5. associate-/r* 54.0%

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

      6. metadata-eval 54.0%

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

   5. Applied egg-rr 54.0%

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

      1. associate-*r/ 56.4%

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

      2. *-commutative 56.4%

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

      3. unpow2 56.4%

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

      4. associate-*r* 64.2%

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

      5. clear-num 64.2%

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

      6. un-div-inv 64.1%

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

      7. div-inv 64.1%

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

      8. metadata-eval 64.1%

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

      9. clear-num 64.1%

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

      10. un-div-inv 64.1%

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

      11. div-inv 64.1%

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

      12. metadata-eval 64.1%

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

   7. Applied egg-rr 64.1%

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

   8. Taylor expanded in h around 0 45.5%

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

      1. times-frac 56.3%

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

   10. Simplified 56.3%

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

       1. pow1/2 56.3%

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

       2. add-cube-cbrt 56.3%

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

       3. unpow-prod-down 56.3%

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

   12. Applied egg-rr 58.9%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 4 \cdot 10^{+99}:\\ \;\;\;\;w0 \cdot {\left(1 - \frac{h \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left({\left({\left(\sqrt[3]{1 - \left(\left(0.5 \cdot \frac{D}{d}\right) \cdot \frac{M}{\frac{\ell}{h}}\right) \cdot \left(M \cdot \frac{D}{2 \cdot d}\right)}\right)}^{2}\right)}^{0.5} \cdot {\left(\sqrt[3]{1 - \left(\left(0.5 \cdot \frac{D}{d}\right) \cdot \frac{M}{\frac{\ell}{h}}\right) \cdot \left(M \cdot \frac{D}{2 \cdot d}\right)}\right)}^{0.5}\right)\\ \end{array} \]

Alternative 5: 88.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ w0_s \cdot \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M_m \cdot D_m}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{+239}:\\ \;\;\;\;w0_m \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D_m}{2 \cdot \frac{d_m}{M_m}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0_m \cdot \sqrt{1 + \left(\frac{D_m}{d_m} \cdot -0.5\right) \cdot \left(\frac{M_m}{\ell} \cdot \left(h \cdot \left(M_m \cdot \frac{D_m}{2 \cdot d_m}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0_m = (fabs.f64 w0)
w0_s = (copysign.f64 1 w0)
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l))) 2e+239)
    (* w0_m (sqrt (- 1.0 (* (/ h l) (pow (/ D_m (* 2.0 (/ d_m M_m))) 2.0)))))
    (*
     w0_m
     (sqrt
      (+
       1.0
       (*
        (* (/ D_m d_m) -0.5)
        (* (/ M_m l) (* h (* M_m (/ D_m (* 2.0 d_m))))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0_m = fabs(w0);
w0_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((1.0 - (pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l))) <= 2e+239) {
		tmp = w0_m * sqrt((1.0 - ((h / l) * pow((D_m / (2.0 * (d_m / M_m))), 2.0))));
	} else {
		tmp = w0_m * sqrt((1.0 + (((D_m / d_m) * -0.5) * ((M_m / l) * (h * (M_m * (D_m / (2.0 * d_m))))))));
	}
	return w0_s * tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0d0, w0)
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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 ((1.0d0 - ((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l))) <= 2d+239) then
        tmp = w0_m * sqrt((1.0d0 - ((h / l) * ((d_m / (2.0d0 * (d_m_1 / m_m))) ** 2.0d0))))
    else
        tmp = w0_m * sqrt((1.0d0 + (((d_m / d_m_1) * (-0.5d0)) * ((m_m / l) * (h * (m_m * (d_m / (2.0d0 * d_m_1))))))))
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0_m = Math.abs(w0);
w0_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((1.0 - (Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l))) <= 2e+239) {
		tmp = w0_m * Math.sqrt((1.0 - ((h / l) * Math.pow((D_m / (2.0 * (d_m / M_m))), 2.0))));
	} else {
		tmp = w0_m * Math.sqrt((1.0 + (((D_m / d_m) * -0.5) * ((M_m / l) * (h * (M_m * (D_m / (2.0 * d_m))))))));
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0_m = math.fabs(w0)
w0_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	tmp = 0
	if (1.0 - (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l))) <= 2e+239:
		tmp = w0_m * math.sqrt((1.0 - ((h / l) * math.pow((D_m / (2.0 * (d_m / M_m))), 2.0))))
	else:
		tmp = w0_m * math.sqrt((1.0 + (((D_m / d_m) * -0.5) * ((M_m / l) * (h * (M_m * (D_m / (2.0 * d_m))))))))
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0, w0)
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l))) <= 2e+239)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D_m / Float64(2.0 * Float64(d_m / M_m))) ^ 2.0)))));
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 + Float64(Float64(Float64(D_m / d_m) * -0.5) * Float64(Float64(M_m / l) * Float64(h * Float64(M_m * Float64(D_m / Float64(2.0 * d_m)))))))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0_m = abs(w0);
w0_s = sign(w0) * abs(1.0);
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((1.0 - ((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l))) <= 2e+239)
		tmp = w0_m * sqrt((1.0 - ((h / l) * ((D_m / (2.0 * (d_m / M_m))) ^ 2.0))));
	else
		tmp = w0_m * sqrt((1.0 + (((D_m / d_m) * -0.5) * ((M_m / l) * (h * (M_m * (D_m / (2.0 * d_m))))))));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0_m = N[Abs[w0], $MachinePrecision]
w0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[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], 2e+239], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m / N[(2.0 * N[(d$95$m / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 + N[(N[(N[(D$95$m / d$95$m), $MachinePrecision] * -0.5), $MachinePrecision] * N[(N[(M$95$m / l), $MachinePrecision] * N[(h * N[(M$95$m * N[(D$95$m / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 99.9%

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

      3. un-div-inv 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. times-frac 99.9%

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

      6. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

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

   1. Initial program 42.5%

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

   3. Simplified 44.8%

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

      1. associate-*r/ 63.3%

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

      2. associate-*l/ 61.0%

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

      3. div-inv 61.0%

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

      4. associate-*l* 63.3%

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

      5. associate-/r* 63.3%

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

      6. metadata-eval 63.3%

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

   5. Applied egg-rr 63.3%

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

      1. associate-*r/ 44.8%

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

      2. *-commutative 44.8%

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

      3. unpow2 44.8%

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

      4. associate-*r* 50.5%

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

      5. clear-num 50.5%

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

      6. un-div-inv 50.5%

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

      7. div-inv 50.5%

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

      8. metadata-eval 50.5%

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

      9. clear-num 50.5%

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

      10. un-div-inv 50.5%

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

      11. div-inv 50.5%

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

      12. metadata-eval 50.5%

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

   7. Applied egg-rr 50.5%

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

   8. Taylor expanded in h around 0 61.1%

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

      1. times-frac 58.6%

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

   10. Simplified 58.6%

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

       1. expm1-log1p-u 58.1%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \left(0.5 \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\ell}\right)\right) \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)}\right)\right)} \]

       2. expm1-udef 58.1%

          \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(0.5 \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\ell}\right)\right) \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)}\right)} - 1\right)} \]

       3. associate-*r* 58.1%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \color{blue}{\left(\left(0.5 \cdot \frac{D}{d}\right) \cdot \frac{M \cdot h}{\ell}\right)} \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)}\right)} - 1\right) \]

       4. associate-/l* 51.1%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(\left(0.5 \cdot \frac{D}{d}\right) \cdot \color{blue}{\frac{M}{\frac{\ell}{h}}}\right) \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)}\right)} - 1\right) \]

   12. Applied egg-rr 51.1%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(\left(0.5 \cdot \frac{D}{d}\right) \cdot \frac{M}{\frac{\ell}{h}}\right) \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)}\right)} - 1\right)} \]
   13. Step-by-step derivation

       1. expm1-def 51.1%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \left(\left(0.5 \cdot \frac{D}{d}\right) \cdot \frac{M}{\frac{\ell}{h}}\right) \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)}\right)\right)} \]

       2. expm1-log1p 51.6%

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

       3. associate-*l* 48.2%

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

       4. cancel-sign-sub-inv 48.2%

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

       5. distribute-lft-neg-in 48.2%

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

       6. metadata-eval 48.2%

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

       7. associate-/r/ 55.2%

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

       8. associate-*l* 57.7%

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

       9. *-commutative 57.7%

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

   14. Simplified 57.7%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{+239}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D}{d} \cdot -0.5\right) \cdot \left(\frac{M}{\ell} \cdot \left(h \cdot \left(M \cdot \frac{D}{2 \cdot d}\right)\right)\right)}\\ \end{array} \]

Alternative 6: 87.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ \begin{array}{l} t_0 := M_m \cdot \frac{D_m}{2 \cdot d_m}\\ w0_s \cdot \begin{array}{l} \mathbf{if}\;M_m \cdot D_m \leq 2 \cdot 10^{+98}:\\ \;\;\;\;w0_m \cdot {\left(1 - \frac{h \cdot {\left(\left(M_m \cdot D_m\right) \cdot \frac{0.5}{d_m}\right)}^{2}}{\ell}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;w0_m \cdot \sqrt{1 - t_0 \cdot \left(\frac{h}{\ell} \cdot t_0\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0_m = (fabs.f64 w0)
w0_s = (copysign.f64 1 w0)
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* M_m (/ D_m (* 2.0 d_m)))))
   (*
    w0_s
    (if (<= (* M_m D_m) 2e+98)
      (*
       w0_m
       (pow (- 1.0 (/ (* h (pow (* (* M_m D_m) (/ 0.5 d_m)) 2.0)) l)) 0.5))
      (* w0_m (sqrt (- 1.0 (* t_0 (* (/ h l) t_0)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0_m = fabs(w0);
w0_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = M_m * (D_m / (2.0 * d_m));
	double tmp;
	if ((M_m * D_m) <= 2e+98) {
		tmp = w0_m * pow((1.0 - ((h * pow(((M_m * D_m) * (0.5 / d_m)), 2.0)) / l)), 0.5);
	} else {
		tmp = w0_m * sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	}
	return w0_s * tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0d0, w0)
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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 = m_m * (d_m / (2.0d0 * d_m_1))
    if ((m_m * d_m) <= 2d+98) then
        tmp = w0_m * ((1.0d0 - ((h * (((m_m * d_m) * (0.5d0 / d_m_1)) ** 2.0d0)) / l)) ** 0.5d0)
    else
        tmp = w0_m * sqrt((1.0d0 - (t_0 * ((h / l) * t_0))))
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0_m = Math.abs(w0);
w0_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = M_m * (D_m / (2.0 * d_m));
	double tmp;
	if ((M_m * D_m) <= 2e+98) {
		tmp = w0_m * Math.pow((1.0 - ((h * Math.pow(((M_m * D_m) * (0.5 / d_m)), 2.0)) / l)), 0.5);
	} else {
		tmp = w0_m * Math.sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0_m = math.fabs(w0)
w0_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	t_0 = M_m * (D_m / (2.0 * d_m))
	tmp = 0
	if (M_m * D_m) <= 2e+98:
		tmp = w0_m * math.pow((1.0 - ((h * math.pow(((M_m * D_m) * (0.5 / d_m)), 2.0)) / l)), 0.5)
	else:
		tmp = w0_m * math.sqrt((1.0 - (t_0 * ((h / l) * t_0))))
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0, w0)
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	t_0 = Float64(M_m * Float64(D_m / Float64(2.0 * d_m)))
	tmp = 0.0
	if (Float64(M_m * D_m) <= 2e+98)
		tmp = Float64(w0_m * (Float64(1.0 - Float64(Float64(h * (Float64(Float64(M_m * D_m) * Float64(0.5 / d_m)) ^ 2.0)) / l)) ^ 0.5));
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(h / l) * t_0)))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0_m = abs(w0);
w0_s = sign(w0) * abs(1.0);
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	t_0 = M_m * (D_m / (2.0 * d_m));
	tmp = 0.0;
	if ((M_m * D_m) <= 2e+98)
		tmp = w0_m * ((1.0 - ((h * (((M_m * D_m) * (0.5 / d_m)) ^ 2.0)) / l)) ^ 0.5);
	else
		tmp = w0_m * sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0_m = N[Abs[w0], $MachinePrecision]
w0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(D$95$m / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 2e+98], N[(w0$95$m * N[Power[N[(1.0 - N[(N[(h * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2e98

   1. Initial program 85.3%

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

   3. Simplified 85.9%

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

      1. associate-*r/ 92.6%

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

      2. add-sqr-sqrt 46.3%

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

      3. associate-/r* 46.3%

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

      4. associate-*l/ 46.3%

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

      5. div-inv 46.3%

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

      6. associate-*l* 45.9%

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

      7. associate-/r* 45.9%

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

      8. metadata-eval 45.9%

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

   5. Applied egg-rr 45.9%

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

      1. pow1/2 45.9%

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

      2. associate-/l/ 45.9%

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

      3. *-commutative 45.9%

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

      4. associate-*r* 46.3%

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

      5. add-sqr-sqrt 92.1%

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

   7. Applied egg-rr 92.1%

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

   if 2e98 < (*.f64 M D)

   1. Initial program 53.8%

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

   3. Simplified 56.3%

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

      1. associate-*r/ 54.0%

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

      2. associate-*l/ 51.4%

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

      3. div-inv 51.4%

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

      4. associate-*l* 54.0%

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

      5. associate-/r* 54.0%

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

      6. metadata-eval 54.0%

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

   5. Applied egg-rr 54.0%

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

      1. associate-*r/ 56.4%

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

      2. *-commutative 56.4%

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

      3. unpow2 56.4%

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

      4. associate-*r* 64.2%

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

      5. clear-num 64.2%

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

      6. un-div-inv 64.1%

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

      7. div-inv 64.1%

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

      8. metadata-eval 64.1%

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

      9. clear-num 64.1%

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

      10. un-div-inv 64.1%

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

      11. div-inv 64.1%

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

      12. metadata-eval 64.1%

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

   7. Applied egg-rr 64.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{+98}:\\ \;\;\;\;w0 \cdot {\left(1 - \frac{h \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \frac{D}{2 \cdot d}\right) \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot \frac{D}{2 \cdot d}\right)\right)}\\ \end{array} \]

Alternative 7: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ w0_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{+43}:\\ \;\;\;\;w0_m \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D_m \cdot \frac{M_m}{2 \cdot d_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0_m \cdot \sqrt{1 + \left(\frac{D_m}{d_m} \cdot -0.5\right) \cdot \left(\frac{M_m}{\ell} \cdot \left(h \cdot \left(M_m \cdot \frac{D_m}{2 \cdot d_m}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0_m = (fabs.f64 w0)
w0_s = (copysign.f64 1 w0)
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= l -2e+43)
    (* w0_m (sqrt (- 1.0 (* (/ h l) (pow (* D_m (/ M_m (* 2.0 d_m))) 2.0)))))
    (*
     w0_m
     (sqrt
      (+
       1.0
       (*
        (* (/ D_m d_m) -0.5)
        (* (/ M_m l) (* h (* M_m (/ D_m (* 2.0 d_m))))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0_m = fabs(w0);
w0_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (l <= -2e+43) {
		tmp = w0_m * sqrt((1.0 - ((h / l) * pow((D_m * (M_m / (2.0 * d_m))), 2.0))));
	} else {
		tmp = w0_m * sqrt((1.0 + (((D_m / d_m) * -0.5) * ((M_m / l) * (h * (M_m * (D_m / (2.0 * d_m))))))));
	}
	return w0_s * tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0d0, w0)
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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 (l <= (-2d+43)) then
        tmp = w0_m * sqrt((1.0d0 - ((h / l) * ((d_m * (m_m / (2.0d0 * d_m_1))) ** 2.0d0))))
    else
        tmp = w0_m * sqrt((1.0d0 + (((d_m / d_m_1) * (-0.5d0)) * ((m_m / l) * (h * (m_m * (d_m / (2.0d0 * d_m_1))))))))
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0_m = Math.abs(w0);
w0_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (l <= -2e+43) {
		tmp = w0_m * Math.sqrt((1.0 - ((h / l) * Math.pow((D_m * (M_m / (2.0 * d_m))), 2.0))));
	} else {
		tmp = w0_m * Math.sqrt((1.0 + (((D_m / d_m) * -0.5) * ((M_m / l) * (h * (M_m * (D_m / (2.0 * d_m))))))));
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0_m = math.fabs(w0)
w0_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	tmp = 0
	if l <= -2e+43:
		tmp = w0_m * math.sqrt((1.0 - ((h / l) * math.pow((D_m * (M_m / (2.0 * d_m))), 2.0))))
	else:
		tmp = w0_m * math.sqrt((1.0 + (((D_m / d_m) * -0.5) * ((M_m / l) * (h * (M_m * (D_m / (2.0 * d_m))))))))
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0, w0)
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (l <= -2e+43)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D_m * Float64(M_m / Float64(2.0 * d_m))) ^ 2.0)))));
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 + Float64(Float64(Float64(D_m / d_m) * -0.5) * Float64(Float64(M_m / l) * Float64(h * Float64(M_m * Float64(D_m / Float64(2.0 * d_m)))))))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0_m = abs(w0);
w0_s = sign(w0) * abs(1.0);
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (l <= -2e+43)
		tmp = w0_m * sqrt((1.0 - ((h / l) * ((D_m * (M_m / (2.0 * d_m))) ^ 2.0))));
	else
		tmp = w0_m * sqrt((1.0 + (((D_m / d_m) * -0.5) * ((M_m / l) * (h * (M_m * (D_m / (2.0 * d_m))))))));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0_m = N[Abs[w0], $MachinePrecision]
w0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[l, -2e+43], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M$95$m / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 + N[(N[(N[(D$95$m / d$95$m), $MachinePrecision] * -0.5), $MachinePrecision] * N[(N[(M$95$m / l), $MachinePrecision] * N[(h * N[(M$95$m * N[(D$95$m / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < -2.00000000000000003e43

   1. Initial program 91.0%

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

   3. Simplified 93.2%

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

   if -2.00000000000000003e43 < l

   1. Initial program 77.9%

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

   3. Simplified 78.4%

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

      1. associate-*r/ 85.3%

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

      2. associate-*l/ 84.9%

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

      3. div-inv 84.8%

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

      4. associate-*l* 84.9%

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

      5. associate-/r* 84.9%

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

      6. metadata-eval 84.9%

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

   5. Applied egg-rr 84.9%

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

      1. associate-*r/ 77.9%

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

      2. *-commutative 77.9%

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

      3. unpow2 77.9%

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

      4. associate-*r* 79.9%

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

      5. clear-num 79.9%

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

      6. un-div-inv 79.9%

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

      7. div-inv 79.9%

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

      8. metadata-eval 79.9%

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

      9. clear-num 79.9%

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

      10. un-div-inv 79.9%

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

      11. div-inv 79.9%

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

      12. metadata-eval 79.9%

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

   7. Applied egg-rr 79.9%

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

   8. Taylor expanded in h around 0 76.0%

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

      1. times-frac 78.6%

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

   10. Simplified 78.6%

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

       1. expm1-log1p-u 78.1%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \left(0.5 \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\ell}\right)\right) \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)}\right)\right)} \]

       2. expm1-udef 78.1%

          \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(0.5 \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\ell}\right)\right) \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)}\right)} - 1\right)} \]

       3. associate-*r* 78.1%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \color{blue}{\left(\left(0.5 \cdot \frac{D}{d}\right) \cdot \frac{M \cdot h}{\ell}\right)} \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)}\right)} - 1\right) \]

       4. associate-/l* 77.5%

          \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(\left(0.5 \cdot \frac{D}{d}\right) \cdot \color{blue}{\frac{M}{\frac{\ell}{h}}}\right) \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)}\right)} - 1\right) \]

   12. Applied egg-rr 77.5%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(\left(0.5 \cdot \frac{D}{d}\right) \cdot \frac{M}{\frac{\ell}{h}}\right) \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)}\right)} - 1\right)} \]
   13. Step-by-step derivation

       1. expm1-def 77.5%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \left(\left(0.5 \cdot \frac{D}{d}\right) \cdot \frac{M}{\frac{\ell}{h}}\right) \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)}\right)\right)} \]

       2. expm1-log1p 77.9%

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

       3. associate-*l* 76.0%

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

       4. cancel-sign-sub-inv 76.0%

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

       5. distribute-lft-neg-in 76.0%

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

       6. metadata-eval 76.0%

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

       7. associate-/r/ 75.8%

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

       8. associate-*l* 76.9%

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

       9. *-commutative 76.9%

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

   14. Simplified 76.9%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{+43}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D}{d} \cdot -0.5\right) \cdot \left(\frac{M}{\ell} \cdot \left(h \cdot \left(M \cdot \frac{D}{2 \cdot d}\right)\right)\right)}\\ \end{array} \]

Alternative 8: 85.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ w0_s \cdot \left(w0_m \cdot \sqrt{1 + \left(\frac{D_m}{d_m} \cdot -0.5\right) \cdot \left(\frac{M_m}{\ell} \cdot \left(h \cdot \left(M_m \cdot \frac{D_m}{2 \cdot d_m}\right)\right)\right)}\right) \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0_m = (fabs.f64 w0)
w0_s = (copysign.f64 1 w0)
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (*
   w0_m
   (sqrt
    (+
     1.0
     (*
      (* (/ D_m d_m) -0.5)
      (* (/ M_m l) (* h (* M_m (/ D_m (* 2.0 d_m)))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0_m = fabs(w0);
w0_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	return w0_s * (w0_m * sqrt((1.0 + (((D_m / d_m) * -0.5) * ((M_m / l) * (h * (M_m * (D_m / (2.0 * d_m)))))))));
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0d0, w0)
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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_s * (w0_m * sqrt((1.0d0 + (((d_m / d_m_1) * (-0.5d0)) * ((m_m / l) * (h * (m_m * (d_m / (2.0d0 * d_m_1)))))))))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0_m = Math.abs(w0);
w0_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	return w0_s * (w0_m * Math.sqrt((1.0 + (((D_m / d_m) * -0.5) * ((M_m / l) * (h * (M_m * (D_m / (2.0 * d_m)))))))));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0_m = math.fabs(w0)
w0_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	return w0_s * (w0_m * math.sqrt((1.0 + (((D_m / d_m) * -0.5) * ((M_m / l) * (h * (M_m * (D_m / (2.0 * d_m)))))))))

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0, w0)
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	return Float64(w0_s * Float64(w0_m * sqrt(Float64(1.0 + Float64(Float64(Float64(D_m / d_m) * -0.5) * Float64(Float64(M_m / l) * Float64(h * Float64(M_m * Float64(D_m / Float64(2.0 * d_m))))))))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0_m = abs(w0);
w0_s = sign(w0) * abs(1.0);
function tmp = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = w0_s * (w0_m * sqrt((1.0 + (((D_m / d_m) * -0.5) * ((M_m / l) * (h * (M_m * (D_m / (2.0 * d_m)))))))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0_m = N[Abs[w0], $MachinePrecision]
w0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0$95$s * N[(w0$95$m * N[Sqrt[N[(1.0 + N[(N[(N[(D$95$m / d$95$m), $MachinePrecision] * -0.5), $MachinePrecision] * N[(N[(M$95$m / l), $MachinePrecision] * N[(h * N[(M$95$m * N[(D$95$m / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

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

3. Simplified 81.6%

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

   1. associate-*r/ 87.0%

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

   2. associate-*l/ 86.2%

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

   3. div-inv 86.2%

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

   4. associate-*l* 85.9%

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

   5. associate-/r* 85.9%

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

   6. metadata-eval 85.9%

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

5. Applied egg-rr 85.9%

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

   1. associate-*r/ 80.4%

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

   2. *-commutative 80.4%

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

   3. unpow2 80.4%

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

   4. associate-*r* 82.3%

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

   5. clear-num 82.3%

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

   6. un-div-inv 82.3%

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

   7. div-inv 82.3%

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

   8. metadata-eval 82.3%

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

   9. clear-num 82.3%

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

   10. un-div-inv 82.3%

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

   11. div-inv 82.3%

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

   12. metadata-eval 82.3%

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

7. Applied egg-rr 82.3%

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

8. Taylor expanded in h around 0 76.2%

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

   1. times-frac 78.7%

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

10. Simplified 78.7%

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

    1. expm1-log1p-u 78.3%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \left(0.5 \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\ell}\right)\right) \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)}\right)\right)} \]

    2. expm1-udef 78.3%

       \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(0.5 \cdot \left(\frac{D}{d} \cdot \frac{M \cdot h}{\ell}\right)\right) \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)}\right)} - 1\right)} \]

    3. associate-*r* 78.3%

       \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \color{blue}{\left(\left(0.5 \cdot \frac{D}{d}\right) \cdot \frac{M \cdot h}{\ell}\right)} \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)}\right)} - 1\right) \]

    4. associate-/l* 78.9%

       \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(\left(0.5 \cdot \frac{D}{d}\right) \cdot \color{blue}{\frac{M}{\frac{\ell}{h}}}\right) \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)}\right)} - 1\right) \]

12. Applied egg-rr 78.9%

    \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(\left(0.5 \cdot \frac{D}{d}\right) \cdot \frac{M}{\frac{\ell}{h}}\right) \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)}\right)} - 1\right)} \]
13. Step-by-step derivation

    1. expm1-def 78.9%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \left(\left(0.5 \cdot \frac{D}{d}\right) \cdot \frac{M}{\frac{\ell}{h}}\right) \cdot \left(M \cdot \frac{D}{d \cdot 2}\right)}\right)\right)} \]

    2. expm1-log1p 79.3%

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

    3. associate-*l* 77.8%

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

    4. cancel-sign-sub-inv 77.8%

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

    5. distribute-lft-neg-in 77.8%

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

    6. metadata-eval 77.8%

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

    7. associate-/r/ 77.3%

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

    8. associate-*l* 78.5%

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

    9. *-commutative 78.5%

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

14. Simplified 78.5%

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

15. Final simplification 78.5%

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

Alternative 9: 67.9% accurate, 216.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ w0_s \cdot w0_m \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0_m = (fabs.f64 w0)
w0_s = (copysign.f64 1 w0)
(FPCore (w0_s w0_m M_m D_m h l d_m) :precision binary64 (* w0_s w0_m))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0_m = fabs(w0);
w0_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	return w0_s * w0_m;
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0d0, w0)
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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_s * w0_m
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0_m = Math.abs(w0);
w0_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	return w0_s * w0_m;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0_m = math.fabs(w0)
w0_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	return w0_s * w0_m

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0, w0)
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	return Float64(w0_s * w0_m)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0_m = abs(w0);
w0_s = sign(w0) * abs(1.0);
function tmp = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = w0_s * w0_m;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0_m = N[Abs[w0], $MachinePrecision]
w0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0$95$s * w0$95$m), $MachinePrecision]

Derivation

1. Initial program 80.7%

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

3. Simplified 81.6%

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

4. Taylor expanded in M around 0 65.5%

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

5. Final simplification 65.5%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023319 
(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))))))