Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 89.8%

Time: 17.6s

Alternatives: 7

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.1% 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: 89.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ \begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell}\\ w0_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;{\left(\sqrt{w0_m} \cdot e^{0.25 \cdot \left(\log \left(0.25 \cdot \frac{{\left(M \cdot D\right)}^{2} \cdot \left(-h\right)}{\ell}\right) + -2 \cdot \log d_m\right)}\right)}^{2}\\ \mathbf{elif}\;t_0 \leq 10^{-6}:\\ \;\;\;\;w0_m \cdot \sqrt{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;w0_m\\ \end{array} \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
w0_m = (fabs.f64 w0)
w0_s = (copysign.f64 1 w0)
(FPCore (w0_s w0_m M D h l d_m)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M D) (* 2.0 d_m)) 2.0) (/ h l))))
   (*
    w0_s
    (if (<= t_0 (- INFINITY))
      (pow
       (*
        (sqrt w0_m)
        (exp
         (*
          0.25
          (+
           (log (* 0.25 (/ (* (pow (* M D) 2.0) (- h)) l)))
           (* -2.0 (log d_m))))))
       2.0)
      (if (<= t_0 1e-6) (* w0_m (sqrt (- 1.0 t_0))) w0_m)))))

C

d_m = fabs(d);
w0_m = fabs(w0);
w0_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d_m) {
	double t_0 = pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = pow((sqrt(w0_m) * exp((0.25 * (log((0.25 * ((pow((M * D), 2.0) * -h) / l))) + (-2.0 * log(d_m)))))), 2.0);
	} else if (t_0 <= 1e-6) {
		tmp = w0_m * sqrt((1.0 - t_0));
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Java

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, double D, double h, double l, double d_m) {
	double t_0 = Math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.pow((Math.sqrt(w0_m) * Math.exp((0.25 * (Math.log((0.25 * ((Math.pow((M * D), 2.0) * -h) / l))) + (-2.0 * Math.log(d_m)))))), 2.0);
	} else if (t_0 <= 1e-6) {
		tmp = w0_m * Math.sqrt((1.0 - t_0));
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Python

d_m = math.fabs(d)
w0_m = math.fabs(w0)
w0_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M, D, h, l, d_m):
	t_0 = math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = math.pow((math.sqrt(w0_m) * math.exp((0.25 * (math.log((0.25 * ((math.pow((M * D), 2.0) * -h) / l))) + (-2.0 * math.log(d_m)))))), 2.0)
	elif t_0 <= 1e-6:
		tmp = w0_m * math.sqrt((1.0 - t_0))
	else:
		tmp = w0_m
	return w0_s * tmp

Julia

d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0, w0)
function code(w0_s, w0_m, M, D, h, l, d_m)
	t_0 = Float64((Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(sqrt(w0_m) * exp(Float64(0.25 * Float64(log(Float64(0.25 * Float64(Float64((Float64(M * D) ^ 2.0) * Float64(-h)) / l))) + Float64(-2.0 * log(d_m)))))) ^ 2.0;
	elseif (t_0 <= 1e-6)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - t_0)));
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

MATLAB

d_m = abs(d);
w0_m = abs(w0);
w0_s = sign(w0) * abs(1.0);
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d_m)
	t_0 = (((M * D) / (2.0 * d_m)) ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (sqrt(w0_m) * exp((0.25 * (log((0.25 * ((((M * D) ^ 2.0) * -h) / l))) + (-2.0 * log(d_m)))))) ^ 2.0;
	elseif (t_0 <= 1e-6)
		tmp = w0_m * sqrt((1.0 - t_0));
	else
		tmp = w0_m;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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_, D_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[t$95$0, (-Infinity)], N[Power[N[(N[Sqrt[w0$95$m], $MachinePrecision] * N[Exp[N[(0.25 * N[(N[Log[N[(0.25 * N[(N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] * (-h)), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$0, 1e-6], N[(w0$95$m * N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0$95$m]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 60.7%

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

   3. Simplified 65.0%

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

   6. Applied egg-rr 36.2%

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

   7. Taylor expanded in d around 0 16.4%

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

      1. expm1-log1p-u 10.4%

         \[\leadsto {\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2} \]

      2. expm1-udef 10.4%

         \[\leadsto {\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)} - 1}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2} \]

      3. associate-*r* 10.4%

         \[\leadsto {\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}\right)} - 1}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2} \]

      4. pow-prod-down 10.4%

         \[\leadsto {\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h\right)} - 1}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2} \]

   9. Applied egg-rr 10.4%

      \[\leadsto {\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\left(D \cdot M\right)}^{2} \cdot h\right)} - 1}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2} \]
   10. Step-by-step derivation

       1. expm1-def 11.8%

          \[\leadsto {\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(D \cdot M\right)}^{2} \cdot h\right)\right)}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2} \]

       2. expm1-log1p 19.0%

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

       3. *-commutative 19.0%

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

   11. Simplified 19.0%

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

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 9.99999999999999955e-7

   1. Initial program 99.9%

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

   if 9.99999999999999955e-7 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 0.0%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in M around 0 67.2%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;{\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(\log \left(0.25 \cdot \frac{{\left(M \cdot D\right)}^{2} \cdot \left(-h\right)}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-6}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ \begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell}\\ w0_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;{\left(\sqrt{w0_m} \cdot e^{0.25 \cdot \left(-2 \cdot \log d_m + \log \left(-0.25 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot D\right)}^{2}\right)\right)\right)}\right)}^{2}\\ \mathbf{elif}\;t_0 \leq 10^{-6}:\\ \;\;\;\;w0_m \cdot \sqrt{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;w0_m\\ \end{array} \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
w0_m = (fabs.f64 w0)
w0_s = (copysign.f64 1 w0)
(FPCore (w0_s w0_m M D h l d_m)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M D) (* 2.0 d_m)) 2.0) (/ h l))))
   (*
    w0_s
    (if (<= t_0 (- INFINITY))
      (pow
       (*
        (sqrt w0_m)
        (exp
         (*
          0.25
          (+
           (* -2.0 (log d_m))
           (log (* -0.25 (* (/ h l) (pow (* M D) 2.0))))))))
       2.0)
      (if (<= t_0 1e-6) (* w0_m (sqrt (- 1.0 t_0))) w0_m)))))

C

d_m = fabs(d);
w0_m = fabs(w0);
w0_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d_m) {
	double t_0 = pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = pow((sqrt(w0_m) * exp((0.25 * ((-2.0 * log(d_m)) + log((-0.25 * ((h / l) * pow((M * D), 2.0)))))))), 2.0);
	} else if (t_0 <= 1e-6) {
		tmp = w0_m * sqrt((1.0 - t_0));
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Java

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, double D, double h, double l, double d_m) {
	double t_0 = Math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.pow((Math.sqrt(w0_m) * Math.exp((0.25 * ((-2.0 * Math.log(d_m)) + Math.log((-0.25 * ((h / l) * Math.pow((M * D), 2.0)))))))), 2.0);
	} else if (t_0 <= 1e-6) {
		tmp = w0_m * Math.sqrt((1.0 - t_0));
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Python

d_m = math.fabs(d)
w0_m = math.fabs(w0)
w0_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M, D, h, l, d_m):
	t_0 = math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = math.pow((math.sqrt(w0_m) * math.exp((0.25 * ((-2.0 * math.log(d_m)) + math.log((-0.25 * ((h / l) * math.pow((M * D), 2.0)))))))), 2.0)
	elif t_0 <= 1e-6:
		tmp = w0_m * math.sqrt((1.0 - t_0))
	else:
		tmp = w0_m
	return w0_s * tmp

Julia

d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0, w0)
function code(w0_s, w0_m, M, D, h, l, d_m)
	t_0 = Float64((Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(sqrt(w0_m) * exp(Float64(0.25 * Float64(Float64(-2.0 * log(d_m)) + log(Float64(-0.25 * Float64(Float64(h / l) * (Float64(M * D) ^ 2.0)))))))) ^ 2.0;
	elseif (t_0 <= 1e-6)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - t_0)));
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

MATLAB

d_m = abs(d);
w0_m = abs(w0);
w0_s = sign(w0) * abs(1.0);
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d_m)
	t_0 = (((M * D) / (2.0 * d_m)) ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (sqrt(w0_m) * exp((0.25 * ((-2.0 * log(d_m)) + log((-0.25 * ((h / l) * ((M * D) ^ 2.0)))))))) ^ 2.0;
	elseif (t_0 <= 1e-6)
		tmp = w0_m * sqrt((1.0 - t_0));
	else
		tmp = w0_m;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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_, D_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[t$95$0, (-Infinity)], N[Power[N[(N[Sqrt[w0$95$m], $MachinePrecision] * N[Exp[N[(0.25 * N[(N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision] + N[Log[N[(-0.25 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$0, 1e-6], N[(w0$95$m * N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0$95$m]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 60.7%

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

   3. Simplified 65.0%

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

   6. Applied egg-rr 36.2%

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

   7. Taylor expanded in d around 0 16.4%

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

   8. Taylor expanded in l around 0 6.1%

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

   10. Simplified 19.0%

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

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 9.99999999999999955e-7

   1. Initial program 99.9%

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

   if 9.99999999999999955e-7 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 0.0%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in M around 0 67.2%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;{\left(\sqrt{w0} \cdot e^{0.25 \cdot \left(-2 \cdot \log d + \log \left(-0.25 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot D\right)}^{2}\right)\right)\right)}\right)}^{2}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-6}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ w0_s \cdot \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d_m}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-6}:\\ \;\;\;\;w0_m \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0_m\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
w0_m = (fabs.f64 w0)
w0_s = (copysign.f64 1 w0)
(FPCore (w0_s w0_m M D h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= (* (pow (/ (* M D) (* 2.0 d_m)) 2.0) (/ h l)) 1e-6)
    (* w0_m (sqrt (- 1.0 (* (/ h l) (pow (* D (/ M (* 2.0 d_m))) 2.0)))))
    w0_m)))

C

d_m = fabs(d);
w0_m = fabs(w0);
w0_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= 1e-6) {
		tmp = w0_m * sqrt((1.0 - ((h / l) * pow((D * (M / (2.0 * d_m))), 2.0))));
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Fortran

d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0d0, w0)
real(8) function code(w0_s, w0_m, m, d, h, l, d_m)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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_m
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_m)) ** 2.0d0) * (h / l)) <= 1d-6) then
        tmp = w0_m * sqrt((1.0d0 - ((h / l) * ((d * (m / (2.0d0 * d_m))) ** 2.0d0))))
    else
        tmp = w0_m
    end if
    code = w0_s * tmp
end function

Java

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, double D, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= 1e-6) {
		tmp = w0_m * Math.sqrt((1.0 - ((h / l) * Math.pow((D * (M / (2.0 * d_m))), 2.0))));
	} else {
		tmp = w0_m;
	}
	return w0_s * tmp;
}

Python

d_m = math.fabs(d)
w0_m = math.fabs(w0)
w0_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M, D, h, l, d_m):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)) <= 1e-6:
		tmp = w0_m * math.sqrt((1.0 - ((h / l) * math.pow((D * (M / (2.0 * d_m))), 2.0))))
	else:
		tmp = w0_m
	return w0_s * tmp

Julia

d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0, w0)
function code(w0_s, w0_m, M, D, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= 1e-6)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D * Float64(M / Float64(2.0 * d_m))) ^ 2.0)))));
	else
		tmp = w0_m;
	end
	return Float64(w0_s * tmp)
end

MATLAB

d_m = abs(d);
w0_m = abs(w0);
w0_s = sign(w0) * abs(1.0);
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d_m)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= 1e-6)
		tmp = w0_m * sqrt((1.0 - ((h / l) * ((D * (M / (2.0 * d_m))) ^ 2.0))));
	else
		tmp = w0_m;
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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_, D_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 1e-6], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 9.99999999999999955e-7

   1. Initial program 88.9%

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

   3. Simplified 88.5%

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

   if 9.99999999999999955e-7 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 0.0%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in M around 0 67.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-6}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ w0_s \cdot \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;w0_m \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d_m}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0_m \cdot \sqrt{1 - \frac{{\left(\frac{D \cdot 0.5}{\frac{d_m}{M}} \cdot \sqrt{h}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
w0_m = (fabs.f64 w0)
w0_s = (copysign.f64 1 w0)
(FPCore (w0_s w0_m M D h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= h -5e-310)
    (* w0_m (sqrt (- 1.0 (/ (* h (pow (* M (* D (/ 0.5 d_m))) 2.0)) l))))
    (*
     w0_m
     (sqrt (- 1.0 (/ (pow (* (/ (* D 0.5) (/ d_m M)) (sqrt h)) 2.0) l)))))))

C

d_m = fabs(d);
w0_m = fabs(w0);
w0_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (h <= -5e-310) {
		tmp = w0_m * sqrt((1.0 - ((h * pow((M * (D * (0.5 / d_m))), 2.0)) / l)));
	} else {
		tmp = w0_m * sqrt((1.0 - (pow((((D * 0.5) / (d_m / M)) * sqrt(h)), 2.0) / l)));
	}
	return w0_s * tmp;
}

Fortran

d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0d0, w0)
real(8) function code(w0_s, w0_m, m, d, h, l, d_m)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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_m
    real(8) :: tmp
    if (h <= (-5d-310)) then
        tmp = w0_m * sqrt((1.0d0 - ((h * ((m * (d * (0.5d0 / d_m))) ** 2.0d0)) / l)))
    else
        tmp = w0_m * sqrt((1.0d0 - (((((d * 0.5d0) / (d_m / m)) * sqrt(h)) ** 2.0d0) / l)))
    end if
    code = w0_s * tmp
end function

Java

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, double D, double h, double l, double d_m) {
	double tmp;
	if (h <= -5e-310) {
		tmp = w0_m * Math.sqrt((1.0 - ((h * Math.pow((M * (D * (0.5 / d_m))), 2.0)) / l)));
	} else {
		tmp = w0_m * Math.sqrt((1.0 - (Math.pow((((D * 0.5) / (d_m / M)) * Math.sqrt(h)), 2.0) / l)));
	}
	return w0_s * tmp;
}

Python

d_m = math.fabs(d)
w0_m = math.fabs(w0)
w0_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M, D, h, l, d_m):
	tmp = 0
	if h <= -5e-310:
		tmp = w0_m * math.sqrt((1.0 - ((h * math.pow((M * (D * (0.5 / d_m))), 2.0)) / l)))
	else:
		tmp = w0_m * math.sqrt((1.0 - (math.pow((((D * 0.5) / (d_m / M)) * math.sqrt(h)), 2.0) / l)))
	return w0_s * tmp

Julia

d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0, w0)
function code(w0_s, w0_m, M, D, h, l, d_m)
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M * Float64(D * Float64(0.5 / d_m))) ^ 2.0)) / l))));
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64((Float64(Float64(Float64(D * 0.5) / Float64(d_m / M)) * sqrt(h)) ^ 2.0) / l))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

d_m = abs(d);
w0_m = abs(w0);
w0_s = sign(w0) * abs(1.0);
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d_m)
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = w0_m * sqrt((1.0 - ((h * ((M * (D * (0.5 / d_m))) ^ 2.0)) / l)));
	else
		tmp = w0_m * sqrt((1.0 - (((((D * 0.5) / (d_m / M)) * sqrt(h)) ^ 2.0) / l)));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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_, D_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[h, -5e-310], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M * N[(D * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(N[(D * 0.5), $MachinePrecision] / N[(d$95$m / M), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   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 85.7%

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

      1. associate-*r/ 87.3%

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

      2. frac-times 86.5%

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

      3. associate-*l/ 85.9%

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

      4. div-inv 85.9%

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

      5. associate-*l* 87.3%

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

      6. associate-/r* 87.3%

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

      7. metadata-eval 87.3%

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

   6. Applied egg-rr 87.3%

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

   if -4.999999999999985e-310 < h

   1. Initial program 83.2%

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

   3. Simplified 87.8%

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

      1. associate-*r/ 87.9%

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

      2. frac-times 84.3%

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

      3. associate-*l/ 86.4%

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

      4. div-inv 86.4%

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

      5. associate-*l* 87.9%

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

      6. associate-/r* 87.9%

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

      7. metadata-eval 87.9%

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

   6. Applied egg-rr 87.9%

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

      1. associate-*r* 86.4%

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

      2. *-commutative 86.4%

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

      3. expm1-log1p-u 69.9%

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

      4. expm1-udef 69.2%

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

      5. *-commutative 69.2%

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

      6. associate-*r* 69.2%

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

      7. associate-*l/ 69.2%

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

   8. Applied egg-rr 69.2%

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

      1. expm1-def 70.6%

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

      2. expm1-log1p 87.9%

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

      3. *-commutative 87.9%

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

   10. Simplified 87.9%

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

       1. add-sqr-sqrt 87.9%

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

       2. pow2 87.9%

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

   12. Applied egg-rr 91.0%

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

       1. associate-*l* 90.2%

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

       2. associate-*r/ 90.2%

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

   14. Simplified 90.2%

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

   15. Taylor expanded in M around 0 88.1%

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

       1. associate-*r* 88.1%

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

       2. associate-/l* 91.7%

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

       3. associate-*r/ 91.7%

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

       4. *-commutative 91.7%

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

   17. Simplified 91.7%

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

4. Final simplification 89.5%

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

Alternative 5: 80.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0d0, w0)
real(8) function code(w0_s, w0_m, m, d, h, l, d_m)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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_m
    code = w0_s * (w0_m * sqrt((1.0d0 - ((h / l) * (((m / 2.0d0) * (d / d_m)) ** 2.0d0)))))
end function

Java

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, double D, double h, double l, double d_m) {
	return w0_s * (w0_m * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / 2.0) * (D / d_m)), 2.0)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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_, D_, h_, l_, d$95$m_] := N[(w0$95$s * N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.1%

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

3. Simplified 86.7%

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

5. Final simplification 86.7%

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

Alternative 6: 66.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ w0_m = \left|w0\right| \\ w0_s = \mathsf{copysign}\left(1, w0\right) \\ w0_s \cdot \begin{array}{l} \mathbf{if}\;M \leq 5.8 \cdot 10^{+108}:\\ \;\;\;\;w0_m\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{w0_m}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
w0_m = (fabs.f64 w0)
w0_s = (copysign.f64 1 w0)
(FPCore (w0_s w0_m M D h l d_m)
 :precision binary64
 (* w0_s (if (<= M 5.8e+108) w0_m (log (exp w0_m)))))

C

d_m = fabs(d);
w0_m = fabs(w0);
w0_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (M <= 5.8e+108) {
		tmp = w0_m;
	} else {
		tmp = log(exp(w0_m));
	}
	return w0_s * tmp;
}

Fortran

d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0d0, w0)
real(8) function code(w0_s, w0_m, m, d, h, l, d_m)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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_m
    real(8) :: tmp
    if (m <= 5.8d+108) then
        tmp = w0_m
    else
        tmp = log(exp(w0_m))
    end if
    code = w0_s * tmp
end function

Java

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, double D, double h, double l, double d_m) {
	double tmp;
	if (M <= 5.8e+108) {
		tmp = w0_m;
	} else {
		tmp = Math.log(Math.exp(w0_m));
	}
	return w0_s * tmp;
}

Python

d_m = math.fabs(d)
w0_m = math.fabs(w0)
w0_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M, D, h, l, d_m):
	tmp = 0
	if M <= 5.8e+108:
		tmp = w0_m
	else:
		tmp = math.log(math.exp(w0_m))
	return w0_s * tmp

Julia

d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0, w0)
function code(w0_s, w0_m, M, D, h, l, d_m)
	tmp = 0.0
	if (M <= 5.8e+108)
		tmp = w0_m;
	else
		tmp = log(exp(w0_m));
	end
	return Float64(w0_s * tmp)
end

MATLAB

d_m = abs(d);
w0_m = abs(w0);
w0_s = sign(w0) * abs(1.0);
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d_m)
	tmp = 0.0;
	if (M <= 5.8e+108)
		tmp = w0_m;
	else
		tmp = log(exp(w0_m));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

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_, D_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[M, 5.8e+108], w0$95$m, N[Log[N[Exp[w0$95$m], $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if M < 5.80000000000000015e108

   1. Initial program 83.9%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in M around 0 72.0%

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

   if 5.80000000000000015e108 < M

   1. Initial program 85.2%

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

   3. Simplified 91.1%

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

      1. add-sqr-sqrt 51.2%

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

      2. sqrt-unprod 43.5%

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

      3. *-commutative 43.5%

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

      4. *-commutative 43.5%

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

      5. swap-sqr 34.6%

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

   6. Applied egg-rr 34.6%

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

      1. associate-*r/ 37.7%

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

      2. associate-*l/ 37.7%

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

      3. *-commutative 37.7%

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

      4. associate-*r* 34.7%

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

      5. *-commutative 34.7%

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

   8. Simplified 34.7%

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

   9. Taylor expanded in h around 0 17.8%

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

       1. sqrt-pow1 36.9%

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

       2. metadata-eval 36.9%

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

       3. pow1 36.9%

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

       4. add-log-exp 32.8%

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

   11. Applied egg-rr 32.8%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5.8 \cdot 10^{+108}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{w0}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.7% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

d_m = abs(d)
w0_m = abs(w0)
w0_s = copysign(1.0d0, w0)
real(8) function code(w0_s, w0_m, m, d, h, l, d_m)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    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_m
    code = w0_s * w0_m
end function

Java

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, double D, double h, double l, double d_m) {
	return w0_s * w0_m;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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_, D_, h_, l_, d$95$m_] := N[(w0$95$s * w0$95$m), $MachinePrecision]

Derivation

1. Initial program 84.1%

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

3. Simplified 86.7%

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

5. Taylor expanded in M around 0 67.5%

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

6. Final simplification 67.5%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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))))))