Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.5% → 88.1%

Time: 16.3s

Alternatives: 8

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: 81.5% 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: 88.1% accurate, 0.3× speedup

Math

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

FPCore

d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
(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)) (- INFINITY))
    (pow
     (*
      (sqrt w0_m)
      (pow
       (exp 0.25)
       (+ (log (* -0.25 (/ (* h (pow (* M D) 2.0)) l))) (* -2.0 (log d_m)))))
     2.0)
    (* w0_m (sqrt (- 1.0 (/ h (* l (pow (* 2.0 (/ (/ d_m D) M)) 2.0)))))))))

C

d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d_m);
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)) <= -((double) INFINITY)) {
		tmp = pow((sqrt(w0_m) * pow(exp(0.25), (log((-0.25 * ((h * pow((M * D), 2.0)) / l))) + (-2.0 * log(d_m))))), 2.0);
	} else {
		tmp = w0_m * sqrt((1.0 - (h / (l * pow((2.0 * ((d_m / D) / M)), 2.0)))));
	}
	return w0_s * tmp;
}

Java

d_m = Math.abs(d);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M && M < D && D < h && h < l && l < d_m;
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)) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.pow((Math.sqrt(w0_m) * Math.pow(Math.exp(0.25), (Math.log((-0.25 * ((h * Math.pow((M * D), 2.0)) / l))) + (-2.0 * Math.log(d_m))))), 2.0);
	} else {
		tmp = w0_m * Math.sqrt((1.0 - (h / (l * Math.pow((2.0 * ((d_m / D) / M)), 2.0)))));
	}
	return w0_s * tmp;
}

Python

d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d_m] = sort([w0_m, M, D, h, l, d_m])
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)) <= -math.inf:
		tmp = math.pow((math.sqrt(w0_m) * math.pow(math.exp(0.25), (math.log((-0.25 * ((h * math.pow((M * D), 2.0)) / l))) + (-2.0 * math.log(d_m))))), 2.0)
	else:
		tmp = w0_m * math.sqrt((1.0 - (h / (l * math.pow((2.0 * ((d_m / D) / M)), 2.0)))))
	return w0_s * tmp

Julia

d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d_m = sort([w0_m, M, D, h, l, d_m])
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)) <= Float64(-Inf))
		tmp = Float64(sqrt(w0_m) * (exp(0.25) ^ Float64(log(Float64(-0.25 * Float64(Float64(h * (Float64(M * D) ^ 2.0)) / l))) + Float64(-2.0 * log(d_m))))) ^ 2.0;
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(h / Float64(l * (Float64(2.0 * Float64(Float64(d_m / D) / M)) ^ 2.0))))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d_m = num2cell(sort([w0_m, M, D, h, l, d_m])){:}
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)) <= -Inf)
		tmp = (sqrt(w0_m) * (exp(0.25) ^ (log((-0.25 * ((h * ((M * D) ^ 2.0)) / l))) + (-2.0 * log(d_m))))) ^ 2.0;
	else
		tmp = w0_m * sqrt((1.0 - (h / (l * ((2.0 * ((d_m / D) / M)) ^ 2.0)))));
	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]
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
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], (-Infinity)], N[Power[N[(N[Sqrt[w0$95$m], $MachinePrecision] * N[Power[N[Exp[0.25], $MachinePrecision], N[(N[Log[N[(-0.25 * N[(N[(h * N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(h / N[(l * N[Power[N[(2.0 * N[(N[(d$95$m / D), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.8%

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

   3. Simplified 56.8%

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

      1. associate-*r/ 59.2%

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

      2. add-sqr-sqrt 59.2%

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

      3. pow2 59.2%

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

      4. sqrt-pow1 59.2%

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

      5. metadata-eval 59.2%

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

      6. pow1 59.2%

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

      7. *-un-lft-identity 59.2%

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

      8. times-frac 59.2%

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

      9. metadata-eval 59.2%

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

   6. Applied egg-rr 59.2%

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

   7. Taylor expanded in M around 0 47.1%

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

      1. associate-*r/ 47.1%

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

      2. associate-*r* 49.1%

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

      3. unpow2 49.1%

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

      4. unpow2 49.1%

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

      5. swap-sqr 60.0%

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

      6. unpow2 60.0%

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

   9. Simplified 60.0%

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

       1. add-sqr-sqrt 34.0%

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

       2. pow2 34.0%

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

       3. associate-/l/ 33.7%

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

       4. associate-*r* 33.7%

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

   11. Applied egg-rr 33.7%

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

   12. Taylor expanded in d around 0 11.4%

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

       1. exp-prod 11.4%

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

       2. distribute-lft-neg-in 11.4%

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

       3. metadata-eval 11.4%

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

       4. associate-*r* 11.5%

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

       5. *-commutative 11.5%

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

       6. unpow2 11.5%

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

       7. unpow2 11.5%

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

       8. swap-sqr 14.6%

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

       9. unpow2 14.6%

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

       10. *-commutative 14.6%

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

       11. *-commutative 14.6%

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

   14. Simplified 14.6%

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

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

   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 89.6%

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

      1. associate-*r/ 91.0%

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

      2. unpow2 91.0%

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

      3. clear-num 91.0%

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

      4. clear-num 91.0%

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

      5. frac-times 91.0%

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

      6. metadata-eval 91.0%

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

      7. *-commutative 91.0%

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

      8. associate-/l* 90.9%

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

      9. *-commutative 90.9%

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

      10. associate-/l* 90.9%

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

   6. Applied egg-rr 90.9%

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

      1. frac-times 97.2%

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

      2. *-un-lft-identity 97.2%

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

      3. pow2 97.2%

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

      4. associate-*r/ 97.2%

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

   8. Applied egg-rr 97.2%

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

      1. *-commutative 97.2%

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

      2. *-commutative 97.2%

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

      3. *-commutative 97.2%

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

      4. associate-*r/ 97.2%

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

      5. associate-/r* 96.3%

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

   10. Simplified 96.3%

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

4. Final simplification 78.8%

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

Alternative 2: 87.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d_m] = \mathsf{sort}([w0_m, M, D, h, l, d_m])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}\\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+246}:\\ \;\;\;\;w0\_m \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{h \cdot \left(M \cdot \left(\left(D \cdot 0.5\right) \cdot \left(\left(M \cdot 0.5\right) \cdot \frac{D}{d\_m}\right)\right)\right)}{d\_m \cdot \ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* M D) (* 2.0 d_m)) 2.0) (/ h l)))))
   (*
    w0_s
    (if (<= t_0 1e+246)
      (* w0_m (sqrt t_0))
      (*
       w0_m
       (sqrt
        (-
         1.0
         (/ (* h (* M (* (* D 0.5) (* (* M 0.5) (/ D d_m))))) (* d_m l)))))))))

C

d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d_m);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d_m) {
	double t_0 = 1.0 - (pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 1e+246) {
		tmp = w0_m * sqrt(t_0);
	} else {
		tmp = w0_m * sqrt((1.0 - ((h * (M * ((D * 0.5) * ((M * 0.5) * (D / d_m))))) / (d_m * l))));
	}
	return w0_s * tmp;
}

Fortran

d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((((m * d) / (2.0d0 * d_m)) ** 2.0d0) * (h / l))
    if (t_0 <= 1d+246) then
        tmp = w0_m * sqrt(t_0)
    else
        tmp = w0_m * sqrt((1.0d0 - ((h * (m * ((d * 0.5d0) * ((m * 0.5d0) * (d / d_m))))) / (d_m * 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);
assert w0_m < M && M < D && D < h && h < l && l < d_m;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d_m) {
	double t_0 = 1.0 - (Math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 1e+246) {
		tmp = w0_m * Math.sqrt(t_0);
	} else {
		tmp = w0_m * Math.sqrt((1.0 - ((h * (M * ((D * 0.5) * ((M * 0.5) * (D / d_m))))) / (d_m * l))));
	}
	return w0_s * tmp;
}

Python

d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d_m] = sort([w0_m, M, D, h, l, d_m])
def code(w0_s, w0_m, M, D, h, l, d_m):
	t_0 = 1.0 - (math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 1e+246:
		tmp = w0_m * math.sqrt(t_0)
	else:
		tmp = w0_m * math.sqrt((1.0 - ((h * (M * ((D * 0.5) * ((M * 0.5) * (D / d_m))))) / (d_m * l))))
	return w0_s * tmp

Julia

d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d_m = sort([w0_m, M, D, h, l, d_m])
function code(w0_s, w0_m, M, D, h, l, d_m)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 1e+246)
		tmp = Float64(w0_m * sqrt(t_0));
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h * Float64(M * Float64(Float64(D * 0.5) * Float64(Float64(M * 0.5) * Float64(D / d_m))))) / Float64(d_m * l)))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d_m = num2cell(sort([w0_m, M, D, h, l, d_m])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d_m)
	t_0 = 1.0 - ((((M * D) / (2.0 * d_m)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 1e+246)
		tmp = w0_m * sqrt(t_0);
	else
		tmp = w0_m * sqrt((1.0 - ((h * (M * ((D * 0.5) * ((M * 0.5) * (D / d_m))))) / (d_m * 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]
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[t$95$0, 1e+246], N[(w0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h * N[(M * N[(N[(D * 0.5), $MachinePrecision] * N[(N[(M * 0.5), $MachinePrecision] * N[(D / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 1.00000000000000007e246

   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 1.00000000000000007e246 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 43.6%

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

   3. Simplified 44.9%

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

      1. unpow2 44.9%

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

      2. associate-*r/ 43.6%

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

      3. associate-/r* 43.6%

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

      4. associate-*l/ 43.6%

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

      5. associate-/l* 43.6%

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

      6. div-inv 43.6%

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

      7. metadata-eval 43.6%

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

      8. *-un-lft-identity 43.6%

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

      9. times-frac 43.6%

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

      10. metadata-eval 43.6%

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

   6. Applied egg-rr 43.6%

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

      1. frac-times 66.7%

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

      2. associate-*l* 65.4%

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

      3. associate-*r* 65.4%

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

   8. Applied egg-rr 65.4%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{+246}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(M \cdot \left(\left(D \cdot 0.5\right) \cdot \left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)\right)\right)}{d \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d_m] = \mathsf{sort}([w0_m, M, D, h, l, d_m])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{h \cdot \left(M \cdot \left(\left(D \cdot 0.5\right) \cdot \left(\left(M \cdot 0.5\right) \cdot \frac{D}{d\_m}\right)\right)\right)}{d\_m \cdot \ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-296}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{h}{\ell} \cdot \frac{\left(M \cdot \left(D \cdot 0.5\right)\right) \cdot \left(0.5 \cdot \frac{M \cdot D}{d\_m}\right)}{d\_m}}\\ \mathbf{else}:\\ \;\;\;\;w0\_m\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= (/ h l) (- INFINITY))
    (*
     w0_m
     (sqrt
      (- 1.0 (/ (* h (* M (* (* D 0.5) (* (* M 0.5) (/ D d_m))))) (* d_m l)))))
    (if (<= (/ h l) -1e-296)
      (*
       w0_m
       (sqrt
        (-
         1.0
         (* (/ h l) (/ (* (* M (* D 0.5)) (* 0.5 (/ (* M D) d_m))) d_m)))))
      w0_m))))

C

d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d_m);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0_m * sqrt((1.0 - ((h * (M * ((D * 0.5) * ((M * 0.5) * (D / d_m))))) / (d_m * l))));
	} else if ((h / l) <= -1e-296) {
		tmp = w0_m * sqrt((1.0 - ((h / l) * (((M * (D * 0.5)) * (0.5 * ((M * D) / d_m))) / d_m))));
	} 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);
assert w0_m < M && M < D && D < h && h < l && l < d_m;
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 / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0_m * Math.sqrt((1.0 - ((h * (M * ((D * 0.5) * ((M * 0.5) * (D / d_m))))) / (d_m * l))));
	} else if ((h / l) <= -1e-296) {
		tmp = w0_m * Math.sqrt((1.0 - ((h / l) * (((M * (D * 0.5)) * (0.5 * ((M * D) / d_m))) / d_m))));
	} 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)
[w0_m, M, D, h, l, d_m] = sort([w0_m, M, D, h, l, d_m])
def code(w0_s, w0_m, M, D, h, l, d_m):
	tmp = 0
	if (h / l) <= -math.inf:
		tmp = w0_m * math.sqrt((1.0 - ((h * (M * ((D * 0.5) * ((M * 0.5) * (D / d_m))))) / (d_m * l))))
	elif (h / l) <= -1e-296:
		tmp = w0_m * math.sqrt((1.0 - ((h / l) * (((M * (D * 0.5)) * (0.5 * ((M * D) / d_m))) / d_m))))
	else:
		tmp = w0_m
	return w0_s * tmp

Julia

d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d_m = sort([w0_m, M, D, h, l, d_m])
function code(w0_s, w0_m, M, D, h, l, d_m)
	tmp = 0.0
	if (Float64(h / l) <= Float64(-Inf))
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h * Float64(M * Float64(Float64(D * 0.5) * Float64(Float64(M * 0.5) * Float64(D / d_m))))) / Float64(d_m * l)))));
	elseif (Float64(h / l) <= -1e-296)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(M * Float64(D * 0.5)) * Float64(0.5 * Float64(Float64(M * D) / d_m))) / d_m)))));
	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);
w0_m, M, D, h, l, d_m = num2cell(sort([w0_m, M, D, h, l, d_m])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d_m)
	tmp = 0.0;
	if ((h / l) <= -Inf)
		tmp = w0_m * sqrt((1.0 - ((h * (M * ((D * 0.5) * ((M * 0.5) * (D / d_m))))) / (d_m * l))));
	elseif ((h / l) <= -1e-296)
		tmp = w0_m * sqrt((1.0 - ((h / l) * (((M * (D * 0.5)) * (0.5 * ((M * D) / d_m))) / d_m))));
	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]
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h * N[(M * N[(N[(D * 0.5), $MachinePrecision] * N[(N[(M * 0.5), $MachinePrecision] * N[(D / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-296], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(M * N[(D * 0.5), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[(M * D), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0$95$m]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -inf.0

   1. Initial program 39.5%

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

   3. Simplified 39.5%

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

      1. unpow2 39.5%

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

      2. associate-*r/ 39.5%

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

      3. associate-/r* 39.5%

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

      4. associate-*l/ 39.5%

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

      5. associate-/l* 39.5%

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

      6. div-inv 39.5%

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

      7. metadata-eval 39.5%

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

      8. *-un-lft-identity 39.5%

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

      9. times-frac 39.5%

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

      10. metadata-eval 39.5%

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

   6. Applied egg-rr 39.5%

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

      1. frac-times 78.9%

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

      2. associate-*l* 78.9%

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

      3. associate-*r* 78.9%

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

   8. Applied egg-rr 78.9%

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

   if -inf.0 < (/.f64 h l) < -1e-296

   1. Initial program 83.7%

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

   3. Simplified 82.3%

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

      1. unpow2 82.3%

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

      2. associate-*r/ 82.3%

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

      3. associate-/r* 82.3%

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

      4. associate-*l/ 82.4%

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

      5. associate-/l* 82.4%

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

      6. div-inv 82.4%

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

      7. metadata-eval 82.4%

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

      8. *-un-lft-identity 82.4%

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

      9. times-frac 82.4%

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

      10. metadata-eval 82.4%

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

   6. Applied egg-rr 82.4%

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

   7. Taylor expanded in M around 0 83.7%

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

   if -1e-296 < (/.f64 h l)

   1. Initial program 90.8%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in M around 0 98.1%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(M \cdot \left(\left(D \cdot 0.5\right) \cdot \left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)\right)\right)}{d \cdot \ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-296}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \frac{\left(M \cdot \left(D \cdot 0.5\right)\right) \cdot \left(0.5 \cdot \frac{M \cdot D}{d}\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d_m] = \mathsf{sort}([w0_m, M, D, h, l, d_m])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;D \leq 4.5 \cdot 10^{+60}:\\ \;\;\;\;w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{h}{\ell} \cdot \frac{\left(M \cdot \left(D \cdot 0.5\right)\right) \cdot \left(0.5 \cdot \frac{M \cdot D}{d\_m}\right)}{d\_m}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= D 4.5e+60)
    w0_m
    (*
     w0_m
     (sqrt
      (-
       1.0
       (* (/ h l) (/ (* (* M (* D 0.5)) (* 0.5 (/ (* M D) d_m))) d_m))))))))

C

d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d_m);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (D <= 4.5e+60) {
		tmp = w0_m;
	} else {
		tmp = w0_m * sqrt((1.0 - ((h / l) * (((M * (D * 0.5)) * (0.5 * ((M * D) / d_m))) / d_m))));
	}
	return w0_s * tmp;
}

Fortran

d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
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 (d <= 4.5d+60) then
        tmp = w0_m
    else
        tmp = w0_m * sqrt((1.0d0 - ((h / l) * (((m * (d * 0.5d0)) * (0.5d0 * ((m * d) / d_m))) / d_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);
assert w0_m < M && M < D && D < h && h < l && l < d_m;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (D <= 4.5e+60) {
		tmp = w0_m;
	} else {
		tmp = w0_m * Math.sqrt((1.0 - ((h / l) * (((M * (D * 0.5)) * (0.5 * ((M * D) / d_m))) / d_m))));
	}
	return w0_s * tmp;
}

Python

d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d_m] = sort([w0_m, M, D, h, l, d_m])
def code(w0_s, w0_m, M, D, h, l, d_m):
	tmp = 0
	if D <= 4.5e+60:
		tmp = w0_m
	else:
		tmp = w0_m * math.sqrt((1.0 - ((h / l) * (((M * (D * 0.5)) * (0.5 * ((M * D) / d_m))) / d_m))))
	return w0_s * tmp

Julia

d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d_m = sort([w0_m, M, D, h, l, d_m])
function code(w0_s, w0_m, M, D, h, l, d_m)
	tmp = 0.0
	if (D <= 4.5e+60)
		tmp = w0_m;
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(M * Float64(D * 0.5)) * Float64(0.5 * Float64(Float64(M * D) / d_m))) / d_m)))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d_m = num2cell(sort([w0_m, M, D, h, l, d_m])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d_m)
	tmp = 0.0;
	if (D <= 4.5e+60)
		tmp = w0_m;
	else
		tmp = w0_m * sqrt((1.0 - ((h / l) * (((M * (D * 0.5)) * (0.5 * ((M * D) / d_m))) / d_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]
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[D, 4.5e+60], w0$95$m, N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(M * N[(D * 0.5), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[(M * D), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if D < 4.50000000000000013e60

   1. Initial program 85.0%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in M around 0 76.5%

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

   if 4.50000000000000013e60 < D

   1. Initial program 76.7%

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

   3. Simplified 71.9%

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

      1. unpow2 71.9%

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

      2. associate-*r/ 71.9%

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

      3. associate-/r* 71.9%

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

      4. associate-*l/ 72.0%

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

      5. associate-/l* 72.0%

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

      6. div-inv 72.0%

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

      7. metadata-eval 72.0%

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

      8. *-un-lft-identity 72.0%

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

      9. times-frac 72.0%

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

      10. metadata-eval 72.0%

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

   6. Applied egg-rr 72.0%

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

   7. Taylor expanded in M around 0 76.7%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 4.5 \cdot 10^{+60}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \frac{\left(M \cdot \left(D \cdot 0.5\right)\right) \cdot \left(0.5 \cdot \frac{M \cdot D}{d}\right)}{d}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d_m] = \mathsf{sort}([w0_m, M, D, h, l, d_m])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;M \leq 1.5 \cdot 10^{-213}:\\ \;\;\;\;w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \left(\left(M \cdot \left(D \cdot 0.5\right)\right) \cdot \left(M \cdot \frac{D \cdot 0.5}{d\_m}\right)\right) \cdot \frac{h}{d\_m \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= M 1.5e-213)
    w0_m
    (*
     w0_m
     (sqrt
      (-
       1.0
       (* (* (* M (* D 0.5)) (* M (/ (* D 0.5) d_m))) (/ h (* d_m l)))))))))

C

d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d_m);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (M <= 1.5e-213) {
		tmp = w0_m;
	} else {
		tmp = w0_m * sqrt((1.0 - (((M * (D * 0.5)) * (M * ((D * 0.5) / d_m))) * (h / (d_m * l)))));
	}
	return w0_s * tmp;
}

Fortran

d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
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 <= 1.5d-213) then
        tmp = w0_m
    else
        tmp = w0_m * sqrt((1.0d0 - (((m * (d * 0.5d0)) * (m * ((d * 0.5d0) / d_m))) * (h / (d_m * 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);
assert w0_m < M && M < D && D < h && h < l && l < d_m;
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 <= 1.5e-213) {
		tmp = w0_m;
	} else {
		tmp = w0_m * Math.sqrt((1.0 - (((M * (D * 0.5)) * (M * ((D * 0.5) / d_m))) * (h / (d_m * l)))));
	}
	return w0_s * tmp;
}

Python

d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d_m] = sort([w0_m, M, D, h, l, d_m])
def code(w0_s, w0_m, M, D, h, l, d_m):
	tmp = 0
	if M <= 1.5e-213:
		tmp = w0_m
	else:
		tmp = w0_m * math.sqrt((1.0 - (((M * (D * 0.5)) * (M * ((D * 0.5) / d_m))) * (h / (d_m * l)))))
	return w0_s * tmp

Julia

d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d_m = sort([w0_m, M, D, h, l, d_m])
function code(w0_s, w0_m, M, D, h, l, d_m)
	tmp = 0.0
	if (M <= 1.5e-213)
		tmp = w0_m;
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(Float64(M * Float64(D * 0.5)) * Float64(M * Float64(Float64(D * 0.5) / d_m))) * Float64(h / Float64(d_m * l))))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d_m = num2cell(sort([w0_m, M, D, h, l, d_m])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d_m)
	tmp = 0.0;
	if (M <= 1.5e-213)
		tmp = w0_m;
	else
		tmp = w0_m * sqrt((1.0 - (((M * (D * 0.5)) * (M * ((D * 0.5) / d_m))) * (h / (d_m * 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]
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[M, 1.5e-213], w0$95$m, N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(N[(M * N[(D * 0.5), $MachinePrecision]), $MachinePrecision] * N[(M * N[(N[(D * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if M < 1.49999999999999993e-213

   1. Initial program 87.3%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in M around 0 78.0%

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

   if 1.49999999999999993e-213 < M

   1. Initial program 78.7%

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

   3. Simplified 77.8%

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

      1. unpow2 77.8%

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

      2. associate-*r/ 77.8%

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

      3. associate-/r* 77.8%

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

      4. associate-*l/ 76.9%

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

      5. associate-/l* 76.9%

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

      6. div-inv 76.9%

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

      7. metadata-eval 76.9%

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

      8. *-un-lft-identity 76.9%

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

      9. times-frac 76.9%

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

      10. metadata-eval 76.9%

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

   6. Applied egg-rr 76.9%

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

      1. frac-times 83.4%

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

      2. associate-*l* 81.7%

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

      3. associate-*r* 81.7%

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

   8. Applied egg-rr 81.7%

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

      1. associate-/l* 75.3%

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

      2. associate-*r* 77.1%

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

      3. associate-*l* 77.1%

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

      4. associate-*r/ 77.1%

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

      5. *-commutative 77.1%

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

   10. Simplified 77.1%

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

Alternative 6: 86.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d_m] = \mathsf{sort}([w0_m, M, D, h, l, d_m])\\ \\ \begin{array}{l} t_0 := M \cdot \frac{D \cdot 0.5}{d\_m}\\ w0\_s \cdot \left(w0\_m \cdot \sqrt{1 - \frac{h \cdot \left(t\_0 \cdot t\_0\right)}{\ell}}\right) \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d_m)
 :precision binary64
 (let* ((t_0 (* M (/ (* D 0.5) d_m))))
   (* w0_s (* w0_m (sqrt (- 1.0 (/ (* h (* t_0 t_0)) l)))))))

C

d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d_m);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d_m) {
	double t_0 = M * ((D * 0.5) / d_m);
	return w0_s * (w0_m * sqrt((1.0 - ((h * (t_0 * t_0)) / l))));
}

Fortran

d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
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) :: t_0
    t_0 = m * ((d * 0.5d0) / d_m)
    code = w0_s * (w0_m * sqrt((1.0d0 - ((h * (t_0 * t_0)) / l))))
end function

Java

d_m = Math.abs(d);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
assert w0_m < M && M < D && D < h && h < l && l < d_m;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d_m) {
	double t_0 = M * ((D * 0.5) / d_m);
	return w0_s * (w0_m * Math.sqrt((1.0 - ((h * (t_0 * t_0)) / l))));
}

Python

d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d_m] = sort([w0_m, M, D, h, l, d_m])
def code(w0_s, w0_m, M, D, h, l, d_m):
	t_0 = M * ((D * 0.5) / d_m)
	return w0_s * (w0_m * math.sqrt((1.0 - ((h * (t_0 * t_0)) / l))))

Julia

d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d_m = sort([w0_m, M, D, h, l, d_m])
function code(w0_s, w0_m, M, D, h, l, d_m)
	t_0 = Float64(M * Float64(Float64(D * 0.5) / d_m))
	return Float64(w0_s * Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h * Float64(t_0 * t_0)) / l)))))
end

MATLAB

d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d_m = num2cell(sort([w0_m, M, D, h, l, d_m])){:}
function tmp = code(w0_s, w0_m, M, D, h, l, d_m)
	t_0 = M * ((D * 0.5) / d_m);
	tmp = w0_s * (w0_m * sqrt((1.0 - ((h * (t_0 * t_0)) / l))));
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]
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(M * N[(N[(D * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 83.6%

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

3. Simplified 82.5%

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

   1. associate-*r/ 87.9%

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

   2. add-sqr-sqrt 87.9%

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

   3. pow2 87.9%

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

   4. sqrt-pow1 87.9%

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

   5. metadata-eval 87.9%

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

   6. pow1 87.9%

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

   7. *-un-lft-identity 87.9%

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

   8. times-frac 87.9%

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

   9. metadata-eval 87.9%

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

6. Applied egg-rr 87.9%

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

   1. unpow2 87.9%

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

   2. associate-*r/ 87.9%

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

   3. associate-*r/ 87.9%

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

8. Applied egg-rr 87.9%

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

9. Final simplification 87.9%

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

Alternative 7: 71.0% accurate, 8.3× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d_m] = \mathsf{sort}([w0_m, M, D, h, l, d_m])\\ \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;D \leq 5.5 \cdot 10^{+126}:\\ \;\;\;\;w0\_m\\ \mathbf{else}:\\ \;\;\;\;w0\_m + \frac{-0.125 \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \left(h \cdot w0\_m\right)\right)}{\ell \cdot \left(d\_m \cdot d\_m\right)}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= D 5.5e+126)
    w0_m
    (+
     w0_m
     (/ (* -0.125 (* (* (* M D) (* M D)) (* h w0_m))) (* l (* d_m d_m)))))))

C

d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D && D < h && h < l && l < d_m);
double code(double w0_s, double w0_m, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (D <= 5.5e+126) {
		tmp = w0_m;
	} else {
		tmp = w0_m + ((-0.125 * (((M * D) * (M * D)) * (h * w0_m))) / (l * (d_m * d_m)));
	}
	return w0_s * tmp;
}

Fortran

d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
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 (d <= 5.5d+126) then
        tmp = w0_m
    else
        tmp = w0_m + (((-0.125d0) * (((m * d) * (m * d)) * (h * w0_m))) / (l * (d_m * d_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);
assert w0_m < M && M < D && D < h && h < l && l < d_m;
public static double code(double w0_s, double w0_m, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (D <= 5.5e+126) {
		tmp = w0_m;
	} else {
		tmp = w0_m + ((-0.125 * (((M * D) * (M * D)) * (h * w0_m))) / (l * (d_m * d_m)));
	}
	return w0_s * tmp;
}

Python

d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
[w0_m, M, D, h, l, d_m] = sort([w0_m, M, D, h, l, d_m])
def code(w0_s, w0_m, M, D, h, l, d_m):
	tmp = 0
	if D <= 5.5e+126:
		tmp = w0_m
	else:
		tmp = w0_m + ((-0.125 * (((M * D) * (M * D)) * (h * w0_m))) / (l * (d_m * d_m)))
	return w0_s * tmp

Julia

d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D, h, l, d_m = sort([w0_m, M, D, h, l, d_m])
function code(w0_s, w0_m, M, D, h, l, d_m)
	tmp = 0.0
	if (D <= 5.5e+126)
		tmp = w0_m;
	else
		tmp = Float64(w0_m + Float64(Float64(-0.125 * Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(h * w0_m))) / Float64(l * Float64(d_m * d_m))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
w0_m, M, D, h, l, d_m = num2cell(sort([w0_m, M, D, h, l, d_m])){:}
function tmp_2 = code(w0_s, w0_m, M, D, h, l, d_m)
	tmp = 0.0;
	if (D <= 5.5e+126)
		tmp = w0_m;
	else
		tmp = w0_m + ((-0.125 * (((M * D) * (M * D)) * (h * w0_m))) / (l * (d_m * d_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]
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[D, 5.5e+126], w0$95$m, N[(w0$95$m + N[(N[(-0.125 * N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(h * w0$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if D < 5.5000000000000004e126

   1. Initial program 84.3%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in M around 0 74.8%

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

   if 5.5000000000000004e126 < D

   1. Initial program 77.3%

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

   3. Simplified 73.3%

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

   5. Taylor expanded in M around 0 28.6%

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

      1. associate-*r/ 28.6%

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

      2. associate-*r* 28.6%

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

      3. unpow-prod-down 56.9%

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

      4. *-commutative 56.9%

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

      5. *-commutative 56.9%

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

   7. Applied egg-rr 56.9%

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

      1. *-commutative 56.9%

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

      2. unpow2 56.9%

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

   9. Applied egg-rr 56.9%

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

       1. unpow2 56.9%

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

   11. Applied egg-rr 56.9%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 5.5 \cdot 10^{+126}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + \frac{-0.125 \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \left(h \cdot w0\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.5% accurate, 216.0× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D, h, l, d_m] = \mathsf{sort}([w0_m, M, D, h, l, d_m])\\ \\ w0\_s \cdot w0\_m \end{array} \]

FPCore

d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
(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);
assert(w0_m < M && M < D && D < h && h < l && l < d_m);
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)
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
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);
assert w0_m < M && M < D && D < h && h < l && l < d_m;
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)
[w0_m, M, D, h, l, d_m] = sort([w0_m, M, D, h, l, d_m])
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)
w0_m, M, D, h, l, d_m = sort([w0_m, M, D, h, l, d_m])
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);
w0_m, M, D, h, l, d_m = num2cell(sort([w0_m, M, D, h, l, d_m])){:}
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]
NOTE: w0_m, M, D, h, l, and d_m should be sorted in increasing order before calling this function.
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 83.6%

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

3. Simplified 82.5%

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

5. Taylor expanded in M around 0 71.7%

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

Reproduce

herbie shell --seed 2024123 
(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))))))