Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 89.3%

Time: 14.9s

Alternatives: 4

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: 81.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ [w0_m, M, D_m, h, l, d] = \mathsf{sort}([w0_m, M, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := M \cdot \left(D\_m \cdot \frac{0.5}{d}\right)\\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;w0\_m \cdot \sqrt{1 - {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 5 \cdot 10^{+183}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - \frac{h \cdot t\_0}{\frac{\ell}{t\_0}}}\\ \end{array} \end{array} \end{array} \]

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_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D_m h l d)
 :precision binary64
 (let* ((t_0 (* M (* D_m (/ 0.5 d)))))
   (*
    w0_s
    (if (<=
         (* w0_m (sqrt (- 1.0 (* (pow (/ (* M D_m) (* 2.0 d)) 2.0) (/ h l)))))
         5e+183)
      (* w0_m (sqrt (- 1.0 (* (/ h l) (pow (* D_m (/ (/ M 2.0) d)) 2.0)))))
      (* w0_m (sqrt (- 1.0 (/ (* h t_0) (/ l t_0)))))))))

C

D_m = fabs(D);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
assert(w0_m < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M, double D_m, double h, double l, double d) {
	double t_0 = M * (D_m * (0.5 / d));
	double tmp;
	if ((w0_m * sqrt((1.0 - (pow(((M * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 5e+183) {
		tmp = w0_m * sqrt((1.0 - ((h / l) * pow((D_m * ((M / 2.0) / d)), 2.0))));
	} else {
		tmp = w0_m * sqrt((1.0 - ((h * t_0) / (l / t_0))));
	}
	return w0_s * tmp;
}

Fortran

D_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
NOTE: w0_m, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0_s, w0_m, m, d_m, h, l, d)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * (d_m * (0.5d0 / d))
    if ((w0_m * sqrt((1.0d0 - ((((m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l))))) <= 5d+183) then
        tmp = w0_m * sqrt((1.0d0 - ((h / l) * ((d_m * ((m / 2.0d0) / d)) ** 2.0d0))))
    else
        tmp = w0_m * sqrt((1.0d0 - ((h * t_0) / (l / t_0))))
    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_m && D_m < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D_m, double h, double l, double d) {
	double t_0 = M * (D_m * (0.5 / d));
	double tmp;
	if ((w0_m * Math.sqrt((1.0 - (Math.pow(((M * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 5e+183) {
		tmp = w0_m * Math.sqrt((1.0 - ((h / l) * Math.pow((D_m * ((M / 2.0) / d)), 2.0))));
	} else {
		tmp = w0_m * Math.sqrt((1.0 - ((h * t_0) / (l / t_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_m, h, l, d] = sort([w0_m, M, D_m, h, l, d])
def code(w0_s, w0_m, M, D_m, h, l, d):
	t_0 = M * (D_m * (0.5 / d))
	tmp = 0
	if (w0_m * math.sqrt((1.0 - (math.pow(((M * D_m) / (2.0 * d)), 2.0) * (h / l))))) <= 5e+183:
		tmp = w0_m * math.sqrt((1.0 - ((h / l) * math.pow((D_m * ((M / 2.0) / d)), 2.0))))
	else:
		tmp = w0_m * math.sqrt((1.0 - ((h * t_0) / (l / t_0))))
	return w0_s * tmp

Julia

D_m = abs(D)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D_m, h, l, d = sort([w0_m, M, D_m, h, l, d])
function code(w0_s, w0_m, M, D_m, h, l, d)
	t_0 = Float64(M * Float64(D_m * Float64(0.5 / d)))
	tmp = 0.0
	if (Float64(w0_m * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))) <= 5e+183)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D_m * Float64(Float64(M / 2.0) / d)) ^ 2.0)))));
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(Float64(h * t_0) / Float64(l / t_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_m, h, l, d = num2cell(sort([w0_m, M, D_m, h, l, d])){:}
function tmp_2 = code(w0_s, w0_m, M, D_m, h, l, d)
	t_0 = M * (D_m * (0.5 / d));
	tmp = 0.0;
	if ((w0_m * sqrt((1.0 - ((((M * D_m) / (2.0 * d)) ^ 2.0) * (h / l))))) <= 5e+183)
		tmp = w0_m * sqrt((1.0 - ((h / l) * ((D_m * ((M / 2.0) / d)) ^ 2.0))));
	else
		tmp = w0_m * sqrt((1.0 - ((h * t_0) / (l / t_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_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(M * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e+183], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(N[(h * t$95$0), $MachinePrecision] / N[(l / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.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))))) < 5.00000000000000009e183

   1. Initial program 95.7%

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

   3. Simplified 95.1%

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

   if 5.00000000000000009e183 < (*.f64 w0 (sqrt.f64 (-.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 57.1%

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

   3. Simplified 58.8%

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

      1. clear-num 58.8%

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

      2. un-div-inv 62.5%

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

      3. clear-num 62.5%

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

      4. un-div-inv 62.5%

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

      5. associate-/r/ 62.5%

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

   6. Applied egg-rr 62.5%

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

      1. associate-/r/ 79.0%

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

      2. associate-*l/ 79.0%

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

      3. *-commutative 79.0%

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

      4. associate-/l* 79.0%

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

      5. associate-*l/ 79.0%

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

      6. *-commutative 79.0%

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

      7. associate-*l/ 79.0%

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

      8. associate-/l/ 79.0%

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

      9. associate-/r/ 79.0%

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

      10. associate-*l/ 79.0%

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

      11. *-commutative 79.0%

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

      12. associate-*r/ 77.3%

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

      13. *-commutative 77.3%

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

      14. associate-/l/ 77.3%

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

      15. associate-/l* 79.0%

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

   8. Simplified 79.0%

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

      1. unpow2 79.0%

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

      2. associate-/l* 82.5%

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

      3. div-inv 82.5%

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

      4. associate-/r* 82.5%

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

      5. metadata-eval 82.5%

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

      6. div-inv 82.6%

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

      7. associate-/r* 82.6%

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

      8. metadata-eval 82.6%

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

   10. Applied egg-rr 82.6%

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

   11. Taylor expanded in D around 0 75.5%

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

       1. associate-*r/ 75.5%

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

       2. associate-/r* 79.1%

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

       3. associate-*l/ 79.1%

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

       4. *-commutative 79.1%

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

       5. associate-*r* 82.6%

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

       6. associate-*r/ 80.9%

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

       7. associate-*r/ 80.9%

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

       8. associate-/l/ 77.6%

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

       9. *-commutative 77.6%

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

       10. *-commutative 77.6%

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

   13. Simplified 77.6%

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

       1. associate-*r* 77.2%

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

       2. div-inv 77.2%

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

       3. associate-*r* 77.2%

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

       4. metadata-eval 77.2%

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

       5. div-inv 77.2%

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

       6. clear-num 77.2%

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

       7. div-inv 77.2%

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

       8. associate-/r* 77.2%

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

       9. associate-/r* 77.4%

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

       10. *-commutative 77.4%

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

       11. associate-*r/ 77.4%

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

       12. associate-/l* 79.1%

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

       13. associate-*r* 79.1%

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

       14. clear-num 79.1%

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

   15. Applied egg-rr 86.1%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 5 \cdot 10^{+183}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}{\frac{\ell}{M \cdot \left(D \cdot \frac{0.5}{d}\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.6% accurate, 1.8× speedup

Math

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

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_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D_m h l d)
 :precision binary64
 (let* ((t_0 (* D_m (* M (/ 0.5 d)))))
   (* 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_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M * (0.5 / d));
	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_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0_s, w0_m, m, d_m, h, l, d)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    t_0 = d_m * (m * (0.5d0 / d))
    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_m && D_m < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M * (0.5 / d));
	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_m, h, l, d] = sort([w0_m, M, D_m, h, l, d])
def code(w0_s, w0_m, M, D_m, h, l, d):
	t_0 = D_m * (M * (0.5 / d))
	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_m, h, l, d = sort([w0_m, M, D_m, h, l, d])
function code(w0_s, w0_m, M, D_m, h, l, d)
	t_0 = Float64(D_m * Float64(M * Float64(0.5 / d)))
	return Float64(w0_s * Float64(w0_m * sqrt(Float64(1.0 - Float64(h * Float64(t_0 * Float64(t_0 / l)))))))
end

MATLAB

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

Derivation

1. Initial program 87.4%

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

3. Simplified 87.3%

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

   1. clear-num 86.9%

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

   2. un-div-inv 87.7%

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

   3. clear-num 87.7%

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

   4. un-div-inv 87.7%

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

   5. associate-/r/ 87.7%

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

6. Applied egg-rr 87.7%

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

   1. associate-/r/ 90.9%

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

   2. associate-*l/ 88.7%

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

   3. *-commutative 88.7%

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

   4. associate-/l* 90.9%

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

   5. associate-*l/ 90.9%

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

   6. *-commutative 90.9%

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

   7. associate-*l/ 90.9%

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

   8. associate-/l/ 90.3%

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

   9. associate-/r/ 90.3%

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

   10. associate-*l/ 90.3%

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

   11. *-commutative 90.3%

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

   12. associate-*r/ 91.4%

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

   13. *-commutative 91.4%

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

   14. associate-/l/ 91.4%

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

   15. associate-/l* 90.9%

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

8. Simplified 90.9%

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

   1. unpow2 90.9%

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

   2. associate-/l* 91.7%

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

   3. div-inv 91.7%

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

   4. associate-/r* 91.7%

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

   5. metadata-eval 91.7%

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

   6. div-inv 91.7%

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

   7. associate-/r* 91.7%

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

   8. metadata-eval 91.7%

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

10. Applied egg-rr 91.7%

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

Alternative 3: 84.4% accurate, 1.8× speedup

Math

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

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_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D_m h l d)
 :precision binary64
 (*
  w0_s
  (*
   w0_m
   (sqrt
    (- 1.0 (* h (* (* D_m (* M (/ 0.5 d))) (* D_m (/ (* M 0.5) (* d l))))))))))

C

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

Fortran

D_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
NOTE: w0_m, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0_s, w0_m, m, d_m, h, l, d)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0_s * (w0_m * sqrt((1.0d0 - (h * ((d_m * (m * (0.5d0 / d))) * (d_m * ((m * 0.5d0) / (d * 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_m && D_m < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D_m, double h, double l, double d) {
	return w0_s * (w0_m * Math.sqrt((1.0 - (h * ((D_m * (M * (0.5 / d))) * (D_m * ((M * 0.5) / (d * l))))))));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 87.4%

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

3. Simplified 87.3%

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

   1. clear-num 86.9%

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

   2. un-div-inv 87.7%

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

   3. clear-num 87.7%

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

   4. un-div-inv 87.7%

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

   5. associate-/r/ 87.7%

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

6. Applied egg-rr 87.7%

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

   1. associate-/r/ 90.9%

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

   2. associate-*l/ 88.7%

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

   3. *-commutative 88.7%

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

   4. associate-/l* 90.9%

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

   5. associate-*l/ 90.9%

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

   6. *-commutative 90.9%

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

   7. associate-*l/ 90.9%

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

   8. associate-/l/ 90.3%

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

   9. associate-/r/ 90.3%

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

   10. associate-*l/ 90.3%

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

   11. *-commutative 90.3%

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

   12. associate-*r/ 91.4%

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

   13. *-commutative 91.4%

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

   14. associate-/l/ 91.4%

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

   15. associate-/l* 90.9%

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

8. Simplified 90.9%

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

   1. unpow2 90.9%

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

   2. associate-/l* 91.7%

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

   3. div-inv 91.7%

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

   4. associate-/r* 91.7%

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

   5. metadata-eval 91.7%

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

   6. div-inv 91.7%

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

   7. associate-/r* 91.7%

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

   8. metadata-eval 91.7%

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

10. Applied egg-rr 91.7%

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

11. Taylor expanded in D around 0 88.7%

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

    1. associate-*r/ 88.7%

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

    2. associate-/r* 90.6%

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

    3. associate-*l/ 90.6%

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

    4. *-commutative 90.6%

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

    5. associate-*r* 91.7%

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

    6. associate-*r/ 89.1%

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

    7. associate-*r/ 89.1%

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

    8. associate-/l/ 88.4%

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

    9. *-commutative 88.4%

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

    10. *-commutative 88.4%

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

13. Simplified 88.4%

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

14. Final simplification 88.4%

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

Alternative 4: 68.2% accurate, 216.0× speedup

Math

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

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_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0_s w0_m M D_m h l d) :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_m && D_m < h && h < l && l < d);
double code(double w0_s, double w0_m, double M, double D_m, double h, double l, double d) {
	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_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0_s, w0_m, m, d_m, h, l, d)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    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_m && D_m < h && h < l && l < d;
public static double code(double w0_s, double w0_m, double M, double D_m, double h, double l, double d) {
	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_m, h, l, d] = sort([w0_m, M, D_m, h, l, d])
def code(w0_s, w0_m, M, D_m, h, l, d):
	return w0_s * w0_m

Julia

D_m = abs(D)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
w0_m, M, D_m, h, l, d = sort([w0_m, M, D_m, h, l, d])
function code(w0_s, w0_m, M, D_m, h, l, d)
	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_m, h, l, d = num2cell(sort([w0_m, M, D_m, h, l, d])){:}
function tmp = code(w0_s, w0_m, M, D_m, h, l, d)
	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_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0$95$s_, w0$95$m_, M_, D$95$m_, h_, l_, d_] := N[(w0$95$s * w0$95$m), $MachinePrecision]

Derivation

1. Initial program 87.4%

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

3. Simplified 87.3%

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

5. Taylor expanded in D around 0 73.1%

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

Reproduce

herbie shell --seed 2024093 
(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))))))