Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.8% → 87.6%

Time: 14.7s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 10^{+132}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M\_m \cdot \frac{D\_m}{2 \cdot d\_m}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{h \cdot {\left(M\_m \cdot D\_m\right)}^{2}}{\ell}\right) + -2 \cdot \log d\_m\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ (* M_m D_m) (* 2.0 d_m)) 1e+132)
   (* w0 (sqrt (- 1.0 (* h (/ (pow (* M_m (/ D_m (* 2.0 d_m))) 2.0) l)))))
   (pow
    (*
     (cbrt w0)
     (pow
      (exp 0.16666666666666666)
      (+
       (log (* -0.25 (/ (* h (pow (* M_m D_m) 2.0)) l)))
       (* -2.0 (log d_m)))))
    3.0)))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1e+132) {
		tmp = w0 * sqrt((1.0 - (h * (pow((M_m * (D_m / (2.0 * d_m))), 2.0) / l))));
	} else {
		tmp = pow((cbrt(w0) * pow(exp(0.16666666666666666), (log((-0.25 * ((h * pow((M_m * D_m), 2.0)) / l))) + (-2.0 * log(d_m))))), 3.0);
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1e+132) {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow((M_m * (D_m / (2.0 * d_m))), 2.0) / l))));
	} else {
		tmp = Math.pow((Math.cbrt(w0) * Math.pow(Math.exp(0.16666666666666666), (Math.log((-0.25 * ((h * Math.pow((M_m * D_m), 2.0)) / l))) + (-2.0 * Math.log(d_m))))), 3.0);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) <= 1e+132)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(M_m * Float64(D_m / Float64(2.0 * d_m))) ^ 2.0) / l)))));
	else
		tmp = Float64(cbrt(w0) * (exp(0.16666666666666666) ^ Float64(log(Float64(-0.25 * Float64(Float64(h * (Float64(M_m * D_m) ^ 2.0)) / l))) + Float64(-2.0 * log(d_m))))) ^ 3.0;
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 1e+132], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(M$95$m * N[(D$95$m / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Power[N[Exp[0.16666666666666666], $MachinePrecision], N[(N[Log[N[(-0.25 * N[(N[(h * N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 9.99999999999999991e131

   1. Initial program 82.2%

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

   3. Simplified 81.8%

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

      1. unpow2 81.8%

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

      2. associate-*l* 84.0%

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

      3. associate-/r* 84.0%

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

      4. associate-*r/ 83.2%

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

      5. associate-*l/ 82.6%

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

      6. *-commutative 82.6%

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

      7. associate-/r* 82.6%

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

      8. associate-*r/ 82.6%

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

      9. associate-*l/ 83.4%

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

      10. *-commutative 83.4%

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

      11. associate-*l* 81.3%

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

      12. unpow2 81.3%

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

      13. expm1-log1p-u 68.7%

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

      14. expm1-undefine 68.7%

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

   6. Applied egg-rr 82.2%

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

      1. associate--l+ 82.2%

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

      2. metadata-eval 82.2%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} + \color{blue}{0}\right)} \]

      3. +-rgt-identity 82.2%

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

      4. associate-*l/ 88.0%

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

      5. associate-/l* 87.5%

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

      6. *-rgt-identity 87.5%

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

      7. times-frac 86.7%

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

      8. /-rgt-identity 86.7%

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

   8. Simplified 86.7%

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

   if 9.99999999999999991e131 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 52.4%

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

   3. Simplified 52.4%

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

      1. unpow2 52.4%

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

      2. associate-*l* 54.8%

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

      3. associate-/r* 54.8%

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

      4. associate-*r/ 52.4%

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

      5. associate-*l/ 54.8%

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

      6. *-commutative 54.8%

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

      7. associate-/r* 54.8%

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

      8. associate-*r/ 52.4%

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

      9. associate-*l/ 54.8%

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

      10. *-commutative 54.8%

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

      11. associate-*l* 52.4%

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

      12. unpow2 52.4%

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

      13. expm1-log1p-u 0.0%

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

      14. expm1-undefine 0.0%

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

   6. Applied egg-rr 52.4%

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

      1. associate--l+ 52.4%

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

      2. metadata-eval 52.4%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} + \color{blue}{0}\right)} \]

      3. +-rgt-identity 52.4%

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

      4. associate-*l/ 52.5%

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

      5. associate-/l* 52.5%

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

      6. *-rgt-identity 52.5%

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

      7. times-frac 52.5%

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

      8. /-rgt-identity 52.5%

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

   8. Simplified 52.5%

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

      1. add-cube-cbrt 52.5%

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

      2. pow3 52.5%

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

   10. Applied egg-rr 52.5%

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

   11. Taylor expanded in d around 0 16.2%

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

       1. exp-prod 16.2%

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

       2. distribute-lft-neg-in 16.2%

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

       3. metadata-eval 16.2%

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

       4. associate-*r* 16.3%

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

       5. unpow2 16.3%

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

       6. unpow2 16.3%

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

       7. swap-sqr 20.2%

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

       8. unpow2 20.2%

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

       9. *-commutative 20.2%

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

   13. Simplified 20.2%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{+132}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{h \cdot {\left(M \cdot D\right)}^{2}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.3% accurate, 1.0× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* h (/ (pow (* M_m (/ D_m (* 2.0 d_m))) 2.0) l))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0 * sqrt((1.0 - (h * (pow((M_m * (D_m / (2.0 * d_m))), 2.0) / l))));
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0 * sqrt((1.0d0 - (h * (((m_m * (d_m / (2.0d0 * d_m_1))) ** 2.0d0) / l))))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0 * Math.sqrt((1.0 - (h * (Math.pow((M_m * (D_m / (2.0 * d_m))), 2.0) / l))));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	return w0 * math.sqrt((1.0 - (h * (math.pow((M_m * (D_m / (2.0 * d_m))), 2.0) / l))))

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(M_m * Float64(D_m / Float64(2.0 * d_m))) ^ 2.0) / l)))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp = code(w0, M_m, D_m, h, l, d_m)
	tmp = w0 * sqrt((1.0 - (h * (((M_m * (D_m / (2.0 * d_m))) ^ 2.0) / l))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(M$95$m * N[(D$95$m / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.1%

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

3. Simplified 78.8%

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

   1. unpow2 78.8%

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

   2. associate-*l* 81.0%

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

   3. associate-/r* 81.0%

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

   4. associate-*r/ 80.0%

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

   5. associate-*l/ 79.7%

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

   6. *-commutative 79.7%

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

   7. associate-/r* 79.7%

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

   8. associate-*r/ 79.5%

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

   9. associate-*l/ 80.5%

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

   10. *-commutative 80.5%

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

   11. associate-*l* 78.3%

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

   12. unpow2 78.3%

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

   13. expm1-log1p-u 61.7%

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

   14. expm1-undefine 61.7%

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

6. Applied egg-rr 79.1%

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

   1. associate--l+ 79.1%

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

   2. metadata-eval 79.1%

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} + \color{blue}{0}\right)} \]

   3. +-rgt-identity 79.1%

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

   4. associate-*l/ 84.4%

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

   5. associate-/l* 84.0%

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

   6. *-rgt-identity 84.0%

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

   7. times-frac 83.2%

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

   8. /-rgt-identity 83.2%

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

8. Simplified 83.2%

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

9. Final simplification 83.2%

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

Alternative 3: 73.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 1.4 \cdot 10^{+163}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + \frac{-0.125 \cdot \left(\left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)\right) \cdot \left(w0 \cdot h\right)\right)}{\ell \cdot {d\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= D_m 1.4e+163)
   w0
   (+
    w0
    (/
     (* -0.125 (* (* (* M_m D_m) (* M_m D_m)) (* w0 h)))
     (* l (pow d_m 2.0))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 1.4e+163) {
		tmp = w0;
	} else {
		tmp = w0 + ((-0.125 * (((M_m * D_m) * (M_m * D_m)) * (w0 * h))) / (l * pow(d_m, 2.0)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (d_m <= 1.4d+163) then
        tmp = w0
    else
        tmp = w0 + (((-0.125d0) * (((m_m * d_m) * (m_m * d_m)) * (w0 * h))) / (l * (d_m_1 ** 2.0d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 1.4e+163) {
		tmp = w0;
	} else {
		tmp = w0 + ((-0.125 * (((M_m * D_m) * (M_m * D_m)) * (w0 * h))) / (l * Math.pow(d_m, 2.0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if D_m <= 1.4e+163:
		tmp = w0
	else:
		tmp = w0 + ((-0.125 * (((M_m * D_m) * (M_m * D_m)) * (w0 * h))) / (l * math.pow(d_m, 2.0)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (D_m <= 1.4e+163)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(Float64(-0.125 * Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) * Float64(w0 * h))) / Float64(l * (d_m ^ 2.0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (D_m <= 1.4e+163)
		tmp = w0;
	else
		tmp = w0 + ((-0.125 * (((M_m * D_m) * (M_m * D_m)) * (w0 * h))) / (l * (d_m ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[D$95$m, 1.4e+163], w0, N[(w0 + N[(N[(-0.125 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(w0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Power[d$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.40000000000000007e163

   1. Initial program 80.4%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in D around 0 70.7%

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

   if 1.40000000000000007e163 < D

   1. Initial program 68.2%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in D around 0 15.2%

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

      1. associate-*r/ 15.2%

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

      2. associate-*r* 15.2%

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

      3. pow-prod-down 67.3%

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

      4. *-commutative 67.3%

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

   7. Applied egg-rr 67.3%

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

      1. unpow2 67.3%

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

   9. Applied egg-rr 67.3%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.4 \cdot 10^{+163}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + \frac{-0.125 \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \left(w0 \cdot h\right)\right)}{\ell \cdot {d}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.3% accurate, 216.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ w0 \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m) :precision binary64 w0)

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	return w0

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	return w0
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp = code(w0, M_m, D_m, h, l, d_m)
	tmp = w0;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := w0

Derivation

1. Initial program 79.1%

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

3. Simplified 78.8%

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

5. Taylor expanded in D around 0 68.1%

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

Reproduce

herbie shell --seed 2024137 
(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))))))