Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.7% → 92.7%

Time: 17.0s

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: 81.7% 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: 92.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+44}:\\ \;\;\;\;w0 \cdot \left(D\_m \cdot \left(\sqrt{\frac{h}{\ell} \cdot -0.25} \cdot \frac{\left|M\_m\right|}{d\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(\frac{D\_m}{d\_m} \cdot \frac{M\_m}{2}\right) \cdot \left(\frac{D\_m}{d\_m} \cdot \left(M\_m \cdot \frac{0.5}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -5e+44)
   (* w0 (* D_m (* (sqrt (* (/ h l) -0.25)) (/ (fabs M_m) d_m))))
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       h
       (* (* (/ D_m d_m) (/ M_m 2.0)) (* (/ D_m d_m) (* M_m (/ 0.5 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) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -5e+44) {
		tmp = w0 * (D_m * (sqrt(((h / l) * -0.25)) * (fabs(M_m) / d_m)));
	} else {
		tmp = w0 * sqrt((1.0 - (h * (((D_m / d_m) * (M_m / 2.0)) * ((D_m / d_m) * (M_m * (0.5 / l)))))));
	}
	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 (((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)) <= (-5d+44)) then
        tmp = w0 * (d_m * (sqrt(((h / l) * (-0.25d0))) * (abs(m_m) / d_m_1)))
    else
        tmp = w0 * sqrt((1.0d0 - (h * (((d_m / d_m_1) * (m_m / 2.0d0)) * ((d_m / d_m_1) * (m_m * (0.5d0 / l)))))))
    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 ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -5e+44) {
		tmp = w0 * (D_m * (Math.sqrt(((h / l) * -0.25)) * (Math.abs(M_m) / d_m)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * (((D_m / d_m) * (M_m / 2.0)) * ((D_m / d_m) * (M_m * (0.5 / l)))))));
	}
	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 (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -5e+44:
		tmp = w0 * (D_m * (math.sqrt(((h / l) * -0.25)) * (math.fabs(M_m) / d_m)))
	else:
		tmp = w0 * math.sqrt((1.0 - (h * (((D_m / d_m) * (M_m / 2.0)) * ((D_m / d_m) * (M_m * (0.5 / l)))))))
	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(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -5e+44)
		tmp = Float64(w0 * Float64(D_m * Float64(sqrt(Float64(Float64(h / l) * -0.25)) * Float64(abs(M_m) / d_m))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(Float64(D_m / d_m) * Float64(M_m / 2.0)) * Float64(Float64(D_m / d_m) * Float64(M_m * Float64(0.5 / l))))))));
	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 (((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -5e+44)
		tmp = w0 * (D_m * (sqrt(((h / l) * -0.25)) * (abs(M_m) / d_m)));
	else
		tmp = w0 * sqrt((1.0 - (h * (((D_m / d_m) * (M_m / 2.0)) * ((D_m / d_m) * (M_m * (0.5 / l)))))));
	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[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -5e+44], N[(w0 * N[(D$95$m * N[(N[Sqrt[N[(N[(h / l), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[M$95$m], $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(M$95$m * N[(0.5 / l), $MachinePrecision]), $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)) < -4.9999999999999996e44

   1. Initial program 56.6%

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

   3. Simplified 56.5%

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

   5. Taylor expanded in D around inf 36.3%

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

      1. *-commutative 36.3%

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

      2. times-frac 33.8%

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

   7. Simplified 33.8%

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

      1. pow1/2 33.8%

         \[\leadsto w0 \cdot \color{blue}{{\left(\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{M}^{2} \cdot h}{\ell}\right) \cdot -0.25\right)}^{0.5}} \]

      2. associate-*l* 33.8%

         \[\leadsto w0 \cdot {\color{blue}{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \left(\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25\right)\right)}}^{0.5} \]

      3. unpow-prod-down 36.2%

         \[\leadsto w0 \cdot \color{blue}{\left({\left(\frac{{D}^{2}}{{d}^{2}}\right)}^{0.5} \cdot {\left(\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25\right)}^{0.5}\right)} \]

      4. pow1/2 36.2%

         \[\leadsto w0 \cdot \left(\color{blue}{\sqrt{\frac{{D}^{2}}{{d}^{2}}}} \cdot {\left(\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25\right)}^{0.5}\right) \]

      5. sqrt-div 38.6%

         \[\leadsto w0 \cdot \left(\color{blue}{\frac{\sqrt{{D}^{2}}}{\sqrt{{d}^{2}}}} \cdot {\left(\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25\right)}^{0.5}\right) \]

      6. sqrt-pow1 21.9%

         \[\leadsto w0 \cdot \left(\frac{\color{blue}{{D}^{\left(\frac{2}{2}\right)}}}{\sqrt{{d}^{2}}} \cdot {\left(\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25\right)}^{0.5}\right) \]

      7. metadata-eval 21.9%

         \[\leadsto w0 \cdot \left(\frac{{D}^{\color{blue}{1}}}{\sqrt{{d}^{2}}} \cdot {\left(\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25\right)}^{0.5}\right) \]

      8. pow1 21.9%

         \[\leadsto w0 \cdot \left(\frac{\color{blue}{D}}{\sqrt{{d}^{2}}} \cdot {\left(\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25\right)}^{0.5}\right) \]

      9. sqrt-pow1 34.0%

         \[\leadsto w0 \cdot \left(\frac{D}{\color{blue}{{d}^{\left(\frac{2}{2}\right)}}} \cdot {\left(\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25\right)}^{0.5}\right) \]

      10. metadata-eval 34.0%

          \[\leadsto w0 \cdot \left(\frac{D}{{d}^{\color{blue}{1}}} \cdot {\left(\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25\right)}^{0.5}\right) \]

      11. pow1 34.0%

          \[\leadsto w0 \cdot \left(\frac{D}{\color{blue}{d}} \cdot {\left(\frac{{M}^{2} \cdot h}{\ell} \cdot -0.25\right)}^{0.5}\right) \]

      12. associate-/l* 34.0%

          \[\leadsto w0 \cdot \left(\frac{D}{d} \cdot {\left(\color{blue}{\left({M}^{2} \cdot \frac{h}{\ell}\right)} \cdot -0.25\right)}^{0.5}\right) \]

   9. Applied egg-rr 34.0%

      \[\leadsto w0 \cdot \color{blue}{\left(\frac{D}{d} \cdot {\left(\left({M}^{2} \cdot \frac{h}{\ell}\right) \cdot -0.25\right)}^{0.5}\right)} \]
   10. Step-by-step derivation

       1. unpow1/2 34.0%

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

       2. associate-*l* 34.0%

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

   11. Simplified 34.0%

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

       1. sqrt-prod 37.6%

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

   13. Applied egg-rr 37.6%

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

       1. unpow2 37.6%

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

       2. rem-sqrt-square 48.7%

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

       3. associate-*l/ 48.7%

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

   15. Simplified 48.7%

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

       1. pow1 48.7%

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

       2. associate-*r* 51.0%

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

       3. associate-/l* 51.0%

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

   17. Applied egg-rr 51.0%

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

       1. unpow1 51.0%

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

       2. associate-*l* 48.7%

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

       3. associate-*l/ 45.1%

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

       4. associate-/l* 47.6%

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

       5. *-commutative 47.6%

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

       6. associate-/l* 47.5%

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

       7. associate-*r/ 47.5%

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

       8. *-commutative 47.5%

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

       9. associate-/l* 47.5%

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

   19. Simplified 47.5%

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

   if -4.9999999999999996e44 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 85.1%

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

   3. Simplified 86.3%

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

   6. Applied egg-rr 86.3%

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

      1. unpow1 86.3%

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

      2. associate-*l/ 94.8%

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

      3. associate-/l* 94.8%

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

      4. associate-*r/ 93.7%

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

      5. *-rgt-identity 93.7%

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

      6. times-frac 94.8%

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

      7. /-rgt-identity 94.8%

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

   8. Simplified 94.8%

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

      1. unpow2 94.8%

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

      2. *-un-lft-identity 94.8%

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

      3. times-frac 97.3%

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

      4. associate-*l/ 96.1%

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

      5. *-commutative 96.1%

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

      6. times-frac 97.3%

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

      7. associate-*l/ 96.1%

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

      8. *-commutative 96.1%

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

      9. times-frac 97.3%

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

   10. Applied egg-rr 97.3%

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

   11. Taylor expanded in D around 0 91.2%

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

       1. *-commutative 91.2%

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

       2. associate-/r* 96.1%

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

       3. associate-*l/ 97.3%

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

       4. associate-*l/ 97.3%

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

       5. associate-*r* 97.3%

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

       6. associate-*r/ 93.8%

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

       7. associate-/l* 93.8%

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

   13. Simplified 93.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+44}:\\ \;\;\;\;w0 \cdot \left(D \cdot \left(\sqrt{\frac{h}{\ell} \cdot -0.25} \cdot \frac{\left|M\right|}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot \frac{0.5}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;d\_m \leq 1.5 \cdot 10^{-128}:\\ \;\;\;\;w0 \cdot \left(1 + h \cdot \left({\left(D\_m \cdot \frac{M\_m}{d\_m}\right)}^{2} \cdot \frac{-0.125}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(0.5 \cdot \frac{M\_m \cdot D\_m}{d\_m}\right) \cdot \left(0.5 \cdot \frac{M\_m \cdot D\_m}{d\_m \cdot \ell}\right)\right)}\\ \end{array} \end{array} \]

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 1.5e-128)
   (* w0 (+ 1.0 (* h (* (pow (* D_m (/ M_m d_m)) 2.0) (/ -0.125 l)))))
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       h
       (* (* 0.5 (/ (* M_m D_m) d_m)) (* 0.5 (/ (* M_m D_m) (* d_m 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) {
	double tmp;
	if (d_m <= 1.5e-128) {
		tmp = w0 * (1.0 + (h * (pow((D_m * (M_m / d_m)), 2.0) * (-0.125 / l))));
	} else {
		tmp = w0 * sqrt((1.0 - (h * ((0.5 * ((M_m * D_m) / d_m)) * (0.5 * ((M_m * D_m) / (d_m * l)))))));
	}
	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 <= 1.5d-128) then
        tmp = w0 * (1.0d0 + (h * (((d_m * (m_m / d_m_1)) ** 2.0d0) * ((-0.125d0) / l))))
    else
        tmp = w0 * sqrt((1.0d0 - (h * ((0.5d0 * ((m_m * d_m) / d_m_1)) * (0.5d0 * ((m_m * d_m) / (d_m_1 * l)))))))
    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 <= 1.5e-128) {
		tmp = w0 * (1.0 + (h * (Math.pow((D_m * (M_m / d_m)), 2.0) * (-0.125 / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * ((0.5 * ((M_m * D_m) / d_m)) * (0.5 * ((M_m * D_m) / (d_m * l)))))));
	}
	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 <= 1.5e-128:
		tmp = w0 * (1.0 + (h * (math.pow((D_m * (M_m / d_m)), 2.0) * (-0.125 / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - (h * ((0.5 * ((M_m * D_m) / d_m)) * (0.5 * ((M_m * D_m) / (d_m * l)))))))
	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 <= 1.5e-128)
		tmp = Float64(w0 * Float64(1.0 + Float64(h * Float64((Float64(D_m * Float64(M_m / d_m)) ^ 2.0) * Float64(-0.125 / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(0.5 * Float64(Float64(M_m * D_m) / d_m)) * Float64(0.5 * Float64(Float64(M_m * D_m) / Float64(d_m * l))))))));
	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 <= 1.5e-128)
		tmp = w0 * (1.0 + (h * (((D_m * (M_m / d_m)) ^ 2.0) * (-0.125 / l))));
	else
		tmp = w0 * sqrt((1.0 - (h * ((0.5 * ((M_m * D_m) / d_m)) * (0.5 * ((M_m * D_m) / (d_m * l)))))));
	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, 1.5e-128], N[(w0 * N[(1.0 + N[(h * N[(N[Power[N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.125 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.49999999999999989e-128

   1. Initial program 71.8%

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

   3. Simplified 73.0%

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

   6. Applied egg-rr 73.0%

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

      1. unpow1 73.0%

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

      2. associate-*l/ 77.6%

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

      3. associate-/l* 78.8%

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

      4. associate-*r/ 78.2%

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

      5. *-rgt-identity 78.2%

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

      6. times-frac 78.8%

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

      7. /-rgt-identity 78.8%

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

   8. Simplified 78.8%

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

   9. Taylor expanded in h around 0 44.3%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 44.3%

          \[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

       2. associate-*r* 45.6%

          \[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}\right) \]

       3. unpow2 45.6%

          \[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h\right)}{{d}^{2} \cdot \ell}\right) \]

       4. unpow2 45.6%

          \[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{{d}^{2} \cdot \ell}\right) \]

       5. swap-sqr 56.9%

          \[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right)}{{d}^{2} \cdot \ell}\right) \]

       6. unpow2 56.9%

          \[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h\right)}{{d}^{2} \cdot \ell}\right) \]

       7. associate-*r* 56.9%

          \[\leadsto w0 \cdot \left(1 + \frac{\color{blue}{\left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right) \]

   11. Simplified 56.9%

       \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{\left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right) \cdot h}{{d}^{2} \cdot \ell}\right)} \]

   12. Taylor expanded in D around 0 44.3%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   13. Step-by-step derivation

       1. associate-*r* 45.6%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right) \]

       2. unpow2 45.6%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{{d}^{2} \cdot \ell}\right) \]

       3. unpow2 45.6%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{{d}^{2} \cdot \ell}\right) \]

       4. swap-sqr 56.9%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h}{{d}^{2} \cdot \ell}\right) \]

       5. unpow2 56.9%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{{d}^{2} \cdot \ell}\right) \]

       6. associate-*r/ 58.8%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{d}^{2} \cdot \ell}\right)}\right) \]

       7. *-commutative 58.8%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{\color{blue}{\ell \cdot {d}^{2}}}\right)\right) \]

       8. associate-*l* 58.8%

          \[\leadsto w0 \cdot \left(1 + \color{blue}{\left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell \cdot {d}^{2}}}\right) \]

       9. associate-*r/ 56.9%

          \[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{\left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right) \cdot h}{\ell \cdot {d}^{2}}}\right) \]

       10. *-commutative 56.9%

           \[\leadsto w0 \cdot \left(1 + \frac{\color{blue}{h \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}}{\ell \cdot {d}^{2}}\right) \]

       11. *-commutative 56.9%

           \[\leadsto w0 \cdot \left(1 + \frac{h \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right) \]

       12. associate-*r/ 59.4%

           \[\leadsto w0 \cdot \left(1 + \color{blue}{h \cdot \frac{-0.125 \cdot {\left(D \cdot M\right)}^{2}}{{d}^{2} \cdot \ell}}\right) \]

       13. *-commutative 59.4%

           \[\leadsto w0 \cdot \left(1 + h \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2} \cdot -0.125}}{{d}^{2} \cdot \ell}\right) \]

       14. times-frac 60.6%

           \[\leadsto w0 \cdot \left(1 + h \cdot \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}} \cdot \frac{-0.125}{\ell}\right)}\right) \]

   14. Simplified 72.3%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{-0.125}{\ell}\right)}\right) \]

   if 1.49999999999999989e-128 < d

   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 84.3%

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

   6. Applied egg-rr 84.3%

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

      1. unpow1 84.3%

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

      2. associate-*l/ 91.3%

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

      3. associate-/l* 90.2%

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

      4. associate-*r/ 90.2%

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

      5. *-rgt-identity 90.2%

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

      6. times-frac 90.2%

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

      7. /-rgt-identity 90.2%

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

   8. Simplified 90.2%

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

      1. unpow2 90.2%

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

      2. *-un-lft-identity 90.2%

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

      3. times-frac 93.4%

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

      4. associate-*l/ 93.4%

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

      5. *-commutative 93.4%

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

      6. times-frac 93.4%

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

      7. associate-*l/ 93.4%

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

      8. *-commutative 93.4%

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

      9. times-frac 93.4%

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

   10. Applied egg-rr 93.4%

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

   11. Taylor expanded in D around 0 93.4%

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

   12. Taylor expanded in D around 0 92.4%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.5 \cdot 10^{-128}:\\ \;\;\;\;w0 \cdot \left(1 + h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{-0.125}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \left(0.5 \cdot \frac{M \cdot D}{d \cdot \ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{d\_m} \cdot \frac{M\_m}{2}\\ w0 \cdot \sqrt{1 - h \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)} \end{array} \end{array} \]

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
 (let* ((t_0 (* (/ D_m d_m) (/ M_m 2.0))))
   (* w0 (sqrt (- 1.0 (* h (* t_0 (/ t_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) {
	double t_0 = (D_m / d_m) * (M_m / 2.0);
	return w0 * sqrt((1.0 - (h * (t_0 * (t_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
    real(8) :: t_0
    t_0 = (d_m / d_m_1) * (m_m / 2.0d0)
    code = w0 * sqrt((1.0d0 - (h * (t_0 * (t_0 / 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) {
	double t_0 = (D_m / d_m) * (M_m / 2.0);
	return w0 * Math.sqrt((1.0 - (h * (t_0 * (t_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):
	t_0 = (D_m / d_m) * (M_m / 2.0)
	return w0 * math.sqrt((1.0 - (h * (t_0 * (t_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)
	t_0 = Float64(Float64(D_m / d_m) * Float64(M_m / 2.0))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(t_0 * Float64(t_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)
	t_0 = (D_m / d_m) * (M_m / 2.0);
	tmp = w0 * sqrt((1.0 - (h * (t_0 * (t_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_] := Block[{t$95$0 = N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.1%

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

3. Simplified 76.8%

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

6. Applied egg-rr 76.8%

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

   1. unpow1 76.8%

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

   2. associate-*l/ 82.3%

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

   3. associate-/l* 82.7%

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

   4. associate-*r/ 82.3%

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

   5. *-rgt-identity 82.3%

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

   6. times-frac 82.7%

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

   7. /-rgt-identity 82.7%

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

8. Simplified 82.7%

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

   1. unpow2 82.7%

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

   2. *-un-lft-identity 82.7%

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

   3. times-frac 85.1%

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

   4. associate-*l/ 84.4%

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

   5. *-commutative 84.4%

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

   6. times-frac 85.1%

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

   7. associate-*l/ 84.4%

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

   8. *-commutative 84.4%

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

   9. times-frac 85.1%

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

10. Applied egg-rr 85.1%

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

11. Final simplification 85.1%

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

Alternative 4: 86.3% accurate, 1.8× speedup

Math

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

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 (* (/ (* (/ D_m d_m) (/ M_m 2.0)) l) (* 0.5 (/ (* M_m D_m) d_m))))))))

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 * ((((D_m / d_m) * (M_m / 2.0)) / l) * (0.5 * ((M_m * D_m) / d_m))))));
}

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 * ((((d_m / d_m_1) * (m_m / 2.0d0)) / l) * (0.5d0 * ((m_m * d_m) / d_m_1))))))
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 * ((((D_m / d_m) * (M_m / 2.0)) / l) * (0.5 * ((M_m * D_m) / d_m))))));
}

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 * ((((D_m / d_m) * (M_m / 2.0)) / l) * (0.5 * ((M_m * D_m) / d_m))))))

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(Float64(Float64(D_m / d_m) * Float64(M_m / 2.0)) / l) * Float64(0.5 * Float64(Float64(M_m * D_m) / d_m)))))))
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 * ((((D_m / d_m) * (M_m / 2.0)) / l) * (0.5 * ((M_m * D_m) / d_m))))));
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[(N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.1%

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

3. Simplified 76.8%

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

6. Applied egg-rr 76.8%

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

   1. unpow1 76.8%

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

   2. associate-*l/ 82.3%

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

   3. associate-/l* 82.7%

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

   4. associate-*r/ 82.3%

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

   5. *-rgt-identity 82.3%

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

   6. times-frac 82.7%

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

   7. /-rgt-identity 82.7%

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

8. Simplified 82.7%

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

   1. unpow2 82.7%

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

   2. *-un-lft-identity 82.7%

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

   3. times-frac 85.1%

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

   4. associate-*l/ 84.4%

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

   5. *-commutative 84.4%

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

   6. times-frac 85.1%

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

   7. associate-*l/ 84.4%

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

   8. *-commutative 84.4%

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

   9. times-frac 85.1%

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

10. Applied egg-rr 85.1%

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

11. Taylor expanded in D around 0 84.4%

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

12. Final simplification 84.4%

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

Alternative 5: 85.2% accurate, 1.8× speedup

Math

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

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 (* (* (/ D_m d_m) (* M_m (/ 0.5 l))) (* 0.5 (/ (* M_m D_m) d_m))))))))

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 * (((D_m / d_m) * (M_m * (0.5 / l))) * (0.5 * ((M_m * D_m) / d_m))))));
}

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 * (((d_m / d_m_1) * (m_m * (0.5d0 / l))) * (0.5d0 * ((m_m * d_m) / d_m_1))))))
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 * (((D_m / d_m) * (M_m * (0.5 / l))) * (0.5 * ((M_m * D_m) / d_m))))));
}

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 * (((D_m / d_m) * (M_m * (0.5 / l))) * (0.5 * ((M_m * D_m) / d_m))))))

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(Float64(D_m / d_m) * Float64(M_m * Float64(0.5 / l))) * Float64(0.5 * Float64(Float64(M_m * D_m) / d_m)))))))
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 * (((D_m / d_m) * (M_m * (0.5 / l))) * (0.5 * ((M_m * D_m) / d_m))))));
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[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(M$95$m * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.1%

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

3. Simplified 76.8%

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

6. Applied egg-rr 76.8%

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

   1. unpow1 76.8%

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

   2. associate-*l/ 82.3%

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

   3. associate-/l* 82.7%

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

   4. associate-*r/ 82.3%

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

   5. *-rgt-identity 82.3%

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

   6. times-frac 82.7%

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

   7. /-rgt-identity 82.7%

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

8. Simplified 82.7%

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

   1. unpow2 82.7%

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

   2. *-un-lft-identity 82.7%

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

   3. times-frac 85.1%

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

   4. associate-*l/ 84.4%

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

   5. *-commutative 84.4%

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

   6. times-frac 85.1%

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

   7. associate-*l/ 84.4%

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

   8. *-commutative 84.4%

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

   9. times-frac 85.1%

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

10. Applied egg-rr 85.1%

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

11. Taylor expanded in D around 0 84.4%

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

12. Taylor expanded in D around 0 81.0%

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

    1. *-commutative 80.3%

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

    2. associate-/r* 84.4%

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

    3. associate-*l/ 85.1%

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

    4. associate-*l/ 85.1%

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

    5. associate-*r* 85.1%

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

    6. associate-*r/ 81.6%

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

    7. associate-/l* 81.6%

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

14. Simplified 80.8%

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

15. Final simplification 80.8%

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

Alternative 6: 72.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 6 \cdot 10^{+50}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left({\left(D\_m \cdot \frac{M\_m}{d\_m}\right)}^{2} \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \end{array} \end{array} \]

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 6e+50)
   w0
   (* -0.125 (* (pow (* D_m (/ M_m d_m)) 2.0) (* h (/ w0 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) {
	double tmp;
	if (M_m <= 6e+50) {
		tmp = w0;
	} else {
		tmp = -0.125 * (pow((D_m * (M_m / d_m)), 2.0) * (h * (w0 / l)));
	}
	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 (m_m <= 6d+50) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d_m * (m_m / d_m_1)) ** 2.0d0) * (h * (w0 / l)))
    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 (M_m <= 6e+50) {
		tmp = w0;
	} else {
		tmp = -0.125 * (Math.pow((D_m * (M_m / d_m)), 2.0) * (h * (w0 / l)));
	}
	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 M_m <= 6e+50:
		tmp = w0
	else:
		tmp = -0.125 * (math.pow((D_m * (M_m / d_m)), 2.0) * (h * (w0 / l)))
	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 (M_m <= 6e+50)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64((Float64(D_m * Float64(M_m / d_m)) ^ 2.0) * Float64(h * Float64(w0 / l))));
	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 (M_m <= 6e+50)
		tmp = w0;
	else
		tmp = -0.125 * (((D_m * (M_m / d_m)) ^ 2.0) * (h * (w0 / l)));
	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[M$95$m, 6e+50], w0, N[(-0.125 * N[(N[Power[N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * N[(w0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.9999999999999996e50

   1. Initial program 79.0%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in D around 0 72.9%

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

   if 5.9999999999999996e50 < M

   1. Initial program 61.2%

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

   3. Simplified 61.2%

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

   6. Applied egg-rr 61.2%

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

      1. unpow1 61.2%

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

      2. associate-*l/ 61.6%

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

      3. associate-/l* 66.1%

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

      4. associate-*r/ 66.2%

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

      5. *-rgt-identity 66.2%

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

      6. times-frac 66.2%

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

      7. /-rgt-identity 66.2%

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

   8. Simplified 66.2%

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

   9. Taylor expanded in h around 0 37.4%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 37.4%

          \[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

       2. associate-*r* 39.6%

          \[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}\right) \]

       3. unpow2 39.6%

          \[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h\right)}{{d}^{2} \cdot \ell}\right) \]

       4. unpow2 39.6%

          \[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{{d}^{2} \cdot \ell}\right) \]

       5. swap-sqr 54.3%

          \[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right)}{{d}^{2} \cdot \ell}\right) \]

       6. unpow2 54.3%

          \[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h\right)}{{d}^{2} \cdot \ell}\right) \]

       7. associate-*r* 54.3%

          \[\leadsto w0 \cdot \left(1 + \frac{\color{blue}{\left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right) \]

   11. Simplified 54.3%

       \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{\left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right) \cdot h}{{d}^{2} \cdot \ell}\right)} \]

   12. Taylor expanded in D around inf 27.9%

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

       1. associate-*r* 27.9%

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

       2. unpow2 27.9%

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

       3. unpow2 27.9%

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

       4. swap-sqr 33.7%

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

       5. unpow2 33.7%

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

       6. times-frac 33.8%

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

       7. unpow2 33.8%

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

       8. unpow2 33.8%

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

       9. times-frac 34.1%

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

       10. associate-*r/ 34.1%

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

       11. associate-*r/ 34.1%

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

       12. unpow1 34.1%

           \[\leadsto -0.125 \cdot \left(\left(\color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{1}} \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{h \cdot w0}{\ell}\right) \]

       13. pow-plus 34.1%

           \[\leadsto -0.125 \cdot \left(\color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{\left(1 + 1\right)}} \cdot \frac{h \cdot w0}{\ell}\right) \]

       14. metadata-eval 34.1%

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

       15. associate-/l* 34.3%

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

   14. Simplified 34.3%

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

Alternative 7: 81.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ w0 \cdot \left(1 + h \cdot \left({\left(D\_m \cdot \frac{M\_m}{d\_m}\right)}^{2} \cdot \frac{-0.125}{\ell}\right)\right) \end{array} \]

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 (+ 1.0 (* h (* (pow (* D_m (/ M_m d_m)) 2.0) (/ -0.125 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 * (1.0 + (h * (pow((D_m * (M_m / d_m)), 2.0) * (-0.125 / 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 * (1.0d0 + (h * (((d_m * (m_m / d_m_1)) ** 2.0d0) * ((-0.125d0) / 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 * (1.0 + (h * (Math.pow((D_m * (M_m / d_m)), 2.0) * (-0.125 / 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 * (1.0 + (h * (math.pow((D_m * (M_m / d_m)), 2.0) * (-0.125 / 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 * Float64(1.0 + Float64(h * Float64((Float64(D_m * Float64(M_m / d_m)) ^ 2.0) * Float64(-0.125 / 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 * (1.0 + (h * (((D_m * (M_m / d_m)) ^ 2.0) * (-0.125 / 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[(1.0 + N[(h * N[(N[Power[N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.125 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.1%

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

3. Simplified 76.8%

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

6. Applied egg-rr 76.8%

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

   1. unpow1 76.8%

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

   2. associate-*l/ 82.3%

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

   3. associate-/l* 82.7%

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

   4. associate-*r/ 82.3%

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

   5. *-rgt-identity 82.3%

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

   6. times-frac 82.7%

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

   7. /-rgt-identity 82.7%

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

8. Simplified 82.7%

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

9. Taylor expanded in h around 0 49.9%

   \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
10. Step-by-step derivation

    1. associate-*r/ 49.9%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

    2. associate-*r* 51.9%

       \[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2} \cdot \ell}\right) \]

    3. unpow2 51.9%

       \[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h\right)}{{d}^{2} \cdot \ell}\right) \]

    4. unpow2 51.9%

       \[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{{d}^{2} \cdot \ell}\right) \]

    5. swap-sqr 64.8%

       \[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right)}{{d}^{2} \cdot \ell}\right) \]

    6. unpow2 64.8%

       \[\leadsto w0 \cdot \left(1 + \frac{-0.125 \cdot \left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h\right)}{{d}^{2} \cdot \ell}\right) \]

    7. associate-*r* 64.8%

       \[\leadsto w0 \cdot \left(1 + \frac{\color{blue}{\left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right) \]

11. Simplified 64.8%

    \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{\left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right) \cdot h}{{d}^{2} \cdot \ell}\right)} \]

12. Taylor expanded in D around 0 49.9%

    \[\leadsto w0 \cdot \left(1 + \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
13. Step-by-step derivation

    1. associate-*r* 51.9%

       \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right) \]

    2. unpow2 51.9%

       \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{{d}^{2} \cdot \ell}\right) \]

    3. unpow2 51.9%

       \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{{d}^{2} \cdot \ell}\right) \]

    4. swap-sqr 64.8%

       \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h}{{d}^{2} \cdot \ell}\right) \]

    5. unpow2 64.8%

       \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{{d}^{2} \cdot \ell}\right) \]

    6. associate-*r/ 66.4%

       \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{d}^{2} \cdot \ell}\right)}\right) \]

    7. *-commutative 66.4%

       \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{\color{blue}{\ell \cdot {d}^{2}}}\right)\right) \]

    8. associate-*l* 66.4%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right) \cdot \frac{h}{\ell \cdot {d}^{2}}}\right) \]

    9. associate-*r/ 64.8%

       \[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{\left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right) \cdot h}{\ell \cdot {d}^{2}}}\right) \]

    10. *-commutative 64.8%

        \[\leadsto w0 \cdot \left(1 + \frac{\color{blue}{h \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}}{\ell \cdot {d}^{2}}\right) \]

    11. *-commutative 64.8%

        \[\leadsto w0 \cdot \left(1 + \frac{h \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right) \]

    12. associate-*r/ 67.2%

        \[\leadsto w0 \cdot \left(1 + \color{blue}{h \cdot \frac{-0.125 \cdot {\left(D \cdot M\right)}^{2}}{{d}^{2} \cdot \ell}}\right) \]

    13. *-commutative 67.2%

        \[\leadsto w0 \cdot \left(1 + h \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2} \cdot -0.125}}{{d}^{2} \cdot \ell}\right) \]

    14. times-frac 68.8%

        \[\leadsto w0 \cdot \left(1 + h \cdot \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}} \cdot \frac{-0.125}{\ell}\right)}\right) \]

14. Simplified 77.7%

    \[\leadsto w0 \cdot \left(1 + \color{blue}{h \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{-0.125}{\ell}\right)}\right) \]
15. Add Preprocessing

Alternative 8: 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 76.1%

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

3. Simplified 76.8%

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

5. Taylor expanded in D around 0 65.4%

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

Reproduce

herbie shell --seed 2024144 
(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))))))