Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 89.5%

Time: 20.7s

Alternatives: 3

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: 81.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\log \left(0.25 \cdot \frac{h \cdot {\left(M\_m \cdot D\_m\right)}^{2}}{-\ell}\right) + -2 \cdot \log d\_m\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{d\_m}\right)\right) \cdot \left(D\_m \cdot \left(\frac{M\_m}{d\_m} \cdot \frac{0.5}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

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)) (- INFINITY))
   (pow
    (*
     (cbrt w0)
     (exp
      (*
       0.16666666666666666
       (+
        (log (* 0.25 (/ (* h (pow (* M_m D_m) 2.0)) (- l))))
        (* -2.0 (log d_m))))))
    3.0)
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       h
       (* (* D_m (* 0.5 (/ M_m d_m))) (* D_m (* (/ M_m d_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)) <= -((double) INFINITY)) {
		tmp = pow((cbrt(w0) * exp((0.16666666666666666 * (log((0.25 * ((h * pow((M_m * D_m), 2.0)) / -l))) + (-2.0 * log(d_m)))))), 3.0);
	} else {
		tmp = w0 * sqrt((1.0 - (h * ((D_m * (0.5 * (M_m / d_m))) * (D_m * ((M_m / d_m) * (0.5 / l)))))));
	}
	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 ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.pow((Math.cbrt(w0) * 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);
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * ((D_m * (0.5 * (M_m / d_m))) * (D_m * ((M_m / d_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)) <= Float64(-Inf))
		tmp = Float64(cbrt(w0) * exp(Float64(0.16666666666666666 * Float64(log(Float64(0.25 * Float64(Float64(h * (Float64(M_m * D_m) ^ 2.0)) / Float64(-l)))) + Float64(-2.0 * log(d_m)))))) ^ 3.0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(D_m * Float64(0.5 * Float64(M_m / d_m))) * Float64(D_m * Float64(Float64(M_m / d_m) * Float64(0.5 / l))))))));
	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[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], (-Infinity)], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * 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]), $MachinePrecision], 3.0], $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(D$95$m * N[(0.5 * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(N[(M$95$m / d$95$m), $MachinePrecision] * 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)) < -inf.0

   1. Initial program 53.9%

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

   3. Simplified 55.3%

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

      1. *-commutative 55.3%

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

      2. add-cbrt-cube 55.3%

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

      3. add-cbrt-cube 46.5%

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

      4. cbrt-unprod 46.5%

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

   6. Applied egg-rr 46.5%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(1 - \frac{h}{\ell} \cdot {\left(\frac{D}{\frac{d}{M} \cdot 2}\right)}^{2}\right)}^{1.5} \cdot {w0}^{3}}} \]
   7. Step-by-step derivation

   8. Simplified 47.9%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(1 - h \cdot \frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}\right)}^{1.5} \cdot {w0}^{3}}} \]

   10. Applied egg-rr 55.3%

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

   11. Taylor expanded in d around 0 25.6%

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

       1. pow1 25.6%

          \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{\color{blue}{{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}^{1}}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

       2. associate-*r* 24.4%

          \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{{\color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}^{1}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

       3. pow-prod-down 29.6%

          \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{{\left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h\right)}^{1}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

   13. Applied egg-rr 29.6%

       \[\leadsto {\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{\color{blue}{{\left({\left(D \cdot M\right)}^{2} \cdot h\right)}^{1}}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]
   14. Step-by-step derivation

       1. unpow1 29.6%

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

       2. *-commutative 29.6%

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

   15. Simplified 29.6%

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

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

   1. Initial program 88.2%

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

   3. Simplified 88.8%

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

      1. add-sqr-sqrt 41.1%

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

      2. sqrt-unprod 25.1%

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

      3. *-commutative 25.1%

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

      4. *-commutative 25.1%

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

      5. swap-sqr 25.1%

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

   6. Applied egg-rr 25.1%

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

      1. associate-*l/ 27.9%

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

      2. associate-/l* 27.9%

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

      3. associate-*l/ 27.9%

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

      4. *-commutative 27.9%

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

      5. associate-*l/ 27.9%

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

      6. associate-/l/ 27.9%

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

      7. associate-/r/ 27.9%

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

      8. associate-*l/ 27.9%

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

      9. *-commutative 27.9%

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

      10. associate-*r/ 27.9%

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

      11. *-commutative 27.9%

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

      12. associate-/l/ 27.9%

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

      13. associate-/l* 27.9%

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

   8. Simplified 27.9%

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

      1. unpow2 27.9%

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

      2. associate-/l* 27.9%

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

      3. *-un-lft-identity 27.9%

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

      4. times-frac 27.9%

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

      5. metadata-eval 27.9%

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

      6. *-un-lft-identity 27.9%

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

      7. times-frac 27.9%

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

      8. metadata-eval 27.9%

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

   10. Applied egg-rr 27.9%

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

       1. associate-/l* 27.4%

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

       2. associate-*r/ 27.4%

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

   12. Applied egg-rr 27.4%

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

       1. *-commutative 27.4%

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

       2. sqrt-prod 27.4%

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

       3. sqrt-pow1 95.6%

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

       4. metadata-eval 95.6%

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

       5. pow1 95.6%

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

       6. associate-*r* 95.6%

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

       7. associate-*r* 95.6%

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

       8. associate-/l/ 93.7%

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

   14. Applied egg-rr 93.7%

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

       1. associate-*l* 93.7%

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

       2. associate-*l* 93.7%

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

       3. times-frac 95.6%

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

   16. Simplified 95.6%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\log \left(0.25 \cdot \frac{h \cdot {\left(M \cdot D\right)}^{2}}{-\ell}\right) + -2 \cdot \log d\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right) \cdot \left(D \cdot \left(\frac{M}{d} \cdot \frac{0.5}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.7% accurate, 1.8× speedup

Math

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

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 (* 0.5 (/ M_m d_m))) (* D_m (* (/ M_m d_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) {
	return w0 * sqrt((1.0 - (h * ((D_m * (0.5 * (M_m / d_m))) * (D_m * ((M_m / d_m) * (0.5 / 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 * ((d_m * (0.5d0 * (m_m / d_m_1))) * (d_m * ((m_m / d_m_1) * (0.5d0 / 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 * ((D_m * (0.5 * (M_m / d_m))) * (D_m * ((M_m / d_m) * (0.5 / 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 * ((D_m * (0.5 * (M_m / d_m))) * (D_m * ((M_m / d_m) * (0.5 / 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(D_m * Float64(0.5 * Float64(M_m / d_m))) * Float64(D_m * Float64(Float64(M_m / d_m) * Float64(0.5 / 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 * ((D_m * (0.5 * (M_m / d_m))) * (D_m * ((M_m / d_m) * (0.5 / 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[(D$95$m * N[(0.5 * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(N[(M$95$m / d$95$m), $MachinePrecision] * N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 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. Step-by-step derivation

   1. add-sqr-sqrt 38.8%

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

   2. sqrt-unprod 26.7%

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

   3. *-commutative 26.7%

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

   4. *-commutative 26.7%

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

   5. swap-sqr 26.5%

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

6. Applied egg-rr 26.5%

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

   1. associate-*l/ 29.0%

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

   2. associate-/l* 29.0%

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

   3. associate-*l/ 29.0%

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

   4. *-commutative 29.0%

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

   5. associate-*l/ 29.0%

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

   6. associate-/l/ 29.0%

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

   7. associate-/r/ 29.0%

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

   8. associate-*l/ 29.0%

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

   9. *-commutative 29.0%

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

   10. associate-*r/ 29.0%

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

   11. *-commutative 29.0%

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

   12. associate-/l/ 29.0%

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

   13. associate-/l* 29.0%

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

8. Simplified 29.0%

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

   1. unpow2 29.0%

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

   2. associate-/l* 29.1%

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

   3. *-un-lft-identity 29.1%

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

   4. times-frac 29.1%

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

   5. metadata-eval 29.1%

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

   6. *-un-lft-identity 29.1%

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

   7. times-frac 29.1%

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

   8. metadata-eval 29.1%

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

10. Applied egg-rr 29.1%

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

    1. associate-/l* 28.7%

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

    2. associate-*r/ 28.7%

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

12. Applied egg-rr 28.7%

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

    1. *-commutative 28.7%

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

    2. sqrt-prod 29.1%

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

    3. sqrt-pow1 86.0%

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

    4. metadata-eval 86.0%

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

    5. pow1 86.0%

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

    6. associate-*r* 85.7%

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

    7. associate-*r* 85.7%

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

    8. associate-/l/ 83.4%

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

14. Applied egg-rr 83.4%

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

    1. associate-*l* 83.9%

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

    2. associate-*l* 83.9%

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

    3. times-frac 86.0%

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

16. Simplified 86.0%

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

17. Final simplification 86.0%

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

Alternative 3: 68.2% 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 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 66.3%

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

Reproduce

herbie shell --seed 2024102 
(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))))))