Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 88.3%

Time: 19.7s

Alternatives: 6

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: 80.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.3% accurate, 0.2× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D) (* 2.0 d_m)) 2.0) (/ h l)) (- INFINITY))
   (pow
    (*
     (cbrt w0)
     (pow
      (exp 0.16666666666666666)
      (fma -2.0 (log d_m) (log (* -0.25 (/ (* h (pow (* M_m D) 2.0)) l))))))
    3.0)
   (*
    w0
    (sqrt
     (-
      1.0
      (* (/ (* M_m (/ 0.5 (/ d_m D))) l) (* h (* M_m (* D (/ 0.5 d_m))))))))))

C

M_m = fabs(M);
d_m = fabs(d);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d_m);
double code(double w0, double M_m, double D, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D) / (2.0 * d_m)), 2.0) * (h / l)) <= -((double) INFINITY)) {
		tmp = pow((cbrt(w0) * pow(exp(0.16666666666666666), fma(-2.0, log(d_m), log((-0.25 * ((h * pow((M_m * D), 2.0)) / l)))))), 3.0);
	} else {
		tmp = w0 * sqrt((1.0 - (((M_m * (0.5 / (d_m / D))) / l) * (h * (M_m * (D * (0.5 / d_m)))))));
	}
	return tmp;
}

Julia

M_m = abs(M)
d_m = abs(d)
w0, M_m, D, h, l, d_m = sort([w0, M_m, D, h, l, d_m])
function code(w0, M_m, D, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= Float64(-Inf))
		tmp = Float64(cbrt(w0) * (exp(0.16666666666666666) ^ fma(-2.0, log(d_m), log(Float64(-0.25 * Float64(Float64(h * (Float64(M_m * D) ^ 2.0)) / l)))))) ^ 3.0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(M_m * Float64(0.5 / Float64(d_m / D))) / l) * Float64(h * Float64(M_m * Float64(D * Float64(0.5 / d_m))))))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D), $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[Power[N[Exp[0.16666666666666666], $MachinePrecision], N[(-2.0 * N[Log[d$95$m], $MachinePrecision] + N[Log[N[(-0.25 * N[(N[(h * N[Power[N[(M$95$m * D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(M$95$m * N[(0.5 / N[(d$95$m / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(h * N[(M$95$m * N[(D * N[(0.5 / d$95$m), $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 2 d)) 2) (/.f64 h l)) < -inf.0

   1. Initial program 59.8%

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

   3. Simplified 59.8%

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

      1. add-cube-cbrt 59.8%

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

      2. pow3 59.8%

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

   6. Applied egg-rr 59.8%

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

   7. Taylor expanded in d around 0 20.3%

      \[\leadsto {\color{blue}{\left({\left(1 \cdot w0\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right) + -2 \cdot \log d\right)}\right)}}^{3} \]
   8. Step-by-step derivation

      1. unpow1/3 32.2%

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

      2. *-lft-identity 32.2%

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

      3. exp-prod 32.0%

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

      4. +-commutative 32.0%

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

      5. fma-define 32.0%

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

      6. distribute-lft-neg-in 32.0%

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

      7. metadata-eval 32.0%

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

      8. associate-*r* 32.0%

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

      9. unpow2 32.0%

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

      10. unpow2 32.0%

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

      11. swap-sqr 42.0%

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

      12. unpow2 42.0%

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

   9. Simplified 42.0%

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

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

   1. Initial program 94.4%

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

   3. Simplified 94.0%

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

      1. associate-*r/ 96.2%

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

      2. add-sqr-sqrt 96.2%

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

      3. pow2 96.2%

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

      4. sqrt-pow1 96.2%

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

      5. metadata-eval 96.2%

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

      6. pow1 96.2%

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

      7. *-un-lft-identity 96.2%

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

      8. times-frac 96.2%

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

      9. metadata-eval 96.2%

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

   6. Applied egg-rr 96.2%

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

      1. associate-*r/ 94.0%

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

      2. clear-num 94.0%

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

      3. div-inv 94.5%

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

      4. unpow2 94.5%

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

      5. div-inv 94.5%

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

      6. times-frac 98.1%

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

      7. clear-num 98.1%

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

      8. un-div-inv 98.1%

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

      9. clear-num 98.1%

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

      10. un-div-inv 98.1%

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

   8. Applied egg-rr 98.1%

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

      1. div-inv 98.1%

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

      2. inv-pow 98.1%

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

      3. pow-flip 98.1%

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

      4. metadata-eval 98.1%

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

      5. pow1 98.1%

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

      6. associate-/r/ 98.1%

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

      7. *-commutative 98.1%

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

   10. Applied egg-rr 98.1%

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

4. Final simplification 85.2%

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

Alternative 2: 75.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ d_m = \left|d\right| \\ [w0, M_m, D, h, l, d_m] = \mathsf{sort}([w0, M_m, D, h, l, d_m])\\ \\ \begin{array}{l} t_0 := D \cdot \frac{0.5}{d\_m}\\ \mathbf{if}\;M\_m \cdot D \leq 5.5 \cdot 10^{-18}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M\_m \cdot \frac{t\_0}{\ell}\right) \cdot \left(h \cdot \left(M\_m \cdot t\_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d_m)
 :precision binary64
 (let* ((t_0 (* D (/ 0.5 d_m))))
   (if (<= (* M_m D) 5.5e-18)
     w0
     (* w0 (sqrt (- 1.0 (* (* M_m (/ t_0 l)) (* h (* M_m t_0)))))))))

C

M_m = fabs(M);
d_m = fabs(d);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d_m);
double code(double w0, double M_m, double D, double h, double l, double d_m) {
	double t_0 = D * (0.5 / d_m);
	double tmp;
	if ((M_m * D) <= 5.5e-18) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((M_m * (t_0 / l)) * (h * (M_m * t_0)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
d_m = abs(d)
NOTE: w0, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_m)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * (0.5d0 / d_m)
    if ((m_m * d) <= 5.5d-18) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((m_m * (t_0 / l)) * (h * (m_m * t_0)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D && D < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D, double h, double l, double d_m) {
	double t_0 = D * (0.5 / d_m);
	double tmp;
	if ((M_m * D) <= 5.5e-18) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((M_m * (t_0 / l)) * (h * (M_m * t_0)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
d_m = math.fabs(d)
[w0, M_m, D, h, l, d_m] = sort([w0, M_m, D, h, l, d_m])
def code(w0, M_m, D, h, l, d_m):
	t_0 = D * (0.5 / d_m)
	tmp = 0
	if (M_m * D) <= 5.5e-18:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((M_m * (t_0 / l)) * (h * (M_m * t_0)))))
	return tmp

Julia

M_m = abs(M)
d_m = abs(d)
w0, M_m, D, h, l, d_m = sort([w0, M_m, D, h, l, d_m])
function code(w0, M_m, D, h, l, d_m)
	t_0 = Float64(D * Float64(0.5 / d_m))
	tmp = 0.0
	if (Float64(M_m * D) <= 5.5e-18)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m * Float64(t_0 / l)) * Float64(h * Float64(M_m * t_0))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d_m = abs(d);
w0, M_m, D, h, l, d_m = num2cell(sort([w0, M_m, D, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D, h, l, d_m)
	t_0 = D * (0.5 / d_m);
	tmp = 0.0;
	if ((M_m * D) <= 5.5e-18)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((M_m * (t_0 / l)) * (h * (M_m * t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(D * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M$95$m * D), $MachinePrecision], 5.5e-18], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 5.5e-18

   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 87.8%

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

   5. Taylor expanded in M around 0 77.4%

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

   if 5.5e-18 < (*.f64 M D)

   1. Initial program 78.9%

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

   3. Simplified 78.9%

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

      1. associate-*r/ 75.1%

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

      2. add-sqr-sqrt 75.1%

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

      3. pow2 75.1%

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

      4. sqrt-pow1 75.1%

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

      5. metadata-eval 75.1%

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

      6. pow1 75.1%

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

      7. *-un-lft-identity 75.1%

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

      8. times-frac 75.1%

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

      9. metadata-eval 75.1%

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

   6. Applied egg-rr 75.1%

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

      1. associate-*r/ 78.9%

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

      2. clear-num 78.9%

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

      3. div-inv 78.9%

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

      4. unpow2 78.9%

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

      5. div-inv 78.9%

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

      6. times-frac 79.0%

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

      7. clear-num 79.0%

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

      8. un-div-inv 79.0%

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

      9. clear-num 79.0%

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

      10. un-div-inv 79.0%

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

   8. Applied egg-rr 79.0%

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

      1. associate-/l* 79.0%

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

      2. associate-/r/ 79.0%

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

      3. associate-/r/ 79.0%

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

      4. /-rgt-identity 79.0%

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

      5. associate-/r/ 79.0%

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

   10. Simplified 79.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 5.5 \cdot 10^{-18}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \frac{D \cdot \frac{0.5}{d}}{\ell}\right) \cdot \left(h \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ d_m = \left|d\right| \\ [w0, M_m, D, h, l, d_m] = \mathsf{sort}([w0, M_m, D, h, l, d_m])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{0.5}{\frac{d\_m}{D}}\\ \mathbf{if}\;D \leq 5.3 \cdot 10^{+82}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(\frac{h}{\ell} \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d_m)
 :precision binary64
 (let* ((t_0 (* M_m (/ 0.5 (/ d_m D)))))
   (if (<= D 5.3e+82) w0 (* w0 (sqrt (- 1.0 (* t_0 (* (/ h l) t_0))))))))

C

M_m = fabs(M);
d_m = fabs(d);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d_m);
double code(double w0, double M_m, double D, double h, double l, double d_m) {
	double t_0 = M_m * (0.5 / (d_m / D));
	double tmp;
	if (D <= 5.3e+82) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
d_m = abs(d)
NOTE: w0, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_m)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m_m * (0.5d0 / (d_m / d))
    if (d <= 5.3d+82) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - (t_0 * ((h / l) * t_0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D && D < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D, double h, double l, double d_m) {
	double t_0 = M_m * (0.5 / (d_m / D));
	double tmp;
	if (D <= 5.3e+82) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
d_m = math.fabs(d)
[w0, M_m, D, h, l, d_m] = sort([w0, M_m, D, h, l, d_m])
def code(w0, M_m, D, h, l, d_m):
	t_0 = M_m * (0.5 / (d_m / D))
	tmp = 0
	if D <= 5.3e+82:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - (t_0 * ((h / l) * t_0))))
	return tmp

Julia

M_m = abs(M)
d_m = abs(d)
w0, M_m, D, h, l, d_m = sort([w0, M_m, D, h, l, d_m])
function code(w0, M_m, D, h, l, d_m)
	t_0 = Float64(M_m * Float64(0.5 / Float64(d_m / D)))
	tmp = 0.0
	if (D <= 5.3e+82)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(h / l) * t_0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d_m = abs(d);
w0, M_m, D, h, l, d_m = num2cell(sort([w0, M_m, D, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D, h, l, d_m)
	t_0 = M_m * (0.5 / (d_m / D));
	tmp = 0.0;
	if (D <= 5.3e+82)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(0.5 / N[(d$95$m / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, 5.3e+82], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 5.29999999999999977e82

   1. Initial program 86.6%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in M around 0 74.0%

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

   if 5.29999999999999977e82 < D

   1. Initial program 85.0%

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

   3. Simplified 84.9%

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

      1. associate-*r/ 84.9%

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

      2. add-sqr-sqrt 84.9%

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

      3. pow2 84.9%

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

      4. sqrt-pow1 84.9%

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

      5. metadata-eval 84.9%

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

      6. pow1 84.9%

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

      7. *-un-lft-identity 84.9%

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

      8. times-frac 84.9%

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

      9. metadata-eval 84.9%

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

   6. Applied egg-rr 84.9%

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

      1. associate-*r/ 84.9%

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

      2. *-commutative 84.9%

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

      3. unpow2 84.9%

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

      4. associate-*r* 90.1%

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

      5. clear-num 90.1%

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

      6. un-div-inv 90.1%

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

      7. clear-num 90.1%

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

      8. un-div-inv 90.1%

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

   8. Applied egg-rr 90.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 5.3 \cdot 10^{+82}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \frac{0.5}{\frac{d}{D}}\right) \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot \frac{0.5}{\frac{d}{D}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ d_m = \left|d\right| \\ [w0, M_m, D, h, l, d_m] = \mathsf{sort}([w0, M_m, D, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 7.2 \cdot 10^{+114}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{\left(h \cdot w0\right) \cdot \left(\left(M\_m \cdot D\right) \cdot \left(M\_m \cdot D\right)\right)}{\ell \cdot {d\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d_m)
 :precision binary64
 (if (<= D 7.2e+114)
   w0
   (+
    w0
    (* -0.125 (/ (* (* h w0) (* (* M_m D) (* M_m D))) (* l (pow d_m 2.0)))))))

C

M_m = fabs(M);
d_m = fabs(d);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d_m);
double code(double w0, double M_m, double D, double h, double l, double d_m) {
	double tmp;
	if (D <= 7.2e+114) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((h * w0) * ((M_m * D) * (M_m * D))) / (l * pow(d_m, 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
d_m = abs(d)
NOTE: w0, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_m)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 7.2d+114) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((h * w0) * ((m_m * d) * (m_m * d))) / (l * (d_m ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D && D < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D, double h, double l, double d_m) {
	double tmp;
	if (D <= 7.2e+114) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((h * w0) * ((M_m * D) * (M_m * D))) / (l * Math.pow(d_m, 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
d_m = math.fabs(d)
[w0, M_m, D, h, l, d_m] = sort([w0, M_m, D, h, l, d_m])
def code(w0, M_m, D, h, l, d_m):
	tmp = 0
	if D <= 7.2e+114:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (((h * w0) * ((M_m * D) * (M_m * D))) / (l * math.pow(d_m, 2.0))))
	return tmp

Julia

M_m = abs(M)
d_m = abs(d)
w0, M_m, D, h, l, d_m = sort([w0, M_m, D, h, l, d_m])
function code(w0, M_m, D, h, l, d_m)
	tmp = 0.0
	if (D <= 7.2e+114)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(h * w0) * Float64(Float64(M_m * D) * Float64(M_m * D))) / Float64(l * (d_m ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d_m = abs(d);
w0, M_m, D, h, l, d_m = num2cell(sort([w0, M_m, D, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D, h, l, d_m)
	tmp = 0.0;
	if (D <= 7.2e+114)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((h * w0) * ((M_m * D) * (M_m * D))) / (l * (d_m ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d$95$m_] := If[LessEqual[D, 7.2e+114], w0, N[(w0 + N[(-0.125 * N[(N[(N[(h * w0), $MachinePrecision] * N[(N[(M$95$m * D), $MachinePrecision] * N[(M$95$m * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Power[d$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 7.2000000000000001e114

   1. Initial program 86.7%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in M around 0 73.8%

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

   if 7.2000000000000001e114 < D

   1. Initial program 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in M around 0 35.1%

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

      1. pow1 35.1%

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

      2. associate-*r* 35.1%

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

      3. pow-prod-down 73.4%

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

   7. Applied egg-rr 73.4%

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

      1. unpow1 73.4%

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

      2. *-commutative 73.4%

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

   9. Simplified 73.4%

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

       1. unpow2 73.4%

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

       2. *-commutative 73.4%

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

       3. *-commutative 73.4%

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

   11. Applied egg-rr 73.4%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 7.2 \cdot 10^{+114}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{\left(h \cdot w0\right) \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)}{\ell \cdot {d}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.0% accurate, 1.8× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d_m)
 :precision binary64
 (*
  w0
  (sqrt
   (-
    1.0
    (* (/ (* M_m (/ 0.5 (/ d_m D))) l) (* h (* M_m (* D (/ 0.5 d_m)))))))))

C

M_m = fabs(M);
d_m = fabs(d);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d_m);
double code(double w0, double M_m, double D, double h, double l, double d_m) {
	return w0 * sqrt((1.0 - (((M_m * (0.5 / (d_m / D))) / l) * (h * (M_m * (D * (0.5 / d_m)))))));
}

Fortran

M_m = abs(m)
d_m = abs(d)
NOTE: w0, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_m)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m
    code = w0 * sqrt((1.0d0 - (((m_m * (0.5d0 / (d_m / d))) / l) * (h * (m_m * (d * (0.5d0 / d_m)))))))
end function

Java

M_m = Math.abs(M);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D && D < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D, double h, double l, double d_m) {
	return w0 * Math.sqrt((1.0 - (((M_m * (0.5 / (d_m / D))) / l) * (h * (M_m * (D * (0.5 / d_m)))))));
}

Python

M_m = math.fabs(M)
d_m = math.fabs(d)
[w0, M_m, D, h, l, d_m] = sort([w0, M_m, D, h, l, d_m])
def code(w0, M_m, D, h, l, d_m):
	return w0 * math.sqrt((1.0 - (((M_m * (0.5 / (d_m / D))) / l) * (h * (M_m * (D * (0.5 / d_m)))))))

Julia

M_m = abs(M)
d_m = abs(d)
w0, M_m, D, h, l, d_m = sort([w0, M_m, D, h, l, d_m])
function code(w0, M_m, D, h, l, d_m)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(M_m * Float64(0.5 / Float64(d_m / D))) / l) * Float64(h * Float64(M_m * Float64(D * Float64(0.5 / d_m))))))))
end

MATLAB

M_m = abs(M);
d_m = abs(d);
w0, M_m, D, h, l, d_m = num2cell(sort([w0, M_m, D, h, l, d_m])){:}
function tmp = code(w0, M_m, D, h, l, d_m)
	tmp = w0 * sqrt((1.0 - (((M_m * (0.5 / (d_m / D))) / l) * (h * (M_m * (D * (0.5 / d_m)))))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d$95$m_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(M$95$m * N[(0.5 / N[(d$95$m / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(h * N[(M$95$m * N[(D * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.4%

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

3. Simplified 86.1%

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

   1. associate-*r/ 88.1%

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

   2. add-sqr-sqrt 88.1%

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

   3. pow2 88.1%

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

   4. sqrt-pow1 88.1%

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

   5. metadata-eval 88.1%

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

   6. pow1 88.1%

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

   7. *-un-lft-identity 88.1%

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

   8. times-frac 88.1%

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

   9. metadata-eval 88.1%

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

6. Applied egg-rr 88.1%

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

   1. associate-*r/ 86.1%

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

   2. clear-num 86.1%

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

   3. div-inv 86.5%

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

   4. unpow2 86.5%

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

   5. div-inv 86.5%

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

   6. times-frac 91.1%

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

   7. clear-num 91.1%

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

   8. un-div-inv 91.1%

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

   9. clear-num 91.1%

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

   10. un-div-inv 91.1%

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

8. Applied egg-rr 91.1%

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

   1. div-inv 91.1%

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

   2. inv-pow 91.1%

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

   3. pow-flip 91.1%

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

   4. metadata-eval 91.1%

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

   5. pow1 91.1%

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

   6. associate-/r/ 91.1%

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

   7. *-commutative 91.1%

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

10. Applied egg-rr 91.1%

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

11. Final simplification 91.1%

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

Alternative 6: 67.4% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

3. Simplified 86.1%

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

5. Taylor expanded in M around 0 70.1%

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

6. Final simplification 70.1%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(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))))))