Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 89.6%

Time: 18.2s

Alternatives: 7

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 5 \cdot 10^{+143}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\left(\frac{d\_m}{M\_m \cdot \left(D\_m \cdot 0.5\right)} \cdot \frac{2 \cdot d\_m}{M\_m \cdot D\_m}\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot \left(\left({\left(\frac{h \cdot -0.25}{\ell}\right)}^{0.16666666666666666} \cdot \sqrt[3]{M\_m \cdot D\_m}\right) \cdot e^{-0.3333333333333333 \cdot \log d\_m}\right)\right)}^{3}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ (* M_m D_m) (* 2.0 d_m)) 5e+143)
   (*
    w0
    (sqrt
     (-
      1.0
      (/
       h
       (* (* (/ d_m (* M_m (* D_m 0.5))) (/ (* 2.0 d_m) (* M_m D_m))) l)))))
   (pow
    (*
     (cbrt w0)
     (*
      (* (pow (/ (* h -0.25) l) 0.16666666666666666) (cbrt (* M_m D_m)))
      (exp (* -0.3333333333333333 (log d_m)))))
    3.0)))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+143) {
		tmp = w0 * sqrt((1.0 - (h / (((d_m / (M_m * (D_m * 0.5))) * ((2.0 * d_m) / (M_m * D_m))) * l))));
	} else {
		tmp = pow((cbrt(w0) * ((pow(((h * -0.25) / l), 0.16666666666666666) * cbrt((M_m * D_m))) * exp((-0.3333333333333333 * log(d_m))))), 3.0);
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+143) {
		tmp = w0 * Math.sqrt((1.0 - (h / (((d_m / (M_m * (D_m * 0.5))) * ((2.0 * d_m) / (M_m * D_m))) * l))));
	} else {
		tmp = Math.pow((Math.cbrt(w0) * ((Math.pow(((h * -0.25) / l), 0.16666666666666666) * Math.cbrt((M_m * D_m))) * Math.exp((-0.3333333333333333 * Math.log(d_m))))), 3.0);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) <= 5e+143)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h / Float64(Float64(Float64(d_m / Float64(M_m * Float64(D_m * 0.5))) * Float64(Float64(2.0 * d_m) / Float64(M_m * D_m))) * l)))));
	else
		tmp = Float64(cbrt(w0) * Float64(Float64((Float64(Float64(h * -0.25) / l) ^ 0.16666666666666666) * cbrt(Float64(M_m * D_m))) * exp(Float64(-0.3333333333333333 * log(d_m))))) ^ 3.0;
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 5e+143], N[(w0 * N[Sqrt[N[(1.0 - N[(h / N[(N[(N[(d$95$m / N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * d$95$m), $MachinePrecision] / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[(N[(N[Power[N[(N[(h * -0.25), $MachinePrecision] / l), $MachinePrecision], 0.16666666666666666], $MachinePrecision] * N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(-0.3333333333333333 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 5.00000000000000012e143

   1. Initial program 90.6%

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

   3. Simplified 89.4%

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

      1. associate-*r/ 90.6%

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

      2. unpow2 90.6%

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

      3. associate-/r* 90.6%

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

      4. frac-times 70.4%

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

      5. associate-/l* 70.4%

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

      6. div-inv 70.4%

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

      7. metadata-eval 70.4%

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

   6. Applied egg-rr 70.4%

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

      1. clear-num 70.4%

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

      2. frac-times 74.5%

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

      3. *-un-lft-identity 74.5%

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

      4. times-frac 95.0%

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

      5. *-commutative 95.0%

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

      6. *-commutative 95.0%

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

   8. Applied egg-rr 95.0%

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

   if 5.00000000000000012e143 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 70.6%

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

   3. Simplified 70.6%

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

      1. add-cube-cbrt 70.6%

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

      2. pow3 70.6%

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

   6. Applied egg-rr 70.6%

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

   7. Taylor expanded in d around 0 39.1%

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

      1. exp-prod 39.1%

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

      2. unpow-prod-up 39.1%

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

      3. distribute-lft-neg-in 39.1%

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

      4. metadata-eval 39.1%

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

      5. associate-*r* 43.0%

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

      6. pow-prod-down 45.9%

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

      7. *-commutative 45.9%

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

   9. Applied egg-rr 45.9%

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

       1. exp-prod 45.9%

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

       2. exp-prod 46.1%

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

       3. *-commutative 46.1%

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

       4. exp-to-pow 46.2%

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

       5. associate-/l* 46.2%

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

       6. *-commutative 46.2%

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

       7. associate-*r* 46.2%

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

       8. metadata-eval 46.2%

          \[\leadsto {\left(\sqrt[3]{w0} \cdot \left({\left(-0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}^{0.16666666666666666} \cdot e^{\color{blue}{-0.3333333333333333} \cdot \log d}\right)\right)}^{3} \]

   11. Simplified 46.2%

       \[\leadsto {\left(\sqrt[3]{w0} \cdot \color{blue}{\left({\left(-0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}^{0.16666666666666666} \cdot e^{-0.3333333333333333 \cdot \log d}\right)}\right)}^{3} \]

   12. Taylor expanded in D around 0 49.5%

       \[\leadsto {\left(\sqrt[3]{w0} \cdot \left(\color{blue}{\left({\left(\frac{-0.25 \cdot h}{\ell}\right)}^{0.16666666666666666} \cdot \sqrt[3]{D \cdot M}\right)} \cdot e^{-0.3333333333333333 \cdot \log d}\right)\right)}^{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+143}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\left(\frac{d}{M \cdot \left(D \cdot 0.5\right)} \cdot \frac{2 \cdot d}{M \cdot D}\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot \left(\left({\left(\frac{h \cdot -0.25}{\ell}\right)}^{0.16666666666666666} \cdot \sqrt[3]{M \cdot D}\right) \cdot e^{-0.3333333333333333 \cdot \log d}\right)\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 5 \cdot 10^{+143}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\left(\frac{d\_m}{M\_m \cdot \left(D\_m \cdot 0.5\right)} \cdot \frac{2 \cdot d\_m}{M\_m \cdot D\_m}\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(-0.25 \cdot \frac{h}{\ell}\right)}^{0.16666666666666666} \cdot \sqrt[3]{D\_m \cdot \frac{M\_m \cdot w0}{d\_m}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ (* M_m D_m) (* 2.0 d_m)) 5e+143)
   (*
    w0
    (sqrt
     (-
      1.0
      (/
       h
       (* (* (/ d_m (* M_m (* D_m 0.5))) (/ (* 2.0 d_m) (* M_m D_m))) l)))))
   (pow
    (*
     (pow (* -0.25 (/ h l)) 0.16666666666666666)
     (cbrt (* D_m (/ (* M_m w0) d_m))))
    3.0)))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+143) {
		tmp = w0 * sqrt((1.0 - (h / (((d_m / (M_m * (D_m * 0.5))) * ((2.0 * d_m) / (M_m * D_m))) * l))));
	} else {
		tmp = pow((pow((-0.25 * (h / l)), 0.16666666666666666) * cbrt((D_m * ((M_m * w0) / d_m)))), 3.0);
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+143) {
		tmp = w0 * Math.sqrt((1.0 - (h / (((d_m / (M_m * (D_m * 0.5))) * ((2.0 * d_m) / (M_m * D_m))) * l))));
	} else {
		tmp = Math.pow((Math.pow((-0.25 * (h / l)), 0.16666666666666666) * Math.cbrt((D_m * ((M_m * w0) / d_m)))), 3.0);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) <= 5e+143)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h / Float64(Float64(Float64(d_m / Float64(M_m * Float64(D_m * 0.5))) * Float64(Float64(2.0 * d_m) / Float64(M_m * D_m))) * l)))));
	else
		tmp = Float64((Float64(-0.25 * Float64(h / l)) ^ 0.16666666666666666) * cbrt(Float64(D_m * Float64(Float64(M_m * w0) / d_m)))) ^ 3.0;
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 5e+143], N[(w0 * N[Sqrt[N[(1.0 - N[(h / N[(N[(N[(d$95$m / N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * d$95$m), $MachinePrecision] / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(-0.25 * N[(h / l), $MachinePrecision]), $MachinePrecision], 0.16666666666666666], $MachinePrecision] * N[Power[N[(D$95$m * N[(N[(M$95$m * w0), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 5.00000000000000012e143

   1. Initial program 90.6%

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

   3. Simplified 89.4%

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

      1. associate-*r/ 90.6%

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

      2. unpow2 90.6%

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

      3. associate-/r* 90.6%

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

      4. frac-times 70.4%

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

      5. associate-/l* 70.4%

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

      6. div-inv 70.4%

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

      7. metadata-eval 70.4%

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

   6. Applied egg-rr 70.4%

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

      1. clear-num 70.4%

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

      2. frac-times 74.5%

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

      3. *-un-lft-identity 74.5%

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

      4. times-frac 95.0%

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

      5. *-commutative 95.0%

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

      6. *-commutative 95.0%

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

   8. Applied egg-rr 95.0%

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

   if 5.00000000000000012e143 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 70.6%

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

   3. Simplified 70.6%

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

      1. add-cube-cbrt 70.6%

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

      2. pow3 70.6%

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

   6. Applied egg-rr 70.6%

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

   7. Taylor expanded in d around 0 39.1%

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

      1. exp-prod 39.1%

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

      2. unpow-prod-up 39.1%

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

      3. distribute-lft-neg-in 39.1%

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

      4. metadata-eval 39.1%

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

      5. associate-*r* 43.0%

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

      6. pow-prod-down 45.9%

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

      7. *-commutative 45.9%

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

   9. Applied egg-rr 45.9%

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

       1. exp-prod 45.9%

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

       2. exp-prod 46.1%

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

       3. *-commutative 46.1%

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

       4. exp-to-pow 46.2%

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

       5. associate-/l* 46.2%

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

       6. *-commutative 46.2%

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

       7. associate-*r* 46.2%

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

       8. metadata-eval 46.2%

          \[\leadsto {\left(\sqrt[3]{w0} \cdot \left({\left(-0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}^{0.16666666666666666} \cdot e^{\color{blue}{-0.3333333333333333} \cdot \log d}\right)\right)}^{3} \]

   11. Simplified 46.2%

       \[\leadsto {\left(\sqrt[3]{w0} \cdot \color{blue}{\left({\left(-0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}^{0.16666666666666666} \cdot e^{-0.3333333333333333 \cdot \log d}\right)}\right)}^{3} \]

   12. Taylor expanded in w0 around 0 81.1%

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

       1. associate-/l* 81.1%

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

       2. associate-/l* 84.4%

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

   14. Simplified 84.4%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+143}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\left(\frac{d}{M \cdot \left(D \cdot 0.5\right)} \cdot \frac{2 \cdot d}{M \cdot D}\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(-0.25 \cdot \frac{h}{\ell}\right)}^{0.16666666666666666} \cdot \sqrt[3]{D \cdot \frac{M \cdot w0}{d}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-312}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d\_m}\right)\right) \cdot \frac{\frac{h}{\ell}}{\frac{2 \cdot \frac{d\_m}{D\_m}}{M\_m}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ h l) -5e-312)
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       (* M_m (* 0.5 (/ D_m d_m)))
       (/ (/ h l) (/ (* 2.0 (/ d_m D_m)) M_m))))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -5e-312) {
		tmp = w0 * sqrt((1.0 - ((M_m * (0.5 * (D_m / d_m))) * ((h / l) / ((2.0 * (d_m / D_m)) / M_m)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if ((h / l) <= (-5d-312)) then
        tmp = w0 * sqrt((1.0d0 - ((m_m * (0.5d0 * (d_m / d_m_1))) * ((h / l) / ((2.0d0 * (d_m_1 / d_m)) / m_m)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -5e-312) {
		tmp = w0 * Math.sqrt((1.0 - ((M_m * (0.5 * (D_m / d_m))) * ((h / l) / ((2.0 * (d_m / D_m)) / M_m)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (h / l) <= -5e-312:
		tmp = w0 * math.sqrt((1.0 - ((M_m * (0.5 * (D_m / d_m))) * ((h / l) / ((2.0 * (d_m / D_m)) / M_m)))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(h / l) <= -5e-312)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m * Float64(0.5 * Float64(D_m / d_m))) * Float64(Float64(h / l) / Float64(Float64(2.0 * Float64(d_m / D_m)) / M_m))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((h / l) <= -5e-312)
		tmp = w0 * sqrt((1.0 - ((M_m * (0.5 * (D_m / d_m))) * ((h / l) / ((2.0 * (d_m / D_m)) / M_m)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e-312], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m * N[(0.5 * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] / N[(N[(2.0 * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision] / M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -5.0000000000022e-312

   1. Initial program 84.7%

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

   3. Simplified 83.4%

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

      1. unpow2 83.4%

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

      2. associate-*r/ 83.4%

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

      3. clear-num 83.4%

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

      4. un-div-inv 83.4%

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

      5. *-un-lft-identity 83.4%

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

      6. times-frac 83.4%

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

      7. metadata-eval 83.4%

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

      8. times-frac 83.4%

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

   6. Applied egg-rr 83.4%

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

      1. associate-*l/ 86.0%

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

      2. associate-*r/ 86.0%

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

      3. frac-times 85.4%

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

      4. *-commutative 85.4%

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

      5. times-frac 84.7%

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

   8. Applied egg-rr 84.7%

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

      1. associate-/l* 84.7%

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

      2. associate-/l* 84.7%

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

      3. times-frac 85.4%

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

      4. *-commutative 85.4%

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

      5. *-commutative 85.4%

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

      6. associate-*r/ 85.4%

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

      7. associate-/r* 86.0%

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

      8. associate-/l* 86.0%

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

   10. Simplified 86.0%

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

   if -5.0000000000022e-312 < (/.f64 h l)

   1. Initial program 93.7%

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

   3. Simplified 92.8%

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

   5. Taylor expanded in M around 0 97.2%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 83.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-312}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d\_m}\right)\right) \cdot \left(0.5 \cdot \left(D\_m \cdot \left(\frac{h}{\ell} \cdot \frac{M\_m}{d\_m}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ h l) -5e-312)
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       (* M_m (* 0.5 (/ D_m d_m)))
       (* 0.5 (* D_m (* (/ h l) (/ M_m d_m))))))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -5e-312) {
		tmp = w0 * sqrt((1.0 - ((M_m * (0.5 * (D_m / d_m))) * (0.5 * (D_m * ((h / l) * (M_m / d_m)))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if ((h / l) <= (-5d-312)) then
        tmp = w0 * sqrt((1.0d0 - ((m_m * (0.5d0 * (d_m / d_m_1))) * (0.5d0 * (d_m * ((h / l) * (m_m / d_m_1)))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -5e-312) {
		tmp = w0 * Math.sqrt((1.0 - ((M_m * (0.5 * (D_m / d_m))) * (0.5 * (D_m * ((h / l) * (M_m / d_m)))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (h / l) <= -5e-312:
		tmp = w0 * math.sqrt((1.0 - ((M_m * (0.5 * (D_m / d_m))) * (0.5 * (D_m * ((h / l) * (M_m / d_m)))))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(h / l) <= -5e-312)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m * Float64(0.5 * Float64(D_m / d_m))) * Float64(0.5 * Float64(D_m * Float64(Float64(h / l) * Float64(M_m / d_m))))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((h / l) <= -5e-312)
		tmp = w0 * sqrt((1.0 - ((M_m * (0.5 * (D_m / d_m))) * (0.5 * (D_m * ((h / l) * (M_m / d_m)))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e-312], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m * N[(0.5 * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(D$95$m * N[(N[(h / l), $MachinePrecision] * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -5.0000000000022e-312

   1. Initial program 84.7%

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

   3. Simplified 83.4%

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

      1. unpow2 83.4%

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

      2. associate-*r/ 83.4%

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

      3. clear-num 83.4%

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

      4. un-div-inv 83.4%

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

      5. *-un-lft-identity 83.4%

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

      6. times-frac 83.4%

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

      7. metadata-eval 83.4%

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

      8. times-frac 83.4%

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

   6. Applied egg-rr 83.4%

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

      1. associate-*l/ 86.0%

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

      2. associate-*r/ 86.0%

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

      3. frac-times 85.4%

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

      4. *-commutative 85.4%

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

      5. times-frac 84.7%

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

   8. Applied egg-rr 84.7%

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

      1. associate-/l* 84.7%

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

      2. associate-/l* 84.7%

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

      3. times-frac 85.4%

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

      4. *-commutative 85.4%

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

      5. *-commutative 85.4%

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

      6. associate-*r/ 85.4%

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

      7. associate-/r* 86.0%

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

      8. associate-/l* 86.0%

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

   10. Simplified 86.0%

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

   11. Taylor expanded in h around 0 83.7%

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

       1. associate-/l* 81.1%

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

       2. *-commutative 81.1%

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

       3. *-commutative 81.1%

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

       4. times-frac 82.7%

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

   13. Simplified 82.7%

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

   if -5.0000000000022e-312 < (/.f64 h l)

   1. Initial program 93.7%

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

   3. Simplified 92.8%

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

   5. Taylor expanded in M around 0 97.2%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 86.5% accurate, 1.8× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (*
  w0
  (sqrt
   (-
    1.0
    (/ h (* (/ d_m (* M_m (* D_m 0.5))) (* l (/ (* 2.0 (/ d_m D_m)) M_m))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
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 / (M_m * (D_m * 0.5))) * (l * ((2.0 * (d_m / D_m)) / M_m))))));
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
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_1 / (m_m * (d_m * 0.5d0))) * (l * ((2.0d0 * (d_m_1 / d_m)) / m_m))))))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
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 / (M_m * (D_m * 0.5))) * (l * ((2.0 * (d_m / D_m)) / M_m))))));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
def code(w0, M_m, D_m, h, l, d_m):
	return w0 * math.sqrt((1.0 - (h / ((d_m / (M_m * (D_m * 0.5))) * (l * ((2.0 * (d_m / D_m)) / M_m))))))

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
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(M_m * Float64(D_m * 0.5))) * Float64(l * Float64(Float64(2.0 * Float64(d_m / D_m)) / M_m)))))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
function tmp = code(w0, M_m, D_m, h, l, d_m)
	tmp = w0 * sqrt((1.0 - (h / ((d_m / (M_m * (D_m * 0.5))) * (l * ((2.0 * (d_m / D_m)) / M_m))))));
end

Wolfram

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

Derivation

1. Initial program 88.6%

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

3. Simplified 87.5%

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

   1. associate-*r/ 88.6%

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

   2. unpow2 88.6%

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

   3. associate-/r* 88.6%

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

   4. frac-times 70.4%

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

   5. associate-/l* 70.4%

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

   6. div-inv 70.4%

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

   7. metadata-eval 70.4%

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

6. Applied egg-rr 70.4%

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

   1. clear-num 70.4%

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

   2. frac-times 74.1%

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

   3. *-un-lft-identity 74.1%

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

   4. times-frac 92.5%

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

   5. *-commutative 92.5%

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

   6. *-commutative 92.5%

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

8. Applied egg-rr 92.5%

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

   1. pow1 92.5%

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

10. Applied egg-rr 93.4%

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

    1. unpow1 93.4%

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

    2. associate-*l* 93.4%

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

    3. *-commutative 93.4%

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

    4. *-commutative 93.4%

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

    5. times-frac 93.6%

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

    6. *-commutative 93.6%

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

    7. *-commutative 93.6%

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

    8. associate-*r/ 93.6%

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

    9. associate-/r* 92.2%

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

    10. associate-/l* 92.2%

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

12. Simplified 92.2%

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

Alternative 6: 86.0% accurate, 1.8× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (*
  w0
  (sqrt
   (-
    1.0
    (/ h (* (* (/ d_m (* M_m (* D_m 0.5))) (/ (* 2.0 d_m) (* M_m D_m))) l))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
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 / (M_m * (D_m * 0.5))) * ((2.0 * d_m) / (M_m * D_m))) * l))));
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
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_1 / (m_m * (d_m * 0.5d0))) * ((2.0d0 * d_m_1) / (m_m * d_m))) * l))))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
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 / (M_m * (D_m * 0.5))) * ((2.0 * d_m) / (M_m * D_m))) * l))));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
def code(w0, M_m, D_m, h, l, d_m):
	return w0 * math.sqrt((1.0 - (h / (((d_m / (M_m * (D_m * 0.5))) * ((2.0 * d_m) / (M_m * D_m))) * l))))

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
function code(w0, M_m, D_m, h, l, d_m)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h / Float64(Float64(Float64(d_m / Float64(M_m * Float64(D_m * 0.5))) * Float64(Float64(2.0 * d_m) / Float64(M_m * D_m))) * l)))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
function tmp = code(w0, M_m, D_m, h, l, d_m)
	tmp = w0 * sqrt((1.0 - (h / (((d_m / (M_m * (D_m * 0.5))) * ((2.0 * d_m) / (M_m * D_m))) * l))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0 * N[Sqrt[N[(1.0 - N[(h / N[(N[(N[(d$95$m / N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * d$95$m), $MachinePrecision] / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.6%

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

3. Simplified 87.5%

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

   1. associate-*r/ 88.6%

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

   2. unpow2 88.6%

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

   3. associate-/r* 88.6%

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

   4. frac-times 70.4%

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

   5. associate-/l* 70.4%

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

   6. div-inv 70.4%

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

   7. metadata-eval 70.4%

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

6. Applied egg-rr 70.4%

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

   1. clear-num 70.4%

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

   2. frac-times 74.1%

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

   3. *-un-lft-identity 74.1%

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

   4. times-frac 92.5%

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

   5. *-commutative 92.5%

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

   6. *-commutative 92.5%

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

8. Applied egg-rr 92.5%

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

9. Final simplification 92.5%

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

Alternative 7: 67.5% accurate, 216.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0 \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(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);
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)
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);
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)
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)
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);
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]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := w0

Derivation

1. Initial program 88.6%

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

3. Simplified 87.5%

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

5. Taylor expanded in M around 0 71.1%

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

Reproduce

herbie shell --seed 2024166 
(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))))))