Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 88.3%

Time: 20.6s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.3% 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, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 2 \cdot 10^{+163}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\sqrt[3]{\frac{{\left(\left(M\_m \cdot D\_m\right) \cdot \frac{0.5}{d\_m}\right)}^{2}}{\ell}}\right)}^{3} \cdot h}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot \left(e^{\log \left(-0.25 \cdot \frac{h \cdot {\left(M\_m \cdot D\_m\right)}^{2}}{\ell}\right) \cdot 0.16666666666666666} \cdot e^{0.16666666666666666 \cdot \left(-2 \cdot \log d\_m\right)}\right)\right)}^{3}\\ \end{array} \end{array} \]

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)) 2e+163)
   (*
    w0
    (sqrt
     (-
      1.0
      (* (pow (cbrt (/ (pow (* (* M_m D_m) (/ 0.5 d_m)) 2.0) l)) 3.0) h))))
   (pow
    (*
     (cbrt w0)
     (*
      (exp
       (*
        (log (* -0.25 (/ (* h (pow (* M_m D_m) 2.0)) l)))
        0.16666666666666666))
      (exp (* 0.16666666666666666 (* -2.0 (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)) <= 2e+163) {
		tmp = w0 * sqrt((1.0 - (pow(cbrt((pow(((M_m * D_m) * (0.5 / d_m)), 2.0) / l)), 3.0) * h)));
	} else {
		tmp = pow((cbrt(w0) * (exp((log((-0.25 * ((h * pow((M_m * D_m), 2.0)) / l))) * 0.16666666666666666)) * exp((0.16666666666666666 * (-2.0 * log(d_m)))))), 3.0);
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
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)) <= 2e+163) {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow(Math.cbrt((Math.pow(((M_m * D_m) * (0.5 / d_m)), 2.0) / l)), 3.0) * h)));
	} else {
		tmp = Math.pow((Math.cbrt(w0) * (Math.exp((Math.log((-0.25 * ((h * Math.pow((M_m * D_m), 2.0)) / l))) * 0.16666666666666666)) * Math.exp((0.16666666666666666 * (-2.0 * 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)) <= 2e+163)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((cbrt(Float64((Float64(Float64(M_m * D_m) * Float64(0.5 / d_m)) ^ 2.0) / l)) ^ 3.0) * h))));
	else
		tmp = Float64(cbrt(w0) * Float64(exp(Float64(log(Float64(-0.25 * Float64(Float64(h * (Float64(M_m * D_m) ^ 2.0)) / l))) * 0.16666666666666666)) * exp(Float64(0.16666666666666666 * Float64(-2.0 * log(d_m)))))) ^ 3.0;
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
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], 2e+163], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[Power[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[(N[Exp[N[(N[Log[N[(-0.25 * N[(N[(h * N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $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)) < 1.9999999999999999e163

   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 85.2%

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

      1. clear-num 85.2%

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

      2. un-div-inv 86.3%

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

      3. *-commutative 86.3%

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

      4. associate-*l/ 86.0%

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

      5. associate-*r/ 85.4%

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

      6. div-inv 85.4%

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

      7. metadata-eval 85.4%

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

   6. Applied egg-rr 85.4%

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

      1. associate-/r/ 89.6%

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

      2. associate-*r/ 90.6%

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

      3. *-commutative 90.6%

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

      4. associate-/l* 90.9%

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

      5. associate-*r/ 90.9%

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

   8. Simplified 90.9%

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

      1. add-cube-cbrt 90.9%

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

      2. pow3 90.9%

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

      3. associate-*r* 90.6%

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

   10. Applied egg-rr 90.6%

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

   if 1.9999999999999999e163 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 44.9%

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

   3. Simplified 44.9%

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

      1. add-cube-cbrt 44.9%

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

      2. pow3 44.9%

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

      3. *-commutative 44.9%

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

      4. *-commutative 44.9%

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

      5. associate-*l/ 44.9%

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

      6. associate-*r/ 44.9%

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

      7. div-inv 44.9%

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

      8. metadata-eval 44.9%

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

   6. Applied egg-rr 44.9%

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

   7. Taylor expanded in d around 0 26.9%

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

      1. distribute-rgt-in 26.9%

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

      2. exp-sum 26.9%

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

      3. distribute-lft-neg-in 26.9%

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

      4. metadata-eval 26.9%

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

      5. associate-*r* 30.2%

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

      6. unpow-prod-down 33.9%

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

   9. Applied egg-rr 33.9%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{+163}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\sqrt[3]{\frac{{\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}}\right)}^{3} \cdot h}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot \left(e^{\log \left(-0.25 \cdot \frac{h \cdot {\left(M \cdot D\right)}^{2}}{\ell}\right) \cdot 0.16666666666666666} \cdot e^{0.16666666666666666 \cdot \left(-2 \cdot \log d\right)}\right)\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \leq 5 \cdot 10^{+295}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot {\left({\left(\sqrt[3]{0.5 \cdot \left(M\_m \cdot \frac{D\_m}{d\_m}\right)}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\ell}}\right)}^{3}}\\ \end{array} \end{array} \]

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 (<= (- 1.0 (* (/ h l) (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0))) 5e+295)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* D_m (/ (/ M_m 2.0) d_m)) 2.0)))))
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       h
       (pow
        (* (pow (cbrt (* 0.5 (* M_m (/ D_m d_m)))) 2.0) (cbrt (/ 1.0 l)))
        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 ((1.0 - ((h / l) * pow(((M_m * D_m) / (2.0 * d_m)), 2.0))) <= 5e+295) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((D_m * ((M_m / 2.0) / d_m)), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 - (h * pow((pow(cbrt((0.5 * (M_m * (D_m / d_m)))), 2.0) * cbrt((1.0 / l))), 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 ((1.0 - ((h / l) * Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0))) <= 5e+295) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((D_m * ((M_m / 2.0) / d_m)), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * Math.pow((Math.pow(Math.cbrt((0.5 * (M_m * (D_m / d_m)))), 2.0) * Math.cbrt((1.0 / l))), 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(1.0 - Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0))) <= 5e+295)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D_m * Float64(Float64(M_m / 2.0) / d_m)) ^ 2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * (Float64((cbrt(Float64(0.5 * Float64(M_m * Float64(D_m / d_m)))) ^ 2.0) * cbrt(Float64(1.0 / l))) ^ 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[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+295], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[Power[N[(N[Power[N[Power[N[(0.5 * N[(M$95$m * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(1.0 / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 4.99999999999999991e295

   1. Initial program 99.6%

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

   3. Simplified 99.4%

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

   if 4.99999999999999991e295 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 37.6%

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

   3. Simplified 38.9%

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

      1. clear-num 38.9%

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

      2. un-div-inv 42.1%

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

      3. *-commutative 42.1%

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

      4. associate-*l/ 40.8%

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

      5. associate-*r/ 42.0%

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

      6. div-inv 42.0%

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

      7. metadata-eval 42.0%

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

   6. Applied egg-rr 42.0%

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

      1. associate-/r/ 56.3%

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

      2. associate-*r/ 56.3%

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

      3. *-commutative 56.3%

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

      4. associate-/l* 57.5%

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

      5. associate-*r/ 57.5%

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

   8. Simplified 57.5%

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

      1. add-cube-cbrt 57.6%

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

      2. pow3 57.6%

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

      3. associate-*r* 56.3%

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

   10. Applied egg-rr 56.3%

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

       1. pow1/3 32.7%

          \[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left({\left(\frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}\right)}^{0.3333333333333333}\right)}}^{3} \cdot h} \]

       2. div-inv 32.7%

          \[\leadsto w0 \cdot \sqrt{1 - {\left({\color{blue}{\left({\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \frac{1}{\ell}\right)}}^{0.3333333333333333}\right)}^{3} \cdot h} \]

       3. associate-*r* 33.9%

          \[\leadsto w0 \cdot \sqrt{1 - {\left({\left({\color{blue}{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}}^{2} \cdot \frac{1}{\ell}\right)}^{0.3333333333333333}\right)}^{3} \cdot h} \]

       4. unpow-prod-down 29.9%

          \[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left({\left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{\ell}\right)}^{0.3333333333333333}\right)}}^{3} \cdot h} \]

   12. Applied egg-rr 32.3%

       \[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left({\left(\sqrt[3]{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}\right)}^{2} \cdot {\left(\frac{1}{\ell}\right)}^{0.3333333333333333}\right)}}^{3} \cdot h} \]
   13. Step-by-step derivation

       1. unpow1/3 62.3%

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

       2. associate-*r* 62.3%

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

       3. *-commutative 62.3%

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

       4. associate-*r* 62.3%

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

   14. Simplified 62.3%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+295}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot {\left({\left(\sqrt[3]{0.5 \cdot \left(M \cdot \frac{D}{d}\right)}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\ell}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \leq 10^{+306}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{2}}{d\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot {\left(\frac{{\left(\sqrt[3]{0.5 \cdot \left(M\_m \cdot \frac{D\_m}{d\_m}\right)}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \end{array} \end{array} \]

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 (<= (- 1.0 (* (/ h l) (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0))) 1e+306)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* D_m (/ (/ M_m 2.0) d_m)) 2.0)))))
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       h
       (pow
        (/ (pow (cbrt (* 0.5 (* M_m (/ D_m d_m)))) 2.0) (cbrt l))
        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 ((1.0 - ((h / l) * pow(((M_m * D_m) / (2.0 * d_m)), 2.0))) <= 1e+306) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((D_m * ((M_m / 2.0) / d_m)), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 - (h * pow((pow(cbrt((0.5 * (M_m * (D_m / d_m)))), 2.0) / cbrt(l)), 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 ((1.0 - ((h / l) * Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0))) <= 1e+306) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((D_m * ((M_m / 2.0) / d_m)), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * Math.pow((Math.pow(Math.cbrt((0.5 * (M_m * (D_m / d_m)))), 2.0) / Math.cbrt(l)), 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(1.0 - Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0))) <= 1e+306)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D_m * Float64(Float64(M_m / 2.0) / d_m)) ^ 2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * (Float64((cbrt(Float64(0.5 * Float64(M_m * Float64(D_m / d_m)))) ^ 2.0) / cbrt(l)) ^ 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[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+306], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(N[(M$95$m / 2.0), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[Power[N[(N[Power[N[Power[N[(0.5 * N[(M$95$m * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 1.00000000000000002e306

   1. Initial program 99.6%

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

   3. Simplified 99.4%

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

   if 1.00000000000000002e306 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 36.8%

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

   3. Simplified 38.0%

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

      1. clear-num 38.0%

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

      2. un-div-inv 41.3%

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

      3. *-commutative 41.3%

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

      4. associate-*l/ 40.0%

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

      5. associate-*r/ 41.3%

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

      6. div-inv 41.3%

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

      7. metadata-eval 41.3%

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

   6. Applied egg-rr 41.3%

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

      1. associate-/r/ 55.8%

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

      2. associate-*r/ 55.7%

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

      3. *-commutative 55.7%

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

      4. associate-/l* 57.0%

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

      5. associate-*r/ 57.0%

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

   8. Simplified 57.0%

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

      1. add-cube-cbrt 57.0%

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

      2. pow3 57.0%

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

      3. associate-*r* 55.7%

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

   10. Applied egg-rr 55.7%

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

       1. cbrt-div 55.7%

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

       2. associate-*r* 57.0%

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

       3. unpow2 57.0%

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

       4. cbrt-prod 63.3%

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

       5. pow2 63.3%

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

       6. *-commutative 63.3%

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

       7. associate-*r/ 63.3%

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

       8. metadata-eval 63.3%

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

       9. div-inv 63.3%

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

       10. associate-/r/ 61.9%

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

       11. div-inv 61.8%

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

       12. div-inv 61.8%

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

       13. metadata-eval 61.8%

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

       14. clear-num 61.8%

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

       15. associate-*l* 61.8%

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

   12. Applied egg-rr 61.8%

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

       1. associate-*r* 61.8%

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

       2. *-commutative 61.8%

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

       3. associate-*r* 61.8%

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

   14. Simplified 61.8%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 10^{+306}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot {\left(\frac{{\left(\sqrt[3]{0.5 \cdot \left(M \cdot \frac{D}{d}\right)}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 2 \cdot 10^{+163}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\sqrt[3]{\frac{{\left(\left(M\_m \cdot D\_m\right) \cdot \frac{0.5}{d\_m}\right)}^{2}}{\ell}}\right)}^{3} \cdot h}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(-2 \cdot \log d\_m + \log \left(0.25 \cdot \frac{h \cdot {\left(M\_m \cdot D\_m\right)}^{2}}{-\ell}\right)\right)}\right)}^{3}\\ \end{array} \end{array} \]

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)) 2e+163)
   (*
    w0
    (sqrt
     (-
      1.0
      (* (pow (cbrt (/ (pow (* (* M_m D_m) (/ 0.5 d_m)) 2.0) l)) 3.0) h))))
   (pow
    (*
     (cbrt w0)
     (exp
      (*
       0.16666666666666666
       (+
        (* -2.0 (log d_m))
        (log (* 0.25 (/ (* h (pow (* M_m D_m) 2.0)) (- l))))))))
    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)) <= 2e+163) {
		tmp = w0 * sqrt((1.0 - (pow(cbrt((pow(((M_m * D_m) * (0.5 / d_m)), 2.0) / l)), 3.0) * h)));
	} else {
		tmp = pow((cbrt(w0) * exp((0.16666666666666666 * ((-2.0 * log(d_m)) + log((0.25 * ((h * pow((M_m * D_m), 2.0)) / -l))))))), 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)) <= 2e+163) {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow(Math.cbrt((Math.pow(((M_m * D_m) * (0.5 / d_m)), 2.0) / l)), 3.0) * h)));
	} else {
		tmp = Math.pow((Math.cbrt(w0) * Math.exp((0.16666666666666666 * ((-2.0 * Math.log(d_m)) + Math.log((0.25 * ((h * Math.pow((M_m * D_m), 2.0)) / -l))))))), 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)) <= 2e+163)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((cbrt(Float64((Float64(Float64(M_m * D_m) * Float64(0.5 / d_m)) ^ 2.0) / l)) ^ 3.0) * h))));
	else
		tmp = Float64(cbrt(w0) * exp(Float64(0.16666666666666666 * Float64(Float64(-2.0 * log(d_m)) + log(Float64(0.25 * Float64(Float64(h * (Float64(M_m * D_m) ^ 2.0)) / Float64(-l)))))))) ^ 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], 2e+163], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[Power[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[(N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision] + N[Log[N[(0.25 * N[(N[(h * N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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)) < 1.9999999999999999e163

   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 85.2%

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

      1. clear-num 85.2%

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

      2. un-div-inv 86.3%

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

      3. *-commutative 86.3%

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

      4. associate-*l/ 86.0%

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

      5. associate-*r/ 85.4%

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

      6. div-inv 85.4%

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

      7. metadata-eval 85.4%

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

   6. Applied egg-rr 85.4%

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

      1. associate-/r/ 89.6%

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

      2. associate-*r/ 90.6%

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

      3. *-commutative 90.6%

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

      4. associate-/l* 90.9%

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

      5. associate-*r/ 90.9%

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

   8. Simplified 90.9%

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

      1. add-cube-cbrt 90.9%

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

      2. pow3 90.9%

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

      3. associate-*r* 90.6%

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

   10. Applied egg-rr 90.6%

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

   if 1.9999999999999999e163 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 44.9%

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

   3. Simplified 44.9%

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

      1. add-cube-cbrt 44.9%

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

      2. pow3 44.9%

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

      3. *-commutative 44.9%

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

      4. *-commutative 44.9%

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

      5. associate-*l/ 44.9%

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

      6. associate-*r/ 44.9%

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

      7. div-inv 44.9%

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

      8. metadata-eval 44.9%

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

   6. Applied egg-rr 44.9%

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

   7. Taylor expanded in d around 0 26.9%

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

      1. pow1 26.9%

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

      2. associate-*r* 30.3%

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

      3. unpow-prod-down 33.9%

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

   9. Applied egg-rr 33.9%

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

       1. unpow1 33.9%

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

       2. *-commutative 33.9%

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

   11. Simplified 33.9%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{+163}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\sqrt[3]{\frac{{\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}}\right)}^{3} \cdot h}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(-2 \cdot \log d + \log \left(0.25 \cdot \frac{h \cdot {\left(M \cdot D\right)}^{2}}{-\ell}\right)\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 5 \cdot 10^{+131}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\sqrt[3]{\frac{{\left(\left(M\_m \cdot D\_m\right) \cdot \frac{0.5}{d\_m}\right)}^{2}}{\ell}}\right)}^{3} \cdot h}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot \left({\left(-0.25 \cdot \left({\left(M\_m \cdot D\_m\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}^{0.16666666666666666} \cdot e^{\log d\_m \cdot -0.3333333333333333}\right)\right)}^{3}\\ \end{array} \end{array} \]

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+131)
   (*
    w0
    (sqrt
     (-
      1.0
      (* (pow (cbrt (/ (pow (* (* M_m D_m) (/ 0.5 d_m)) 2.0) l)) 3.0) h))))
   (pow
    (*
     (cbrt w0)
     (*
      (pow (* -0.25 (* (pow (* M_m D_m) 2.0) (/ h l))) 0.16666666666666666)
      (exp (* (log d_m) -0.3333333333333333))))
    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+131) {
		tmp = w0 * sqrt((1.0 - (pow(cbrt((pow(((M_m * D_m) * (0.5 / d_m)), 2.0) / l)), 3.0) * h)));
	} else {
		tmp = pow((cbrt(w0) * (pow((-0.25 * (pow((M_m * D_m), 2.0) * (h / l))), 0.16666666666666666) * exp((log(d_m) * -0.3333333333333333)))), 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+131) {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow(Math.cbrt((Math.pow(((M_m * D_m) * (0.5 / d_m)), 2.0) / l)), 3.0) * h)));
	} else {
		tmp = Math.pow((Math.cbrt(w0) * (Math.pow((-0.25 * (Math.pow((M_m * D_m), 2.0) * (h / l))), 0.16666666666666666) * Math.exp((Math.log(d_m) * -0.3333333333333333)))), 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+131)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((cbrt(Float64((Float64(Float64(M_m * D_m) * Float64(0.5 / d_m)) ^ 2.0) / l)) ^ 3.0) * h))));
	else
		tmp = Float64(cbrt(w0) * Float64((Float64(-0.25 * Float64((Float64(M_m * D_m) ^ 2.0) * Float64(h / l))) ^ 0.16666666666666666) * exp(Float64(log(d_m) * -0.3333333333333333)))) ^ 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+131], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[Power[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[(N[Power[N[(-0.25 * N[(N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.16666666666666666], $MachinePrecision] * N[Exp[N[(N[Log[d$95$m], $MachinePrecision] * -0.3333333333333333), $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)) < 4.99999999999999995e131

   1. Initial program 85.2%

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

   3. Simplified 85.5%

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

      1. clear-num 85.5%

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

      2. un-div-inv 86.5%

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

      3. *-commutative 86.5%

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

      4. associate-*l/ 86.3%

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

      5. associate-*r/ 85.6%

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

      6. div-inv 85.6%

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

      7. metadata-eval 85.6%

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

   6. Applied egg-rr 85.6%

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

      1. associate-/r/ 89.9%

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

      2. associate-*r/ 90.9%

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

      3. *-commutative 90.9%

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

      4. associate-/l* 91.2%

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

      5. associate-*r/ 91.2%

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

   8. Simplified 91.2%

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

      1. add-cube-cbrt 91.2%

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

      2. pow3 91.2%

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

      3. associate-*r* 90.9%

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

   10. Applied egg-rr 90.9%

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

   if 4.99999999999999995e131 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 47.3%

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

   3. Simplified 47.3%

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

      1. add-cube-cbrt 47.3%

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

      2. pow3 47.3%

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

      3. *-commutative 47.3%

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

      4. *-commutative 47.3%

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

      5. associate-*l/ 47.3%

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

      6. associate-*r/ 47.3%

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

      7. div-inv 47.3%

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

      8. metadata-eval 47.3%

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

   6. Applied egg-rr 47.3%

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

   7. Taylor expanded in d around 0 27.8%

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

      1. exp-prod 27.8%

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

      2. unpow-prod-up 27.8%

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

      3. distribute-lft-neg-in 27.8%

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

      4. metadata-eval 27.8%

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

      5. associate-*r* 30.5%

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

      6. unpow-prod-down 33.6%

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

   9. Applied egg-rr 33.6%

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

       1. exp-prod 33.6%

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

       2. exp-prod 34.1%

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

       3. *-commutative 34.1%

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

       4. exp-to-pow 34.1%

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

       5. associate-/l* 30.0%

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

       6. associate-*r* 30.0%

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

       7. metadata-eval 30.0%

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

   11. Simplified 30.0%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+131}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\sqrt[3]{\frac{{\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}}\right)}^{3} \cdot h}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot \left({\left(-0.25 \cdot \left({\left(M \cdot D\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}^{0.16666666666666666} \cdot e^{\log d \cdot -0.3333333333333333}\right)\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 1.95 \cdot 10^{+164}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \left(h \cdot \frac{w0}{\ell}\right) \cdot {\left(D\_m \cdot \frac{M\_m}{d\_m}\right)}^{2}, w0\right)\\ \end{array} \end{array} \]

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 (<= D_m 1.95e+164)
   w0
   (fma -0.125 (* (* h (/ w0 l)) (pow (* D_m (/ M_m d_m)) 2.0)) 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 (D_m <= 1.95e+164) {
		tmp = w0;
	} else {
		tmp = fma(-0.125, ((h * (w0 / l)) * pow((D_m * (M_m / d_m)), 2.0)), 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 (D_m <= 1.95e+164)
		tmp = w0;
	else
		tmp = fma(-0.125, Float64(Float64(h * Float64(w0 / l)) * (Float64(D_m * Float64(M_m / d_m)) ^ 2.0)), w0);
	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[D$95$m, 1.95e+164], w0, N[(-0.125 * N[(N[(h * N[(w0 / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.94999999999999993e164

   1. Initial program 82.2%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in D around 0 73.9%

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

   if 1.94999999999999993e164 < D

   1. Initial program 70.9%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in D around 0 17.8%

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

      1. +-commutative 17.8%

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

      2. fma-define 17.8%

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

      3. associate-*r* 18.0%

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

      4. unpow2 18.0%

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

      5. unpow2 18.0%

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

      6. swap-sqr 53.1%

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

      7. unpow2 53.1%

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

   7. Simplified 53.1%

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

   8. Taylor expanded in D around 0 17.8%

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

      1. associate-*r* 18.0%

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

      2. unpow2 18.0%

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

      3. unpow2 18.0%

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

      4. swap-sqr 53.1%

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

      5. unpow2 53.1%

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

      6. *-commutative 53.1%

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

      7. associate-*r/ 53.1%

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

      8. associate-/r* 57.4%

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

   10. Simplified 57.4%

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

   11. Taylor expanded in D around 0 17.8%

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

       1. associate-*r* 18.0%

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

       2. *-commutative 18.0%

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

       3. unpow2 18.0%

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

       4. unpow2 18.0%

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

       5. swap-sqr 53.1%

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

       6. unpow2 53.1%

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

       7. *-commutative 53.1%

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

       8. times-frac 57.4%

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

       9. associate-*r/ 53.1%

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

       10. unpow2 53.1%

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

       11. unpow2 53.1%

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

       12. times-frac 66.6%

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

       13. unpow1 66.6%

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

       14. pow-plus 66.6%

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

       15. associate-/l* 66.8%

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

       16. metadata-eval 66.8%

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

   13. Simplified 66.8%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.95 \cdot 10^{+164}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \left(h \cdot \frac{w0}{\ell}\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, w0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.4% accurate, 1.0× speedup

Math

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

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 (/ (pow (* D_m (* M_m (/ 0.5 d_m))) 2.0) 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 * (pow((D_m * (M_m * (0.5 / d_m))), 2.0) / 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 * (m_m * (0.5d0 / d_m_1))) ** 2.0d0) / 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 * (Math.pow((D_m * (M_m * (0.5 / d_m))), 2.0) / 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 * (math.pow((D_m * (M_m * (0.5 / d_m))), 2.0) / 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(D_m * Float64(M_m * Float64(0.5 / d_m))) ^ 2.0) / 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 * (0.5 / d_m))) ^ 2.0) / l))));
end

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Simplified 81.4%

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

   1. clear-num 81.4%

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

   2. un-div-inv 82.4%

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

   3. *-commutative 82.4%

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

   4. associate-*l/ 82.2%

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

   5. associate-*r/ 81.6%

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

   6. div-inv 81.6%

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

   7. metadata-eval 81.6%

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

6. Applied egg-rr 81.6%

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

   1. associate-/r/ 85.8%

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

   2. associate-*r/ 86.8%

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

   3. *-commutative 86.8%

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

   4. associate-/l* 87.0%

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

   5. associate-*r/ 87.0%

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

8. Simplified 87.0%

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

9. Final simplification 87.0%

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

Alternative 8: 63.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.1 \cdot 10^{+78}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{w0}\right)\\ \end{array} \end{array} \]

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 1.1e+78) w0 (log (exp 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 (M_m <= 1.1e+78) {
		tmp = w0;
	} else {
		tmp = log(exp(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 (m_m <= 1.1d+78) then
        tmp = w0
    else
        tmp = log(exp(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 (M_m <= 1.1e+78) {
		tmp = w0;
	} else {
		tmp = Math.log(Math.exp(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 M_m <= 1.1e+78:
		tmp = w0
	else:
		tmp = math.log(math.exp(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 (M_m <= 1.1e+78)
		tmp = w0;
	else
		tmp = log(exp(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 (M_m <= 1.1e+78)
		tmp = w0;
	else
		tmp = log(exp(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[M$95$m, 1.1e+78], w0, N[Log[N[Exp[w0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.10000000000000007e78

   1. Initial program 84.8%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in D around 0 76.7%

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

   if 1.10000000000000007e78 < M

   1. Initial program 62.4%

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

   3. Simplified 64.7%

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

   6. Applied egg-rr 19.3%

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

      1. associate-*l/ 19.8%

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

      2. associate-/l* 22.1%

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

      3. associate-*r/ 19.8%

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

      4. *-commutative 19.8%

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

      5. associate-/l* 22.1%

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

      6. associate-*r/ 22.1%

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

   8. Simplified 22.1%

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

   9. Taylor expanded in h around 0 10.7%

      \[\leadsto \sqrt{\color{blue}{{w0}^{2}}} \]
   10. Step-by-step derivation

       1. sqrt-pow1 41.7%

          \[\leadsto \color{blue}{{w0}^{\left(\frac{2}{2}\right)}} \]

       2. metadata-eval 41.7%

          \[\leadsto {w0}^{\color{blue}{1}} \]

       3. pow1 41.7%

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

       4. add-log-exp 18.3%

          \[\leadsto \color{blue}{\log \left(e^{w0}\right)} \]

   11. Applied egg-rr 18.3%

       \[\leadsto \color{blue}{\log \left(e^{w0}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.1 \cdot 10^{+78}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{w0}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.8% 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 81.2%

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

3. Simplified 81.4%

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

5. Taylor expanded in D around 0 71.1%

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

6. Final simplification 71.1%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(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))))))