Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 91.3%

Time: 24.1s

Alternatives: 8

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.4% 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: 91.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+257}:\\ \;\;\;\;w0 \cdot {\left(e^{0.25 \cdot \left(\left(\left(2 \cdot \log M\_m + \log \left(\frac{h}{\ell} \cdot -0.25\right)\right) + 2 \cdot \log D\_m\right) + -2 \cdot \log d\_m\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(\left(M\_m \cdot 0.5\right) \cdot \frac{D\_m}{d\_m}\right)}{\ell} \cdot \left(0.5 \cdot \left(D\_m \cdot \frac{M\_m}{d\_m}\right)\right)}\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -2e+257)
   (*
    w0
    (pow
     (exp
      (*
       0.25
       (+
        (+ (+ (* 2.0 (log M_m)) (log (* (/ h l) -0.25))) (* 2.0 (log D_m)))
        (* -2.0 (log d_m)))))
     2.0))
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       (/ (* h (* (* M_m 0.5) (/ D_m d_m))) l)
       (* 0.5 (* D_m (/ M_m d_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) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+257) {
		tmp = w0 * pow(exp((0.25 * ((((2.0 * log(M_m)) + log(((h / l) * -0.25))) + (2.0 * log(D_m))) + (-2.0 * log(d_m))))), 2.0);
	} else {
		tmp = w0 * sqrt((1.0 - (((h * ((M_m * 0.5) * (D_m / d_m))) / l) * (0.5 * (D_m * (M_m / d_m))))));
	}
	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 * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)) <= (-2d+257)) then
        tmp = w0 * (exp((0.25d0 * ((((2.0d0 * log(m_m)) + log(((h / l) * (-0.25d0)))) + (2.0d0 * log(d_m))) + ((-2.0d0) * log(d_m_1))))) ** 2.0d0)
    else
        tmp = w0 * sqrt((1.0d0 - (((h * ((m_m * 0.5d0) * (d_m / d_m_1))) / l) * (0.5d0 * (d_m * (m_m / d_m_1))))))
    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 ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+257) {
		tmp = w0 * Math.pow(Math.exp((0.25 * ((((2.0 * Math.log(M_m)) + Math.log(((h / l) * -0.25))) + (2.0 * Math.log(D_m))) + (-2.0 * Math.log(d_m))))), 2.0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((h * ((M_m * 0.5) * (D_m / d_m))) / l) * (0.5 * (D_m * (M_m / d_m))))));
	}
	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 (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+257:
		tmp = w0 * math.pow(math.exp((0.25 * ((((2.0 * math.log(M_m)) + math.log(((h / l) * -0.25))) + (2.0 * math.log(D_m))) + (-2.0 * math.log(d_m))))), 2.0)
	else:
		tmp = w0 * math.sqrt((1.0 - (((h * ((M_m * 0.5) * (D_m / d_m))) / l) * (0.5 * (D_m * (M_m / d_m))))))
	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(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e+257)
		tmp = Float64(w0 * (exp(Float64(0.25 * Float64(Float64(Float64(Float64(2.0 * log(M_m)) + log(Float64(Float64(h / l) * -0.25))) + Float64(2.0 * log(D_m))) + Float64(-2.0 * log(d_m))))) ^ 2.0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h * Float64(Float64(M_m * 0.5) * Float64(D_m / d_m))) / l) * Float64(0.5 * Float64(D_m * Float64(M_m / d_m)))))));
	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 * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -2e+257)
		tmp = w0 * (exp((0.25 * ((((2.0 * log(M_m)) + log(((h / l) * -0.25))) + (2.0 * log(D_m))) + (-2.0 * log(d_m))))) ^ 2.0);
	else
		tmp = w0 * sqrt((1.0 - (((h * ((M_m * 0.5) * (D_m / d_m))) / l) * (0.5 * (D_m * (M_m / d_m))))));
	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[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+257], N[(w0 * N[Power[N[Exp[N[(0.25 * N[(N[(N[(N[(2.0 * N[Log[M$95$m], $MachinePrecision]), $MachinePrecision] + N[Log[N[(N[(h / l), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Log[D$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h * N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -2.00000000000000006e257

   1. Initial program 60.5%

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

   3. Simplified 63.2%

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

      1. add-sqr-sqrt 63.2%

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

      2. pow2 63.2%

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

   6. Applied egg-rr 63.2%

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

   7. Taylor expanded in d around 0 35.6%

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

   8. Taylor expanded in D around 0 9.1%

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

      1. distribute-lft-neg-in 9.1%

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

      2. metadata-eval 9.1%

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

      3. associate-/l* 8.3%

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

   10. Simplified 8.3%

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

   11. Taylor expanded in M around 0 2.7%

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

       1. +-commutative 2.7%

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

   13. Simplified 2.7%

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

   if -2.00000000000000006e257 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 89.2%

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

   3. Simplified 90.3%

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

      1. associate-*r/ 97.1%

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

      2. add-sqr-sqrt 97.1%

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

      3. pow2 97.1%

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

      4. unpow2 97.1%

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

      5. sqrt-prod 62.6%

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

      6. add-sqr-sqrt 97.1%

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

      7. *-un-lft-identity 97.1%

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

      8. times-frac 97.1%

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

      9. metadata-eval 97.1%

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

   6. Applied egg-rr 97.1%

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

      1. associate-*r/ 90.3%

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

      2. *-commutative 90.3%

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

      3. unpow2 90.3%

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

      4. associate-*r* 91.5%

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

      5. associate-*r/ 91.5%

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

      6. associate-*r/ 91.5%

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

   8. Applied egg-rr 91.5%

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

      1. associate-*l/ 99.4%

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

      2. associate-*r/ 99.4%

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

      3. associate-*r* 99.4%

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

   10. Applied egg-rr 99.4%

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

   11. Taylor expanded in M around 0 97.0%

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

       1. associate-/l* 98.4%

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

   13. Simplified 98.4%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+257}:\\ \;\;\;\;w0 \cdot {\left(e^{0.25 \cdot \left(\left(\left(2 \cdot \log M + \log \left(\frac{h}{\ell} \cdot -0.25\right)\right) + 2 \cdot \log D\right) + -2 \cdot \log d\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}{\ell} \cdot \left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+265}:\\ \;\;\;\;w0 \cdot {\left(e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M\_m}^{2}}{\ell}\right) + \left(2 \cdot \log D\_m + -2 \cdot \log d\_m\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(\left(M\_m \cdot 0.5\right) \cdot \frac{D\_m}{d\_m}\right)}{\ell} \cdot \left(0.5 \cdot \left(D\_m \cdot \frac{M\_m}{d\_m}\right)\right)}\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -2e+265)
   (*
    w0
    (pow
     (exp
      (*
       0.25
       (+
        (log (* -0.25 (/ (* h (pow M_m 2.0)) l)))
        (+ (* 2.0 (log D_m)) (* -2.0 (log d_m))))))
     2.0))
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       (/ (* h (* (* M_m 0.5) (/ D_m d_m))) l)
       (* 0.5 (* D_m (/ M_m d_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) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+265) {
		tmp = w0 * pow(exp((0.25 * (log((-0.25 * ((h * pow(M_m, 2.0)) / l))) + ((2.0 * log(D_m)) + (-2.0 * log(d_m)))))), 2.0);
	} else {
		tmp = w0 * sqrt((1.0 - (((h * ((M_m * 0.5) * (D_m / d_m))) / l) * (0.5 * (D_m * (M_m / d_m))))));
	}
	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 * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)) <= (-2d+265)) then
        tmp = w0 * (exp((0.25d0 * (log(((-0.25d0) * ((h * (m_m ** 2.0d0)) / l))) + ((2.0d0 * log(d_m)) + ((-2.0d0) * log(d_m_1)))))) ** 2.0d0)
    else
        tmp = w0 * sqrt((1.0d0 - (((h * ((m_m * 0.5d0) * (d_m / d_m_1))) / l) * (0.5d0 * (d_m * (m_m / d_m_1))))))
    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 ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+265) {
		tmp = w0 * Math.pow(Math.exp((0.25 * (Math.log((-0.25 * ((h * Math.pow(M_m, 2.0)) / l))) + ((2.0 * Math.log(D_m)) + (-2.0 * Math.log(d_m)))))), 2.0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((h * ((M_m * 0.5) * (D_m / d_m))) / l) * (0.5 * (D_m * (M_m / d_m))))));
	}
	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 (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -2e+265:
		tmp = w0 * math.pow(math.exp((0.25 * (math.log((-0.25 * ((h * math.pow(M_m, 2.0)) / l))) + ((2.0 * math.log(D_m)) + (-2.0 * math.log(d_m)))))), 2.0)
	else:
		tmp = w0 * math.sqrt((1.0 - (((h * ((M_m * 0.5) * (D_m / d_m))) / l) * (0.5 * (D_m * (M_m / d_m))))))
	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(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -2e+265)
		tmp = Float64(w0 * (exp(Float64(0.25 * Float64(log(Float64(-0.25 * Float64(Float64(h * (M_m ^ 2.0)) / l))) + Float64(Float64(2.0 * log(D_m)) + Float64(-2.0 * log(d_m)))))) ^ 2.0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h * Float64(Float64(M_m * 0.5) * Float64(D_m / d_m))) / l) * Float64(0.5 * Float64(D_m * Float64(M_m / d_m)))))));
	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 * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -2e+265)
		tmp = w0 * (exp((0.25 * (log((-0.25 * ((h * (M_m ^ 2.0)) / l))) + ((2.0 * log(D_m)) + (-2.0 * log(d_m)))))) ^ 2.0);
	else
		tmp = w0 * sqrt((1.0 - (((h * ((M_m * 0.5) * (D_m / d_m))) / l) * (0.5 * (D_m * (M_m / d_m))))));
	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[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+265], N[(w0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[(-0.25 * N[(N[(h * N[Power[M$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(2.0 * N[Log[D$95$m], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h * N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -2.00000000000000013e265

   1. Initial program 59.4%

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

   3. Simplified 62.2%

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

      1. add-sqr-sqrt 62.1%

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

      2. pow2 62.1%

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

   6. Applied egg-rr 62.1%

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

   7. Taylor expanded in d around 0 36.6%

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

   8. Taylor expanded in l around 0 22.4%

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

      1. mul-1-neg 22.4%

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

      2. unsub-neg 22.4%

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

      3. distribute-lft-neg-in 22.4%

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

      4. metadata-eval 22.4%

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

      5. associate-*r* 22.4%

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

      6. associate-*r* 23.7%

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

      7. associate-*r* 23.7%

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

      8. unpow2 23.7%

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

      9. unpow2 23.7%

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

      10. swap-sqr 23.7%

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

      11. unpow2 23.7%

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

      12. associate-*r* 23.7%

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

      13. log-div 37.9%

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

      14. associate-*r/ 37.9%

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

      15. associate-/l* 37.1%

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

   10. Simplified 37.1%

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

   11. Taylor expanded in D around 0 9.4%

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

   if -2.00000000000000013e265 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 89.3%

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

   3. Simplified 90.4%

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

      1. associate-*r/ 97.2%

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

      2. add-sqr-sqrt 97.2%

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

      3. pow2 97.2%

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

      4. unpow2 97.2%

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

      5. sqrt-prod 62.5%

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

      6. add-sqr-sqrt 97.2%

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

      7. *-un-lft-identity 97.2%

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

      8. times-frac 97.2%

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

      9. metadata-eval 97.2%

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

   6. Applied egg-rr 97.2%

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

      1. associate-*r/ 90.4%

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

      2. *-commutative 90.4%

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

      3. unpow2 90.4%

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

      4. associate-*r* 91.6%

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

      5. associate-*r/ 91.6%

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

      6. associate-*r/ 91.6%

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

   8. Applied egg-rr 91.6%

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

      1. associate-*l/ 99.4%

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

      2. associate-*r/ 99.4%

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

      3. associate-*r* 99.4%

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

   10. Applied egg-rr 99.4%

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

   11. Taylor expanded in M around 0 97.1%

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

       1. associate-/l* 97.9%

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

   13. Simplified 97.9%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+265}:\\ \;\;\;\;w0 \cdot {\left(e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell}\right) + \left(2 \cdot \log D + -2 \cdot \log d\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}{\ell} \cdot \left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-307}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(D\_m \cdot \frac{M\_m \cdot 0.5}{d\_m}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\sqrt{h} \cdot \left(D\_m \cdot \left(M\_m \cdot \frac{0.5}{d\_m}\right)\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

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 -5e-307)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* D_m (/ (* M_m 0.5) d_m)) 2.0)) l))))
   (*
    w0
    (sqrt (- 1.0 (/ (pow (* (sqrt h) (* 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) {
	double tmp;
	if (h <= -5e-307) {
		tmp = w0 * sqrt((1.0 - ((h * pow((D_m * ((M_m * 0.5) / d_m)), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (pow((sqrt(h) * (D_m * (M_m * (0.5 / d_m)))), 2.0) / l)));
	}
	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 <= (-5d-307)) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((d_m * ((m_m * 0.5d0) / d_m_1)) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (((sqrt(h) * (d_m * (m_m * (0.5d0 / d_m_1)))) ** 2.0d0) / l)))
    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 <= -5e-307) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((D_m * ((M_m * 0.5) / d_m)), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow((Math.sqrt(h) * (D_m * (M_m * (0.5 / d_m)))), 2.0) / l)));
	}
	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 <= -5e-307:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((D_m * ((M_m * 0.5) / d_m)), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (math.pow((math.sqrt(h) * (D_m * (M_m * (0.5 / d_m)))), 2.0) / l)))
	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 (h <= -5e-307)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(D_m * Float64(Float64(M_m * 0.5) / d_m)) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(sqrt(h) * Float64(D_m * Float64(M_m * Float64(0.5 / d_m)))) ^ 2.0) / l))));
	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 <= -5e-307)
		tmp = w0 * sqrt((1.0 - ((h * ((D_m * ((M_m * 0.5) / d_m)) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 - (((sqrt(h) * (D_m * (M_m * (0.5 / d_m)))) ^ 2.0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[h, -5e-307], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(D$95$m * N[(N[(M$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[Sqrt[h], $MachinePrecision] * N[(D$95$m * N[(M$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -5.00000000000000014e-307

   1. Initial program 82.5%

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

   3. Simplified 83.3%

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

      1. associate-*r/ 88.6%

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

      2. add-sqr-sqrt 88.6%

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

      3. pow2 88.6%

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

      4. unpow2 88.6%

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

      5. sqrt-prod 53.1%

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

      6. add-sqr-sqrt 88.6%

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

      7. *-un-lft-identity 88.6%

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

      8. times-frac 88.6%

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

      9. metadata-eval 88.6%

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

   6. Applied egg-rr 88.6%

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

      1. associate-*r* 88.6%

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

      2. clear-num 88.6%

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

      3. un-div-inv 88.6%

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

   8. Applied egg-rr 88.6%

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

      1. associate-/r/ 86.5%

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

   10. Simplified 86.5%

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

   if -5.00000000000000014e-307 < h

   1. Initial program 79.9%

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

   3. Simplified 82.4%

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

      1. associate-*r/ 88.9%

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

      2. add-sqr-sqrt 88.9%

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

      3. pow2 88.9%

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

      4. unpow2 88.9%

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

      5. sqrt-prod 53.9%

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

      6. add-sqr-sqrt 88.9%

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

      7. *-un-lft-identity 88.9%

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

      8. times-frac 88.9%

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

      9. metadata-eval 88.9%

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

   6. Applied egg-rr 88.9%

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

      1. add-sqr-sqrt 88.9%

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

      2. *-un-lft-identity 88.9%

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

      3. times-frac 88.9%

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

      4. sqrt-prod 88.9%

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

      5. unpow2 88.9%

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

      6. sqrt-prod 53.9%

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

      7. add-sqr-sqrt 77.9%

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

      8. associate-*r/ 77.9%

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

   8. Applied egg-rr 94.3%

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

      1. /-rgt-identity 94.3%

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

      2. associate-*r/ 94.2%

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

      3. unpow2 94.2%

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

      4. *-commutative 94.2%

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

      5. *-commutative 94.2%

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

      6. *-commutative 94.2%

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

      7. associate-/r/ 94.3%

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

      8. associate-*r/ 94.3%

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

      9. associate-/r/ 94.3%

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

   10. Simplified 94.3%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-307}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\sqrt{h} \cdot \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right)}^{2}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D\_m \cdot 0.5}{d\_m}\\ \mathbf{if}\;M\_m \leq 1.45 \cdot 10^{-19}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \frac{\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d\_m}}{\ell}\right) \cdot \frac{D\_m \cdot \left(M\_m \cdot 0.5\right)}{d\_m}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(\frac{h}{\ell} \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* M_m (/ (* D_m 0.5) d_m))))
   (if (<= M_m 1.45e-19)
     (*
      w0
      (sqrt
       (-
        1.0
        (*
         (* h (/ (/ (* M_m (* D_m 0.5)) d_m) l))
         (/ (* D_m (* M_m 0.5)) d_m)))))
     (* w0 (sqrt (- 1.0 (* t_0 (* (/ h l) t_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 t_0 = M_m * ((D_m * 0.5) / d_m);
	double tmp;
	if (M_m <= 1.45e-19) {
		tmp = w0 * sqrt((1.0 - ((h * (((M_m * (D_m * 0.5)) / d_m) / l)) * ((D_m * (M_m * 0.5)) / d_m))));
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = m_m * ((d_m * 0.5d0) / d_m_1)
    if (m_m <= 1.45d-19) then
        tmp = w0 * sqrt((1.0d0 - ((h * (((m_m * (d_m * 0.5d0)) / d_m_1) / l)) * ((d_m * (m_m * 0.5d0)) / d_m_1))))
    else
        tmp = w0 * sqrt((1.0d0 - (t_0 * ((h / l) * t_0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
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 t_0 = M_m * ((D_m * 0.5) / d_m);
	double tmp;
	if (M_m <= 1.45e-19) {
		tmp = w0 * Math.sqrt((1.0 - ((h * (((M_m * (D_m * 0.5)) / d_m) / l)) * ((D_m * (M_m * 0.5)) / d_m))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = M_m * ((D_m * 0.5) / d_m)
	tmp = 0
	if M_m <= 1.45e-19:
		tmp = w0 * math.sqrt((1.0 - ((h * (((M_m * (D_m * 0.5)) / d_m) / l)) * ((D_m * (M_m * 0.5)) / d_m))))
	else:
		tmp = w0 * math.sqrt((1.0 - (t_0 * ((h / l) * t_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)
	t_0 = Float64(M_m * Float64(Float64(D_m * 0.5) / d_m))
	tmp = 0.0
	if (M_m <= 1.45e-19)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(Float64(Float64(M_m * Float64(D_m * 0.5)) / d_m) / l)) * Float64(Float64(D_m * Float64(M_m * 0.5)) / d_m)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(h / l) * t_0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = M_m * ((D_m * 0.5) / d_m);
	tmp = 0.0;
	if (M_m <= 1.45e-19)
		tmp = w0 * sqrt((1.0 - ((h * (((M_m * (D_m * 0.5)) / d_m) / l)) * ((D_m * (M_m * 0.5)) / d_m))));
	else
		tmp = w0 * sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 1.45e-19], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 1.45e-19

   1. Initial program 84.0%

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

   3. Simplified 84.5%

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

      1. associate-*r/ 91.3%

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

      2. add-sqr-sqrt 91.3%

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

      3. pow2 91.3%

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

      4. unpow2 91.3%

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

      5. sqrt-prod 59.1%

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

      6. add-sqr-sqrt 91.3%

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

      7. *-un-lft-identity 91.3%

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

      8. times-frac 91.3%

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

      9. metadata-eval 91.3%

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

   6. Applied egg-rr 91.3%

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

      1. associate-*r/ 84.5%

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

      2. *-commutative 84.5%

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

      3. unpow2 84.5%

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

      4. associate-*r* 86.1%

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

      5. associate-*r/ 86.1%

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

      6. associate-*r/ 86.1%

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

   8. Applied egg-rr 86.1%

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

      1. associate-*l/ 93.5%

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

      2. associate-*r/ 93.5%

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

      3. associate-*r* 93.5%

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

   10. Applied egg-rr 93.5%

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

       1. associate-/l* 93.0%

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

       2. associate-*r/ 91.1%

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

       3. associate-*l* 91.1%

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

   12. Simplified 91.1%

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

       1. associate-*r/ 91.1%

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

       2. associate-*r* 91.1%

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

   14. Applied egg-rr 91.1%

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

   if 1.45e-19 < M

   1. Initial program 73.9%

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

   3. Simplified 78.3%

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

      1. associate-*r/ 81.4%

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

      2. add-sqr-sqrt 81.4%

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

      3. pow2 81.4%

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

      4. unpow2 81.4%

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

      5. sqrt-prod 37.4%

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

      6. add-sqr-sqrt 81.4%

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

      7. *-un-lft-identity 81.4%

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

      8. times-frac 81.4%

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

      9. metadata-eval 81.4%

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

   6. Applied egg-rr 81.4%

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

      1. associate-*r/ 78.3%

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

      2. *-commutative 78.3%

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

      3. unpow2 78.3%

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

      4. associate-*r* 81.3%

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

      5. associate-*r/ 81.3%

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

      6. associate-*r/ 81.3%

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

   8. Applied egg-rr 81.3%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.45 \cdot 10^{-19}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d}}{\ell}\right) \cdot \frac{D \cdot \left(M \cdot 0.5\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \frac{D \cdot 0.5}{d}\right) \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot \frac{D \cdot 0.5}{d}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.5% accurate, 1.8× speedup

Math

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

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 (/ (/ (* M_m (* D_m 0.5)) d_m) l)) (* M_m (/ (* D_m 0.5) d_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 * (((M_m * (D_m * 0.5)) / d_m) / l)) * (M_m * ((D_m * 0.5) / d_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 * (((m_m * (d_m * 0.5d0)) / d_m_1) / l)) * (m_m * ((d_m * 0.5d0) / d_m_1)))))
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 * (((M_m * (D_m * 0.5)) / d_m) / l)) * (M_m * ((D_m * 0.5) / d_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 * (((M_m * (D_m * 0.5)) / d_m) / l)) * (M_m * ((D_m * 0.5) / d_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(Float64(h * Float64(Float64(Float64(M_m * Float64(D_m * 0.5)) / d_m) / l)) * Float64(M_m * Float64(Float64(D_m * 0.5) / d_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 * (((M_m * (D_m * 0.5)) / d_m) / l)) * (M_m * ((D_m * 0.5) / d_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[(N[(h * N[(N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.3%

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

3. Simplified 82.9%

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

   1. associate-*r/ 88.7%

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

   2. add-sqr-sqrt 88.7%

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

   3. pow2 88.7%

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

   4. unpow2 88.7%

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

   5. sqrt-prod 53.4%

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

   6. add-sqr-sqrt 88.7%

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

   7. *-un-lft-identity 88.7%

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

   8. times-frac 88.7%

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

   9. metadata-eval 88.7%

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

6. Applied egg-rr 88.7%

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

   1. associate-*r/ 82.9%

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

   2. *-commutative 82.9%

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

   3. unpow2 82.9%

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

   4. associate-*r* 84.9%

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

   5. associate-*r/ 84.9%

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

   6. associate-*r/ 84.9%

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

8. Applied egg-rr 84.9%

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

   1. associate-*l/ 91.0%

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

   2. associate-*r/ 91.0%

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

   3. associate-*r* 91.0%

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

10. Applied egg-rr 91.0%

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

    1. associate-/l* 90.3%

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

    2. associate-*r/ 87.5%

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

    3. associate-*l* 87.5%

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

12. Simplified 87.5%

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

13. Final simplification 87.5%

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

Alternative 6: 89.0% accurate, 1.8× speedup

Math

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

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 (/ (/ (* M_m (* D_m 0.5)) d_m) l)) (/ (* D_m (* M_m 0.5)) d_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 * (((M_m * (D_m * 0.5)) / d_m) / l)) * ((D_m * (M_m * 0.5)) / d_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 * (((m_m * (d_m * 0.5d0)) / d_m_1) / l)) * ((d_m * (m_m * 0.5d0)) / d_m_1))))
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 * (((M_m * (D_m * 0.5)) / d_m) / l)) * ((D_m * (M_m * 0.5)) / d_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 * (((M_m * (D_m * 0.5)) / d_m) / l)) * ((D_m * (M_m * 0.5)) / d_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(Float64(h * Float64(Float64(Float64(M_m * Float64(D_m * 0.5)) / d_m) / l)) * Float64(Float64(D_m * Float64(M_m * 0.5)) / d_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 * (((M_m * (D_m * 0.5)) / d_m) / l)) * ((D_m * (M_m * 0.5)) / d_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[(N[(h * N[(N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.3%

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

3. Simplified 82.9%

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

   1. associate-*r/ 88.7%

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

   2. add-sqr-sqrt 88.7%

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

   3. pow2 88.7%

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

   4. unpow2 88.7%

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

   5. sqrt-prod 53.4%

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

   6. add-sqr-sqrt 88.7%

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

   7. *-un-lft-identity 88.7%

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

   8. times-frac 88.7%

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

   9. metadata-eval 88.7%

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

6. Applied egg-rr 88.7%

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

   1. associate-*r/ 82.9%

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

   2. *-commutative 82.9%

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

   3. unpow2 82.9%

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

   4. associate-*r* 84.9%

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

   5. associate-*r/ 84.9%

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

   6. associate-*r/ 84.9%

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

8. Applied egg-rr 84.9%

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

   1. associate-*l/ 91.0%

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

   2. associate-*r/ 91.0%

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

   3. associate-*r* 91.0%

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

10. Applied egg-rr 91.0%

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

    1. associate-/l* 90.3%

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

    2. associate-*r/ 87.5%

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

    3. associate-*l* 87.5%

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

12. Simplified 87.5%

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

    1. associate-*r/ 87.5%

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

    2. associate-*r* 87.5%

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

14. Applied egg-rr 87.5%

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

15. Final simplification 87.5%

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

Alternative 7: 87.7% accurate, 1.8× speedup

Math

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

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 (* (* M_m 0.5) (/ D_m d_m))) l) (* 0.5 (* D_m (/ M_m d_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 * ((M_m * 0.5) * (D_m / d_m))) / l) * (0.5 * (D_m * (M_m / d_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 * ((m_m * 0.5d0) * (d_m / d_m_1))) / l) * (0.5d0 * (d_m * (m_m / d_m_1))))))
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 * ((M_m * 0.5) * (D_m / d_m))) / l) * (0.5 * (D_m * (M_m / d_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 * ((M_m * 0.5) * (D_m / d_m))) / l) * (0.5 * (D_m * (M_m / d_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(Float64(Float64(h * Float64(Float64(M_m * 0.5) * Float64(D_m / d_m))) / l) * Float64(0.5 * Float64(D_m * Float64(M_m / d_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 * ((M_m * 0.5) * (D_m / d_m))) / l) * (0.5 * (D_m * (M_m / d_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[(N[(N[(h * N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.3%

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

3. Simplified 82.9%

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

   1. associate-*r/ 88.7%

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

   2. add-sqr-sqrt 88.7%

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

   3. pow2 88.7%

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

   4. unpow2 88.7%

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

   5. sqrt-prod 53.4%

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

   6. add-sqr-sqrt 88.7%

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

   7. *-un-lft-identity 88.7%

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

   8. times-frac 88.7%

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

   9. metadata-eval 88.7%

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

6. Applied egg-rr 88.7%

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

   1. associate-*r/ 82.9%

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

   2. *-commutative 82.9%

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

   3. unpow2 82.9%

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

   4. associate-*r* 84.9%

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

   5. associate-*r/ 84.9%

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

   6. associate-*r/ 84.9%

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

8. Applied egg-rr 84.9%

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

   1. associate-*l/ 91.0%

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

   2. associate-*r/ 91.0%

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

   3. associate-*r* 91.0%

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

10. Applied egg-rr 91.0%

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

11. Taylor expanded in M around 0 88.2%

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

    1. associate-/l* 89.5%

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

13. Simplified 89.5%

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

14. Final simplification 89.5%

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

Alternative 8: 68.3% accurate, 216.0× speedup

Math

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

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 81.3%

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

3. Simplified 82.9%

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

5. Taylor expanded in M around 0 68.6%

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

6. Final simplification 68.6%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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))))))