Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.5% → 88.3%

Time: 20.3s

Alternatives: 7

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.5% 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, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{D\_m \cdot \left(M\_m \cdot 0.5\right)}{d\_m}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \frac{\left(M\_m \cdot D\_m\right) \cdot h}{d\_m \cdot \ell}\right) \cdot \frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d\_m}}\\ \end{array} \end{array} \]

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+152)
   (* w0 (sqrt (- 1.0 (/ (* (pow (/ (* D_m (* M_m 0.5)) d_m) 2.0) h) l))))
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       (* 0.5 (/ (* (* M_m D_m) h) (* 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) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 2e+152) {
		tmp = w0 * sqrt((1.0 - ((pow(((D_m * (M_m * 0.5)) / d_m), 2.0) * h) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - ((0.5 * (((M_m * D_m) * h) / (d_m * l))) * ((M_m * (D_m * 0.5)) / 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)) <= 2d+152) then
        tmp = w0 * sqrt((1.0d0 - (((((d_m * (m_m * 0.5d0)) / d_m_1) ** 2.0d0) * h) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - ((0.5d0 * (((m_m * d_m) * h) / (d_m_1 * l))) * ((m_m * (d_m * 0.5d0)) / 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 (((M_m * D_m) / (2.0 * d_m)) <= 2e+152) {
		tmp = w0 * Math.sqrt((1.0 - ((Math.pow(((D_m * (M_m * 0.5)) / d_m), 2.0) * h) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((0.5 * (((M_m * D_m) * h) / (d_m * l))) * ((M_m * (D_m * 0.5)) / 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 ((M_m * D_m) / (2.0 * d_m)) <= 2e+152:
		tmp = w0 * math.sqrt((1.0 - ((math.pow(((D_m * (M_m * 0.5)) / d_m), 2.0) * h) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - ((0.5 * (((M_m * D_m) * h) / (d_m * l))) * ((M_m * (D_m * 0.5)) / 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(M_m * D_m) / Float64(2.0 * d_m)) <= 2e+152)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * Float64(M_m * 0.5)) / d_m) ^ 2.0) * h) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.5 * Float64(Float64(Float64(M_m * D_m) * h) / Float64(d_m * l))) * Float64(Float64(M_m * Float64(D_m * 0.5)) / 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)) <= 2e+152)
		tmp = w0 * sqrt((1.0 - (((((D_m * (M_m * 0.5)) / d_m) ^ 2.0) * h) / l)));
	else
		tmp = w0 * sqrt((1.0 - ((0.5 * (((M_m * D_m) * h) / (d_m * l))) * ((M_m * (D_m * 0.5)) / 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[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2e+152], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.5 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 2.0000000000000001e152

   1. Initial program 86.4%

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

   3. Simplified 85.9%

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

      1. associate-*r/ 92.2%

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

      2. add-sqr-sqrt 92.2%

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

      3. pow2 92.2%

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

      4. sqrt-pow1 92.2%

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

      5. metadata-eval 92.2%

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

      6. pow1 92.2%

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

      7. *-un-lft-identity 92.2%

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

      8. times-frac 92.2%

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

      9. metadata-eval 92.2%

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

   6. Applied egg-rr 92.2%

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

      1. associate-*r* 92.2%

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

      2. metadata-eval 92.2%

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

      3. div-inv 92.2%

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

      4. associate-*r/ 92.6%

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

      5. div-inv 92.6%

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

      6. metadata-eval 92.6%

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

   8. Applied egg-rr 92.6%

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

   if 2.0000000000000001e152 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 56.7%

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

   3. Simplified 56.7%

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

      1. associate-*r/ 62.9%

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

      2. add-sqr-sqrt 62.9%

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

      3. pow2 62.9%

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

      4. sqrt-pow1 62.9%

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

      5. metadata-eval 62.9%

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

      6. pow1 62.9%

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

      7. *-un-lft-identity 62.9%

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

      8. times-frac 62.9%

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

      9. metadata-eval 62.9%

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

   6. Applied egg-rr 62.9%

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

      1. associate-/l* 56.7%

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

      2. unpow2 56.7%

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

      3. associate-*l* 65.9%

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

      4. associate-*r* 65.9%

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

      5. metadata-eval 65.9%

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

      6. div-inv 65.9%

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

      7. clear-num 65.9%

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

      8. div-inv 65.9%

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

      9. associate-/r* 65.9%

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

      10. associate-*r* 65.9%

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

      11. metadata-eval 65.9%

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

      12. div-inv 65.9%

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

      13. clear-num 65.9%

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

      14. div-inv 65.9%

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

      15. associate-/r* 65.9%

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

      16. associate-*l* 56.7%

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

      17. unpow2 56.7%

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

      18. *-commutative 56.7%

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

      19. unpow2 56.7%

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

   8. Applied egg-rr 65.9%

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

   9. Taylor expanded in h around 0 70.9%

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

       1. pow1 70.9%

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

       2. associate-*r* 74.4%

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

       3. *-commutative 74.4%

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

       4. associate-*r/ 77.2%

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

   11. Applied egg-rr 77.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(0.5 \cdot \frac{\left(M \cdot D\right) \cdot h}{d \cdot \ell}\right) \cdot \frac{M \cdot \left(D \cdot 0.5\right)}{d}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := w0 \cdot \sqrt{1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+270}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(M\_m \cdot \left(\left(M\_m \cdot 0.5\right) \cdot \left(0.5 \cdot \frac{D\_m}{d\_m}\right)\right)\right)}{\ell \cdot \frac{d\_m}{D\_m}}}\\ \end{array} \end{array} \]

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
         (*
          w0
          (sqrt (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)))))))
   (if (<= t_0 5e+270)
     t_0
     (*
      w0
      (sqrt
       (-
        1.0
        (/
         (* h (* M_m (* (* M_m 0.5) (* 0.5 (/ D_m d_m)))))
         (* l (/ d_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 t_0 = w0 * sqrt((1.0 - (pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 5e+270) {
		tmp = t_0;
	} else {
		tmp = w0 * sqrt((1.0 - ((h * (M_m * ((M_m * 0.5) * (0.5 * (D_m / d_m))))) / (l * (d_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) :: t_0
    real(8) :: tmp
    t_0 = w0 * sqrt((1.0d0 - ((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l))))
    if (t_0 <= 5d+270) then
        tmp = t_0
    else
        tmp = w0 * sqrt((1.0d0 - ((h * (m_m * ((m_m * 0.5d0) * (0.5d0 * (d_m / d_m_1))))) / (l * (d_m_1 / d_m)))))
    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 = w0 * Math.sqrt((1.0 - (Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 5e+270) {
		tmp = t_0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * (M_m * ((M_m * 0.5) * (0.5 * (D_m / d_m))))) / (l * (d_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):
	t_0 = w0 * math.sqrt((1.0 - (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l))))
	tmp = 0
	if t_0 <= 5e+270:
		tmp = t_0
	else:
		tmp = w0 * math.sqrt((1.0 - ((h * (M_m * ((M_m * 0.5) * (0.5 * (D_m / d_m))))) / (l * (d_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)
	t_0 = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)))))
	tmp = 0.0
	if (t_0 <= 5e+270)
		tmp = t_0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(M_m * Float64(Float64(M_m * 0.5) * Float64(0.5 * Float64(D_m / d_m))))) / Float64(l * Float64(d_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)
	t_0 = w0 * sqrt((1.0 - ((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l))));
	tmp = 0.0;
	if (t_0 <= 5e+270)
		tmp = t_0;
	else
		tmp = w0 * sqrt((1.0 - ((h * (M_m * ((M_m * 0.5) * (0.5 * (D_m / d_m))))) / (l * (d_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_] := Block[{t$95$0 = N[(w0 * N[Sqrt[N[(1.0 - 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]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+270], t$95$0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(M$95$m * N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(0.5 * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.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))))) < 4.99999999999999976e270

   1. Initial program 93.0%

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

   if 4.99999999999999976e270 < (*.f64 w0 (sqrt.f64 (-.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 41.0%

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

   3. Simplified 42.9%

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

      1. clear-num 42.9%

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

      2. un-div-inv 42.9%

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

      3. associate-/l* 42.9%

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

   6. Applied egg-rr 42.9%

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

      1. unpow2 42.9%

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

      2. associate-/r* 42.9%

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

      3. associate-*r/ 41.0%

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

      4. associate-/r* 41.0%

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

      5. div-inv 41.0%

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

      6. div-inv 41.0%

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

      7. metadata-eval 41.0%

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

      8. clear-num 41.0%

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

      9. associate-*r* 41.0%

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

      10. associate-*r/ 41.0%

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

      11. div-inv 41.0%

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

      12. metadata-eval 41.0%

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

   8. Applied egg-rr 41.0%

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

      1. frac-times 75.8%

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

      2. associate-*l* 75.8%

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

      3. associate-*r/ 75.8%

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

   10. Applied egg-rr 75.8%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 5 \cdot 10^{+270}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(M \cdot \left(\left(M \cdot 0.5\right) \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right)}{\ell \cdot \frac{d}{D}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{D\_m \cdot \left(M\_m \cdot 0.5\right)}{d\_m}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(M\_m \cdot 0.5\right) \cdot \frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d\_m}\right) \cdot \frac{h}{\frac{d\_m \cdot \ell}{D\_m}}}\\ \end{array} \end{array} \]

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+152)
   (* w0 (sqrt (- 1.0 (/ (* (pow (/ (* D_m (* M_m 0.5)) d_m) 2.0) h) l))))
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       (* (* M_m 0.5) (/ (* M_m (* D_m 0.5)) d_m))
       (/ h (/ (* d_m l) 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 (((M_m * D_m) / (2.0 * d_m)) <= 2e+152) {
		tmp = w0 * sqrt((1.0 - ((pow(((D_m * (M_m * 0.5)) / d_m), 2.0) * h) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (((M_m * 0.5) * ((M_m * (D_m * 0.5)) / d_m)) * (h / ((d_m * l) / 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)) <= 2d+152) then
        tmp = w0 * sqrt((1.0d0 - (((((d_m * (m_m * 0.5d0)) / d_m_1) ** 2.0d0) * h) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (((m_m * 0.5d0) * ((m_m * (d_m * 0.5d0)) / d_m_1)) * (h / ((d_m_1 * l) / d_m)))))
    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 * D_m) / (2.0 * d_m)) <= 2e+152) {
		tmp = w0 * Math.sqrt((1.0 - ((Math.pow(((D_m * (M_m * 0.5)) / d_m), 2.0) * h) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((M_m * 0.5) * ((M_m * (D_m * 0.5)) / d_m)) * (h / ((d_m * l) / 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 ((M_m * D_m) / (2.0 * d_m)) <= 2e+152:
		tmp = w0 * math.sqrt((1.0 - ((math.pow(((D_m * (M_m * 0.5)) / d_m), 2.0) * h) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (((M_m * 0.5) * ((M_m * (D_m * 0.5)) / d_m)) * (h / ((d_m * l) / 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(M_m * D_m) / Float64(2.0 * d_m)) <= 2e+152)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(Float64(D_m * Float64(M_m * 0.5)) / d_m) ^ 2.0) * h) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(M_m * 0.5) * Float64(Float64(M_m * Float64(D_m * 0.5)) / d_m)) * Float64(h / Float64(Float64(d_m * l) / 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)) <= 2e+152)
		tmp = w0 * sqrt((1.0 - (((((D_m * (M_m * 0.5)) / d_m) ^ 2.0) * h) / l)));
	else
		tmp = w0 * sqrt((1.0 - (((M_m * 0.5) * ((M_m * (D_m * 0.5)) / d_m)) * (h / ((d_m * l) / 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[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2e+152], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(N[(D$95$m * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] * N[(h / N[(N[(d$95$m * l), $MachinePrecision] / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 2.0000000000000001e152

   1. Initial program 86.4%

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

   3. Simplified 85.9%

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

      1. associate-*r/ 92.2%

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

      2. add-sqr-sqrt 92.2%

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

      3. pow2 92.2%

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

      4. sqrt-pow1 92.2%

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

      5. metadata-eval 92.2%

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

      6. pow1 92.2%

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

      7. *-un-lft-identity 92.2%

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

      8. times-frac 92.2%

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

      9. metadata-eval 92.2%

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

   6. Applied egg-rr 92.2%

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

      1. associate-*r* 92.2%

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

      2. metadata-eval 92.2%

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

      3. div-inv 92.2%

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

      4. associate-*r/ 92.6%

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

      5. div-inv 92.6%

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

      6. metadata-eval 92.6%

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

   8. Applied egg-rr 92.6%

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

   if 2.0000000000000001e152 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 56.7%

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

   3. Simplified 56.7%

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

      1. clear-num 56.7%

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

      2. un-div-inv 56.7%

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

      3. associate-/l* 56.7%

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

   6. Applied egg-rr 56.7%

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

      1. unpow2 56.7%

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

      2. associate-/r* 56.7%

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

      3. associate-*r/ 56.7%

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

      4. associate-/r* 56.7%

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

      5. div-inv 56.7%

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

      6. div-inv 56.7%

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

      7. metadata-eval 56.7%

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

      8. clear-num 56.7%

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

      9. associate-*r* 56.7%

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

      10. associate-*r/ 56.7%

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

      11. div-inv 56.7%

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

      12. metadata-eval 56.7%

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

   8. Applied egg-rr 56.7%

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

      1. frac-times 68.7%

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

      2. associate-*l* 68.7%

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

      3. associate-*r/ 68.7%

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

   10. Applied egg-rr 68.7%

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

       1. associate-/l* 68.6%

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

       2. associate-*r* 68.6%

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

       3. associate-*r/ 68.6%

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

       4. associate-*r/ 68.6%

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

       5. associate-*l/ 71.6%

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

   12. Applied egg-rr 71.6%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(M \cdot 0.5\right) \cdot \frac{M \cdot \left(D \cdot 0.5\right)}{d}\right) \cdot \frac{h}{\frac{d \cdot \ell}{D}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 3.4 \cdot 10^{+74}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M\_m \cdot \left(\left(D\_m \cdot 0.25\right) \cdot \frac{M\_m}{d\_m}\right)\right) \cdot \frac{\frac{h}{\ell}}{\frac{d\_m}{D\_m}}}\\ \end{array} \end{array} \]

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 3.4e+74)
   w0
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* M_m (* (* D_m 0.25) (/ M_m d_m))) (/ (/ h l) (/ d_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 (D_m <= 3.4e+74) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((M_m * ((D_m * 0.25) * (M_m / d_m))) * ((h / l) / (d_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 (d_m <= 3.4d+74) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((m_m * ((d_m * 0.25d0) * (m_m / d_m_1))) * ((h / l) / (d_m_1 / d_m)))))
    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 (D_m <= 3.4e+74) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((M_m * ((D_m * 0.25) * (M_m / d_m))) * ((h / l) / (d_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 D_m <= 3.4e+74:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((M_m * ((D_m * 0.25) * (M_m / d_m))) * ((h / l) / (d_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 (D_m <= 3.4e+74)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M_m * Float64(Float64(D_m * 0.25) * Float64(M_m / d_m))) * Float64(Float64(h / l) / Float64(d_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 (D_m <= 3.4e+74)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((M_m * ((D_m * 0.25) * (M_m / d_m))) * ((h / l) / (d_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[D$95$m, 3.4e+74], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M$95$m * N[(N[(D$95$m * 0.25), $MachinePrecision] * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] / N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 3.3999999999999999e74

   1. Initial program 82.6%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in M around 0 73.1%

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

   if 3.3999999999999999e74 < D

   1. Initial program 81.4%

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

   3. Simplified 76.8%

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

      1. clear-num 76.8%

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

      2. un-div-inv 76.8%

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

      3. associate-/l* 76.8%

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

   6. Applied egg-rr 76.8%

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

      1. unpow2 76.8%

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

      2. associate-/r* 76.8%

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

      3. associate-*r/ 76.7%

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

      4. associate-/r* 76.7%

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

      5. div-inv 76.7%

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

      6. div-inv 76.7%

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

      7. metadata-eval 76.7%

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

      8. clear-num 76.7%

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

      9. associate-*r* 76.7%

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

      10. associate-*r/ 76.7%

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

      11. div-inv 76.7%

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

      12. metadata-eval 76.7%

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

   8. Applied egg-rr 76.7%

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

      1. frac-times 77.0%

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

      2. associate-*l* 77.0%

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

      3. associate-*r/ 77.0%

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

   10. Applied egg-rr 77.0%

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

       1. times-frac 76.7%

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

       2. associate-*r* 76.7%

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

       3. associate-*r/ 76.7%

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

       4. associate-*r/ 76.7%

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

   12. Applied egg-rr 76.7%

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

       1. associate-*l/ 79.0%

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

       2. associate-/l* 76.7%

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

       3. *-commutative 76.7%

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

       4. associate-*l* 76.7%

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

       5. *-commutative 76.7%

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

       6. associate-*r* 76.7%

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

       7. associate-*r/ 76.7%

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

       8. *-commutative 76.7%

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

       9. associate-*r* 76.7%

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

       10. associate-*l* 76.7%

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

       11. associate-*l/ 76.7%

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

       12. metadata-eval 76.7%

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

       13. *-commutative 76.7%

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

       14. associate-/l* 76.7%

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

       15. associate-*r* 76.7%

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

   14. Simplified 76.7%

       \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(M \cdot \left(\left(0.25 \cdot D\right) \cdot \frac{M}{d}\right)\right) \cdot \frac{\frac{h}{\ell}}{\frac{d}{D}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 3.4 \cdot 10^{+74}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \left(\left(D \cdot 0.25\right) \cdot \frac{M}{d}\right)\right) \cdot \frac{\frac{h}{\ell}}{\frac{d}{D}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.5% accurate, 1.8× speedup

Math

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

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 (* (* M_m 0.5) (* 0.5 (/ D_m d_m))))) (* l (/ d_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 * ((M_m * 0.5) * (0.5 * (D_m / d_m))))) / (l * (d_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 * ((m_m * 0.5d0) * (0.5d0 * (d_m / d_m_1))))) / (l * (d_m_1 / d_m)))))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0 * Math.sqrt((1.0 - ((h * (M_m * ((M_m * 0.5) * (0.5 * (D_m / d_m))))) / (l * (d_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 * ((M_m * 0.5) * (0.5 * (D_m / d_m))))) / (l * (d_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(h * Float64(M_m * Float64(Float64(M_m * 0.5) * Float64(0.5 * Float64(D_m / d_m))))) / Float64(l * Float64(d_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 * ((M_m * 0.5) * (0.5 * (D_m / d_m))))) / (l * (d_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[(h * N[(M$95$m * N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(0.5 * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.4%

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

3. Simplified 82.0%

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

   1. clear-num 82.0%

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

   2. un-div-inv 82.0%

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

   3. associate-/l* 82.0%

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

6. Applied egg-rr 82.0%

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

   1. unpow2 82.0%

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

   2. associate-/r* 82.0%

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

   3. associate-*r/ 79.8%

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

   4. associate-/r* 79.8%

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

   5. div-inv 79.8%

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

   6. div-inv 79.8%

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

   7. metadata-eval 79.8%

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

   8. clear-num 79.8%

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

   9. associate-*r* 79.8%

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

   10. associate-*r/ 79.8%

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

   11. div-inv 79.8%

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

   12. metadata-eval 79.8%

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

8. Applied egg-rr 79.8%

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

   1. frac-times 85.2%

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

   2. associate-*l* 85.2%

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

   3. associate-*r/ 85.2%

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

10. Applied egg-rr 85.2%

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

11. Final simplification 85.2%

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

Alternative 6: 83.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0 \cdot \sqrt{1 - \frac{h \cdot \left(M\_m \cdot \left(0.25 \cdot \frac{M\_m \cdot D\_m}{d\_m}\right)\right)}{\ell \cdot \frac{d\_m}{D\_m}}} \end{array} \]

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 (* 0.25 (/ (* M_m D_m) d_m)))) (* l (/ d_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 * (0.25 * ((M_m * D_m) / d_m)))) / (l * (d_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 * (0.25d0 * ((m_m * d_m) / d_m_1)))) / (l * (d_m_1 / d_m)))))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0 * Math.sqrt((1.0 - ((h * (M_m * (0.25 * ((M_m * D_m) / d_m)))) / (l * (d_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 * (0.25 * ((M_m * D_m) / d_m)))) / (l * (d_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(h * Float64(M_m * Float64(0.25 * Float64(Float64(M_m * D_m) / d_m)))) / Float64(l * Float64(d_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 * (0.25 * ((M_m * D_m) / d_m)))) / (l * (d_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[(h * N[(M$95$m * N[(0.25 * N[(N[(M$95$m * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.4%

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

3. Simplified 82.0%

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

   1. clear-num 82.0%

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

   2. un-div-inv 82.0%

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

   3. associate-/l* 82.0%

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

6. Applied egg-rr 82.0%

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

   1. unpow2 82.0%

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

   2. associate-/r* 82.0%

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

   3. associate-*r/ 79.8%

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

   4. associate-/r* 79.8%

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

   5. div-inv 79.8%

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

   6. div-inv 79.8%

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

   7. metadata-eval 79.8%

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

   8. clear-num 79.8%

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

   9. associate-*r* 79.8%

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

   10. associate-*r/ 79.8%

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

   11. div-inv 79.8%

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

   12. metadata-eval 79.8%

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

8. Applied egg-rr 79.8%

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

   1. frac-times 85.2%

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

   2. associate-*l* 85.2%

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

   3. associate-*r/ 85.2%

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

10. Applied egg-rr 85.2%

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

11. Taylor expanded in D around 0 85.2%

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

12. Final simplification 85.2%

    \[\leadsto w0 \cdot \sqrt{1 - \frac{h \cdot \left(M \cdot \left(0.25 \cdot \frac{M \cdot D}{d}\right)\right)}{\ell \cdot \frac{d}{D}}} \]
13. Add Preprocessing

Alternative 7: 68.1% 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 82.4%

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

3. Simplified 82.0%

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

5. Taylor expanded in M around 0 70.0%

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

6. Final simplification 70.0%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(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))))))