Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 87.8%

Time: 19.0s

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: 87.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 10^{+151}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{{\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d\_m}\right)\right)}^{2} \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \frac{\sqrt{\frac{h \cdot \left(-0.25 \cdot {\left(M\_m \cdot D\_m\right)}^{2}\right)}{\ell}}}{d\_m}\\ \end{array} \end{array} \]

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)) 1e+151)
   (*
    w0
    (sqrt (+ 1.0 (/ -1.0 (/ l (* (pow (* M_m (* 0.5 (/ D_m d_m))) 2.0) h))))))
   (* w0 (/ (sqrt (/ (* h (* -0.25 (pow (* M_m D_m) 2.0))) 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)) <= 1e+151) {
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (pow((M_m * (0.5 * (D_m / d_m))), 2.0) * h)))));
	} else {
		tmp = w0 * (sqrt(((h * (-0.25 * pow((M_m * D_m), 2.0))) / 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)) <= 1d+151) then
        tmp = w0 * sqrt((1.0d0 + ((-1.0d0) / (l / (((m_m * (0.5d0 * (d_m / d_m_1))) ** 2.0d0) * h)))))
    else
        tmp = w0 * (sqrt(((h * ((-0.25d0) * ((m_m * d_m) ** 2.0d0))) / l)) / 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)) <= 1e+151) {
		tmp = w0 * Math.sqrt((1.0 + (-1.0 / (l / (Math.pow((M_m * (0.5 * (D_m / d_m))), 2.0) * h)))));
	} else {
		tmp = w0 * (Math.sqrt(((h * (-0.25 * Math.pow((M_m * D_m), 2.0))) / 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)) <= 1e+151:
		tmp = w0 * math.sqrt((1.0 + (-1.0 / (l / (math.pow((M_m * (0.5 * (D_m / d_m))), 2.0) * h)))))
	else:
		tmp = w0 * (math.sqrt(((h * (-0.25 * math.pow((M_m * D_m), 2.0))) / 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)) <= 1e+151)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-1.0 / Float64(l / Float64((Float64(M_m * Float64(0.5 * Float64(D_m / d_m))) ^ 2.0) * h))))));
	else
		tmp = Float64(w0 * Float64(sqrt(Float64(Float64(h * Float64(-0.25 * (Float64(M_m * D_m) ^ 2.0))) / 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)) <= 1e+151)
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (((M_m * (0.5 * (D_m / d_m))) ^ 2.0) * h)))));
	else
		tmp = w0 * (sqrt(((h * (-0.25 * ((M_m * D_m) ^ 2.0))) / 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], 1e+151], N[(w0 * N[Sqrt[N[(1.0 + N[(-1.0 / N[(l / N[(N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[Sqrt[N[(N[(h * N[(-0.25 * N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1.00000000000000002e151

   1. Initial program 84.4%

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

   3. Simplified 85.2%

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

      1. associate-*r/ 90.5%

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

      2. clear-num 90.5%

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

      3. add-sqr-sqrt 90.5%

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

      4. pow2 90.5%

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

      5. sqrt-pow1 90.5%

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

      6. metadata-eval 90.5%

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

      7. pow1 90.5%

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

      8. *-un-lft-identity 90.5%

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

      9. times-frac 90.5%

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

      10. metadata-eval 90.5%

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

   6. Applied egg-rr 90.5%

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

   if 1.00000000000000002e151 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 58.8%

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

   3. Simplified 58.8%

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

      1. associate-*r/ 59.1%

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

      2. clear-num 59.1%

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

      3. add-sqr-sqrt 59.1%

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

      4. pow2 59.1%

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

      5. sqrt-pow1 59.1%

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

      6. metadata-eval 59.1%

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

      7. pow1 59.1%

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

      8. *-un-lft-identity 59.1%

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

      9. times-frac 59.1%

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

      10. metadata-eval 59.1%

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

   6. Applied egg-rr 59.1%

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

   7. Taylor expanded in l around 0 48.2%

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

      1. associate-*r/ 48.2%

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

      2. associate-*r* 48.2%

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

      3. *-commutative 48.2%

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

      4. unpow2 48.2%

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

      5. unpow2 48.2%

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

      6. swap-sqr 57.9%

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

      7. unpow2 57.9%

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

      8. *-commutative 57.9%

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

      9. *-commutative 57.9%

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

   9. Simplified 57.9%

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

       1. unpow2 57.9%

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

       2. *-commutative 57.9%

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

       3. *-commutative 57.9%

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

   11. Applied egg-rr 57.9%

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

       1. associate-/r* 58.3%

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

       2. sqrt-div 61.4%

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

       3. associate-*r* 61.4%

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

       4. pow2 61.4%

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

       5. sqrt-pow1 44.7%

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

       6. metadata-eval 44.7%

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

       7. pow1 44.7%

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

   13. Applied egg-rr 44.7%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{+151}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \frac{\sqrt{\frac{h \cdot \left(-0.25 \cdot {\left(M \cdot D\right)}^{2}\right)}{\ell}}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{D\_m}{d\_m} \cdot \left(M\_m \cdot 0.5\right)\\ \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 10^{+151}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(t\_0 \cdot t\_0\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \frac{\sqrt{\frac{h \cdot \left(-0.25 \cdot {\left(M\_m \cdot D\_m\right)}^{2}\right)}{\ell}}}{d\_m}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (/ D_m d_m) (* M_m 0.5))))
   (if (<= (/ (* M_m D_m) (* 2.0 d_m)) 1e+151)
     (* w0 (sqrt (- 1.0 (/ (* h (* t_0 t_0)) l))))
     (* w0 (/ (sqrt (/ (* h (* -0.25 (pow (* M_m D_m) 2.0))) 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 t_0 = (D_m / d_m) * (M_m * 0.5);
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1e+151) {
		tmp = w0 * sqrt((1.0 - ((h * (t_0 * t_0)) / l)));
	} else {
		tmp = w0 * (sqrt(((h * (-0.25 * pow((M_m * D_m), 2.0))) / 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) :: t_0
    real(8) :: tmp
    t_0 = (d_m / d_m_1) * (m_m * 0.5d0)
    if (((m_m * d_m) / (2.0d0 * d_m_1)) <= 1d+151) then
        tmp = w0 * sqrt((1.0d0 - ((h * (t_0 * t_0)) / l)))
    else
        tmp = w0 * (sqrt(((h * ((-0.25d0) * ((m_m * d_m) ** 2.0d0))) / l)) / 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 t_0 = (D_m / d_m) * (M_m * 0.5);
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1e+151) {
		tmp = w0 * Math.sqrt((1.0 - ((h * (t_0 * t_0)) / l)));
	} else {
		tmp = w0 * (Math.sqrt(((h * (-0.25 * Math.pow((M_m * D_m), 2.0))) / 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):
	t_0 = (D_m / d_m) * (M_m * 0.5)
	tmp = 0
	if ((M_m * D_m) / (2.0 * d_m)) <= 1e+151:
		tmp = w0 * math.sqrt((1.0 - ((h * (t_0 * t_0)) / l)))
	else:
		tmp = w0 * (math.sqrt(((h * (-0.25 * math.pow((M_m * D_m), 2.0))) / 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)
	t_0 = Float64(Float64(D_m / d_m) * Float64(M_m * 0.5))
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) <= 1e+151)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(t_0 * t_0)) / l))));
	else
		tmp = Float64(w0 * Float64(sqrt(Float64(Float64(h * Float64(-0.25 * (Float64(M_m * D_m) ^ 2.0))) / 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)
	t_0 = (D_m / d_m) * (M_m * 0.5);
	tmp = 0.0;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1e+151)
		tmp = w0 * sqrt((1.0 - ((h * (t_0 * t_0)) / l)));
	else
		tmp = w0 * (sqrt(((h * (-0.25 * ((M_m * D_m) ^ 2.0))) / 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_] := Block[{t$95$0 = N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 1e+151], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[Sqrt[N[(N[(h * N[(-0.25 * N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1.00000000000000002e151

   1. Initial program 84.4%

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

   3. Simplified 85.2%

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

      1. unpow2 85.2%

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

      2. *-commutative 85.2%

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

      3. associate-*r* 84.0%

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

      4. *-un-lft-identity 84.0%

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

      5. times-frac 84.0%

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

      6. metadata-eval 84.0%

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

      7. *-un-lft-identity 84.0%

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

      8. times-frac 84.0%

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

      9. metadata-eval 84.0%

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

   6. Applied egg-rr 84.0%

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

      1. associate-*r/ 89.2%

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

      2. associate-*l* 90.5%

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

      3. *-commutative 90.5%

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

      4. unpow2 90.5%

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

      5. associate-*r/ 90.5%

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

   8. Applied egg-rr 90.5%

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

      1. associate-*r/ 90.5%

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

      2. pow2 90.5%

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

      3. associate-*r* 90.5%

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

      4. associate-*r* 90.5%

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

   10. Applied egg-rr 90.5%

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

   if 1.00000000000000002e151 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 58.8%

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

   3. Simplified 58.8%

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

      1. associate-*r/ 59.1%

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

      2. clear-num 59.1%

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

      3. add-sqr-sqrt 59.1%

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

      4. pow2 59.1%

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

      5. sqrt-pow1 59.1%

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

      6. metadata-eval 59.1%

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

      7. pow1 59.1%

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

      8. *-un-lft-identity 59.1%

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

      9. times-frac 59.1%

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

      10. metadata-eval 59.1%

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

   6. Applied egg-rr 59.1%

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

   7. Taylor expanded in l around 0 48.2%

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

      1. associate-*r/ 48.2%

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

      2. associate-*r* 48.2%

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

      3. *-commutative 48.2%

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

      4. unpow2 48.2%

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

      5. unpow2 48.2%

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

      6. swap-sqr 57.9%

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

      7. unpow2 57.9%

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

      8. *-commutative 57.9%

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

      9. *-commutative 57.9%

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

   9. Simplified 57.9%

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

       1. unpow2 57.9%

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

       2. *-commutative 57.9%

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

       3. *-commutative 57.9%

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

   11. Applied egg-rr 57.9%

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

       1. associate-/r* 58.3%

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

       2. sqrt-div 61.4%

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

       3. associate-*r* 61.4%

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

       4. pow2 61.4%

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

       5. sqrt-pow1 44.7%

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

       6. metadata-eval 44.7%

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

       7. pow1 44.7%

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

   13. Applied egg-rr 44.7%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{+151}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \frac{\sqrt{\frac{h \cdot \left(-0.25 \cdot {\left(M \cdot D\right)}^{2}\right)}{\ell}}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{D\_m}{d\_m}\\ \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-254}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(M\_m \cdot \left(t\_0 \cdot \left(M\_m \cdot t\_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ D_m d_m))))
   (if (<= (/ h l) -2e-254)
     (* w0 (sqrt (- 1.0 (* (/ h l) (* M_m (* t_0 (* M_m t_0)))))))
     w0)))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = 0.5 * (D_m / d_m);
	double tmp;
	if ((h / l) <= -2e-254) {
		tmp = w0 * sqrt((1.0 - ((h / l) * (M_m * (t_0 * (M_m * t_0))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (d_m / d_m_1)
    if ((h / l) <= (-2d-254)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (m_m * (t_0 * (m_m * t_0))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = 0.5 * (D_m / d_m);
	double tmp;
	if ((h / l) <= -2e-254) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * (M_m * (t_0 * (M_m * t_0))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = 0.5 * (D_m / d_m)
	tmp = 0
	if (h / l) <= -2e-254:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * (M_m * (t_0 * (M_m * t_0))))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(0.5 * Float64(D_m / d_m))
	tmp = 0.0
	if (Float64(h / l) <= -2e-254)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(M_m * Float64(t_0 * Float64(M_m * t_0)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = 0.5 * (D_m / d_m);
	tmp = 0.0;
	if ((h / l) <= -2e-254)
		tmp = w0 * sqrt((1.0 - ((h / l) * (M_m * (t_0 * (M_m * t_0))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(0.5 * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], -2e-254], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(M$95$m * N[(t$95$0 * N[(M$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.9999999999999998e-254

   1. Initial program 78.1%

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

   3. Simplified 79.5%

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

      1. unpow2 79.5%

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

      2. *-commutative 79.5%

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

      3. associate-*r* 78.2%

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

      4. *-un-lft-identity 78.2%

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

      5. times-frac 78.2%

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

      6. metadata-eval 78.2%

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

      7. *-un-lft-identity 78.2%

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

      8. times-frac 78.2%

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

      9. metadata-eval 78.2%

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

   6. Applied egg-rr 78.2%

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

   if -1.9999999999999998e-254 < (/.f64 h l)

   1. Initial program 85.1%

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

   3. Simplified 85.1%

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

   5. Taylor expanded in M around 0 92.3%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-254}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(M \cdot \left(\left(0.5 \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-229}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \left(\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d\_m}\right)\right) \cdot \frac{0.5}{d\_m}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ h l) -1e-229)
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       (/ h l)
       (* (* M_m D_m) (* (* M_m (* 0.5 (/ D_m d_m))) (/ 0.5 d_m)))))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -1e-229) {
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M_m * D_m) * ((M_m * (0.5 * (D_m / d_m))) * (0.5 / d_m))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if ((h / l) <= (-1d-229)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((m_m * d_m) * ((m_m * (0.5d0 * (d_m / d_m_1))) * (0.5d0 / d_m_1))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -1e-229) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * ((M_m * D_m) * ((M_m * (0.5 * (D_m / d_m))) * (0.5 / d_m))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (h / l) <= -1e-229:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * ((M_m * D_m) * ((M_m * (0.5 * (D_m / d_m))) * (0.5 / d_m))))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(h / l) <= -1e-229)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(M_m * D_m) * Float64(Float64(M_m * Float64(0.5 * Float64(D_m / d_m))) * Float64(0.5 / d_m)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((h / l) <= -1e-229)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M_m * D_m) * ((M_m * (0.5 * (D_m / d_m))) * (0.5 / d_m))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(h / l), $MachinePrecision], -1e-229], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * N[(0.5 * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.00000000000000007e-229

   1. Initial program 78.5%

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

   3. Simplified 79.9%

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

      1. unpow2 79.9%

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

      2. associate-*r/ 78.5%

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

      3. div-inv 78.5%

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

      4. associate-*l* 78.5%

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

      5. associate-/r* 78.5%

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

      6. metadata-eval 78.5%

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

      7. *-un-lft-identity 78.5%

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

      8. times-frac 78.5%

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

      9. metadata-eval 78.5%

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

   6. Applied egg-rr 78.5%

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

   if -1.00000000000000007e-229 < (/.f64 h l)

   1. Initial program 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in M around 0 91.6%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-229}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\left(M \cdot D\right) \cdot \left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \frac{0.5}{d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot D\_m \leq 5.3 \cdot 10^{-102}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125 \cdot \left(h \cdot \left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)\right)\right)}{\ell \cdot {d\_m}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* M_m D_m) 5.3e-102)
   w0
   (*
    w0
    (+
     1.0
     (/ (* -0.125 (* h (* (* M_m D_m) (* M_m D_m)))) (* l (pow d_m 2.0)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((M_m * D_m) <= 5.3e-102) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((-0.125 * (h * ((M_m * D_m) * (M_m * D_m)))) / (l * pow(d_m, 2.0))));
	}
	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) :: tmp
    if ((m_m * d_m) <= 5.3d-102) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + (((-0.125d0) * (h * ((m_m * d_m) * (m_m * d_m)))) / (l * (d_m_1 ** 2.0d0))))
    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) <= 5.3e-102) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((-0.125 * (h * ((M_m * D_m) * (M_m * D_m)))) / (l * Math.pow(d_m, 2.0))));
	}
	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):
	tmp = 0
	if (M_m * D_m) <= 5.3e-102:
		tmp = w0
	else:
		tmp = w0 * (1.0 + ((-0.125 * (h * ((M_m * D_m) * (M_m * D_m)))) / (l * math.pow(d_m, 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 5.3e-102)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 * Float64(h * Float64(Float64(M_m * D_m) * Float64(M_m * D_m)))) / Float64(l * (d_m ^ 2.0)))));
	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)
	tmp = 0.0;
	if ((M_m * D_m) <= 5.3e-102)
		tmp = w0;
	else
		tmp = w0 * (1.0 + ((-0.125 * (h * ((M_m * D_m) * (M_m * D_m)))) / (l * (d_m ^ 2.0))));
	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_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5.3e-102], w0, N[(w0 * N[(1.0 + N[(N[(-0.125 * N[(h * N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Power[d$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 5.3000000000000003e-102

   1. Initial program 85.1%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in M around 0 78.9%

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

   if 5.3000000000000003e-102 < (*.f64 M D)

   1. Initial program 71.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in M around 0 41.3%

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

      1. +-commutative 41.3%

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

      2. fma-define 41.3%

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

      3. associate-*r* 41.3%

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

      4. *-commutative 41.3%

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

      5. unpow2 41.3%

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

      6. unpow2 41.3%

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

      7. swap-sqr 55.2%

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

      8. unpow2 55.2%

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

      9. *-commutative 55.2%

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

   7. Simplified 55.2%

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

   8. Taylor expanded in w0 around 0 42.8%

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

      1. *-commutative 42.8%

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

      2. associate-*r/ 42.8%

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

      3. associate-*r* 42.9%

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

      4. *-commutative 42.9%

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

      5. unpow2 42.9%

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

      6. unpow2 42.9%

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

      7. swap-sqr 58.2%

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

      8. unpow2 58.2%

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

      9. *-commutative 58.2%

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

      10. *-commutative 58.2%

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

   10. Simplified 58.2%

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

       1. unpow2 36.2%

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

       2. *-commutative 36.2%

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

       3. *-commutative 36.2%

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

   12. Applied egg-rr 58.2%

       \[\leadsto \left(1 + \frac{-0.125 \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot h\right)}{\ell \cdot {d}^{2}}\right) \cdot w0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 5.3 \cdot 10^{-102}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125 \cdot \left(h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right)}{\ell \cdot {d}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{D\_m}{d\_m} \cdot \left(M\_m \cdot 0.5\right)\\ w0 \cdot \sqrt{1 - \frac{h \cdot \left(t\_0 \cdot t\_0\right)}{\ell}} \end{array} \end{array} \]

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
 (let* ((t_0 (* (/ D_m d_m) (* M_m 0.5))))
   (* w0 (sqrt (- 1.0 (/ (* h (* t_0 t_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 t_0 = (D_m / d_m) * (M_m * 0.5);
	return w0 * sqrt((1.0 - ((h * (t_0 * t_0)) / l)));
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    t_0 = (d_m / d_m_1) * (m_m * 0.5d0)
    code = w0 * sqrt((1.0d0 - ((h * (t_0 * t_0)) / l)))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (D_m / d_m) * (M_m * 0.5);
	return w0 * Math.sqrt((1.0 - ((h * (t_0 * t_0)) / l)));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = (D_m / d_m) * (M_m * 0.5)
	return w0 * math.sqrt((1.0 - ((h * (t_0 * t_0)) / l)))

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(D_m / d_m) * Float64(M_m * 0.5))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(t_0 * t_0)) / l))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
function tmp = code(w0, M_m, D_m, h, l, d_m)
	t_0 = (D_m / d_m) * (M_m * 0.5);
	tmp = w0 * sqrt((1.0 - ((h * (t_0 * t_0)) / l)));
end

Wolfram

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

Derivation

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 82.1%

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

   1. unpow2 82.1%

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

   2. *-commutative 82.1%

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

   3. associate-*r* 81.0%

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

   4. *-un-lft-identity 81.0%

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

   5. times-frac 81.0%

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

   6. metadata-eval 81.0%

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

   7. *-un-lft-identity 81.0%

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

   8. times-frac 81.0%

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

   9. metadata-eval 81.0%

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

6. Applied egg-rr 81.0%

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

   1. associate-*r/ 85.7%

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

   2. associate-*l* 86.8%

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

   3. *-commutative 86.8%

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

   4. unpow2 86.8%

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

   5. associate-*r/ 86.8%

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

8. Applied egg-rr 86.8%

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

   1. associate-*r/ 86.8%

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

   2. pow2 86.8%

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

   3. associate-*r* 86.8%

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

   4. associate-*r* 86.8%

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

10. Applied egg-rr 86.8%

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

11. Final simplification 86.8%

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

Alternative 7: 85.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 82.1%

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

   1. unpow2 82.1%

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

   2. *-commutative 82.1%

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

   3. associate-*r* 81.0%

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

   4. *-un-lft-identity 81.0%

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

   5. times-frac 81.0%

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

   6. metadata-eval 81.0%

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

   7. *-un-lft-identity 81.0%

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

   8. times-frac 81.0%

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

   9. metadata-eval 81.0%

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

6. Applied egg-rr 81.0%

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

   1. associate-*r/ 85.7%

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

   2. associate-*l* 86.8%

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

   3. *-commutative 86.8%

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

   4. unpow2 86.8%

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

   5. associate-*r/ 86.8%

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

8. Applied egg-rr 86.8%

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

   1. associate-*r/ 86.8%

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

   2. pow2 86.8%

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

   3. associate-*r* 86.8%

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

   4. associate-*r* 86.8%

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

10. Applied egg-rr 86.8%

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

11. Taylor expanded in M around 0 86.1%

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

    1. associate-/l* 86.0%

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

13. Simplified 86.0%

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

14. Final simplification 86.0%

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

Alternative 8: 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 81.4%

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

3. Simplified 82.1%

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

5. Taylor expanded in M around 0 67.1%

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

Reproduce

herbie shell --seed 2024113 
(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))))))