Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 88.2%

Time: 15.3s

Alternatives: 12

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: 80.8% 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: 88.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{+278}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot h\right)\right)\right)}}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0
         (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))))
   (if (<= t_0 5e+278)
     t_0
     (*
      w0
      (sqrt
       (+
        1.0
        (/ -1.0 (/ l (* 0.25 (* (/ D d) (* (* M (/ D d)) (* M h))))))))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 5e+278) {
		tmp = t_0;
	} else {
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (0.25 * ((D / d) * ((M * (D / d)) * (M * h))))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
    if (t_0 <= 5d+278) then
        tmp = t_0
    else
        tmp = w0 * sqrt((1.0d0 + ((-1.0d0) / (l / (0.25d0 * ((d / d_1) * ((m * (d / d_1)) * (m * h))))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 5e+278) {
		tmp = t_0;
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-1.0 / (l / (0.25 * ((D / d) * ((M * (D / d)) * (M * h))))))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))
	tmp = 0
	if t_0 <= 5e+278:
		tmp = t_0
	else:
		tmp = w0 * math.sqrt((1.0 + (-1.0 / (l / (0.25 * ((D / d) * ((M * (D / d)) * (M * h))))))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
	tmp = 0.0
	if (t_0 <= 5e+278)
		tmp = t_0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-1.0 / Float64(l / Float64(0.25 * Float64(Float64(D / d) * Float64(Float64(M * Float64(D / d)) * Float64(M * h)))))))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
	tmp = 0.0;
	if (t_0 <= 5e+278)
		tmp = t_0;
	else
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (0.25 * ((D / d) * ((M * (D / d)) * (M * h))))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 5e+278], t$95$0, N[(w0 * N[Sqrt[N[(1.0 + N[(-1.0 / N[(l / N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < 5.00000000000000029e278

   1. Initial program 90.5%

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

   if 5.00000000000000029e278 < (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))))

   1. Initial program 39.2%

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

   3. Simplified 43.6%

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

      1. associate-*r/ 65.3%

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

      2. frac-times 60.9%

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

      3. *-commutative 60.9%

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

      4. clear-num 60.9%

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

      5. *-commutative 60.9%

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

      6. *-commutative 60.9%

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

      7. frac-times 65.3%

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

      8. div-inv 65.3%

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

      9. metadata-eval 65.3%

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

      10. associate-*l* 65.3%

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

   5. Applied egg-rr 65.3%

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

   6. Taylor expanded in h around 0 40.9%

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

      1. unpow2 40.9%

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

      2. associate-*r* 42.9%

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

      3. associate-/l* 42.3%

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

      4. associate-/r/ 42.9%

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

      5. unpow2 42.9%

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

      6. unpow2 42.9%

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

      7. times-frac 59.2%

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

      8. associate-*l* 63.6%

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

      9. associate-*r* 71.0%

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

   8. Simplified 71.0%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 5 \cdot 10^{+278}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot h\right)\right)\right)}}}\\ \end{array} \]

Alternative 2: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.4 \cdot 10^{-231}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot h\right)\right)\right)}}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 2.4e-231)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ D 2.0) (/ M d)) 2.0)))))
   (*
    w0
    (sqrt
     (+ 1.0 (/ -1.0 (/ l (* 0.25 (* (/ D d) (* (* M (/ D d)) (* M h)))))))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 2.4e-231) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D / 2.0) * (M / d)), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (0.25 * ((D / d) * ((M * (D / d)) * (M * h))))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (d_1 <= 2.4d-231) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((d / 2.0d0) * (m / d_1)) ** 2.0d0))))
    else
        tmp = w0 * sqrt((1.0d0 + ((-1.0d0) / (l / (0.25d0 * ((d / d_1) * ((m * (d / d_1)) * (m * h))))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 2.4e-231) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D / 2.0) * (M / d)), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-1.0 / (l / (0.25 * ((D / d) * ((M * (D / d)) * (M * h))))))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 2.4e-231:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D / 2.0) * (M / d)), 2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 + (-1.0 / (l / (0.25 * ((D / d) * ((M * (D / d)) * (M * h))))))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 2.4e-231)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D / 2.0) * Float64(M / d)) ^ 2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-1.0 / Float64(l / Float64(0.25 * Float64(Float64(D / d) * Float64(Float64(M * Float64(D / d)) * Float64(M * h)))))))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 2.4e-231)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D / 2.0) * (M / d)) ^ 2.0))));
	else
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (0.25 * ((D / d) * ((M * (D / d)) * (M * h))))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 2.4e-231], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / 2.0), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-1.0 / N[(l / N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 2.39999999999999992e-231

   1. Initial program 76.4%

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

   3. Simplified 77.0%

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

   if 2.39999999999999992e-231 < d

   1. Initial program 78.3%

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

   3. Simplified 77.5%

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

      1. associate-*r/ 85.2%

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

      2. frac-times 86.1%

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

      3. *-commutative 86.1%

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

      4. clear-num 86.1%

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

      5. *-commutative 86.1%

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

      6. *-commutative 86.1%

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

      7. frac-times 85.3%

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

      8. div-inv 85.3%

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

      9. metadata-eval 85.3%

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

      10. associate-*l* 85.3%

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

   5. Applied egg-rr 85.3%

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

   6. Taylor expanded in h around 0 62.9%

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

      1. unpow2 62.9%

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

      2. associate-*r* 65.7%

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

      3. associate-/l* 64.6%

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

      4. associate-/r/ 65.8%

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

      5. unpow2 65.8%

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

      6. unpow2 65.8%

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

      7. times-frac 74.9%

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

      8. associate-*l* 79.6%

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

      9. associate-*r* 84.3%

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

   8. Simplified 84.3%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.4 \cdot 10^{-231}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot h\right)\right)\right)}}}\\ \end{array} \]

Alternative 3: 80.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 8 \cdot 10^{-111}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \frac{D}{\frac{\frac{\frac{\ell}{h} \cdot \frac{d}{M}}{M}}{0.25}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot h\right) \cdot \frac{M}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 8e-111)
   (* w0 (sqrt (- 1.0 (* (/ D d) (/ D (/ (/ (* (/ l h) (/ d M)) M) 0.25))))))
   (* w0 (sqrt (- 1.0 (* (/ D d) (* 0.25 (* (/ D d) (* (* M h) (/ M l))))))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 8e-111) {
		tmp = w0 * sqrt((1.0 - ((D / d) * (D / ((((l / h) * (d / M)) / M) / 0.25)))));
	} else {
		tmp = w0 * sqrt((1.0 - ((D / d) * (0.25 * ((D / d) * ((M * h) * (M / l)))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (d_1 <= 8d-111) then
        tmp = w0 * sqrt((1.0d0 - ((d / d_1) * (d / ((((l / h) * (d_1 / m)) / m) / 0.25d0)))))
    else
        tmp = w0 * sqrt((1.0d0 - ((d / d_1) * (0.25d0 * ((d / d_1) * ((m * h) * (m / l)))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 8e-111) {
		tmp = w0 * Math.sqrt((1.0 - ((D / d) * (D / ((((l / h) * (d / M)) / M) / 0.25)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((D / d) * (0.25 * ((D / d) * ((M * h) * (M / l)))))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 8e-111:
		tmp = w0 * math.sqrt((1.0 - ((D / d) * (D / ((((l / h) * (d / M)) / M) / 0.25)))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((D / d) * (0.25 * ((D / d) * ((M * h) * (M / l)))))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 8e-111)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D / d) * Float64(D / Float64(Float64(Float64(Float64(l / h) * Float64(d / M)) / M) / 0.25))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D / d) * Float64(0.25 * Float64(Float64(D / d) * Float64(Float64(M * h) * Float64(M / l))))))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 8e-111)
		tmp = w0 * sqrt((1.0 - ((D / d) * (D / ((((l / h) * (d / M)) / M) / 0.25)))));
	else
		tmp = w0 * sqrt((1.0 - ((D / d) * (0.25 * ((D / d) * ((M * h) * (M / l)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 8e-111], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D / d), $MachinePrecision] * N[(D / N[(N[(N[(N[(l / h), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / M), $MachinePrecision] / 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D / d), $MachinePrecision] * N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] * N[(M / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 8.00000000000000071e-111

   1. Initial program 75.5%

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

   3. Simplified 75.5%

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

   4. Taylor expanded in D around 0 48.4%

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

      1. *-commutative 48.4%

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

      2. times-frac 48.9%

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

      3. unpow2 48.9%

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

      4. unpow2 48.9%

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

      5. unpow2 48.9%

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

   6. Simplified 48.9%

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

      1. *-un-lft-identity 48.9%

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

      2. associate-*l* 48.9%

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

      3. times-frac 65.8%

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

      4. associate-*l* 67.1%

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

   8. Applied egg-rr 67.1%

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

      1. *-lft-identity 67.1%

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

      2. associate-*l* 71.2%

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

      3. associate-*l/ 71.2%

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

   10. Simplified 71.2%

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

   11. Taylor expanded in D around 0 67.4%

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

       1. *-commutative 67.4%

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

       2. times-frac 69.4%

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

       3. associate-*l* 69.4%

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

       4. unpow2 69.4%

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

       5. associate-*r* 71.2%

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

       6. associate-*l/ 71.2%

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

       7. associate-*r* 71.2%

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

       8. times-frac 69.4%

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

       9. *-commutative 69.4%

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

       10. associate-/l* 70.6%

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

       11. associate-*r* 70.6%

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

       12. associate-/r* 70.6%

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

       13. *-commutative 70.6%

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

       14. associate-/r* 74.8%

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

       15. *-commutative 74.8%

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

       16. times-frac 75.9%

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

   13. Simplified 75.9%

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

   if 8.00000000000000071e-111 < d

   1. Initial program 80.2%

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

   3. Simplified 80.2%

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

      1. *-commutative 80.2%

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

      2. frac-times 80.2%

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

      3. *-commutative 80.2%

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

      4. associate-*l/ 88.7%

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

      5. *-commutative 88.7%

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

      6. frac-times 88.7%

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

      7. div-inv 88.7%

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

      8. metadata-eval 88.7%

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

      9. associate-*l* 88.7%

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

   5. Applied egg-rr 88.7%

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

   6. Taylor expanded in h around 0 62.1%

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

      1. unpow2 62.1%

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

      2. associate-*r* 65.6%

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

      3. times-frac 65.3%

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

      4. unpow2 65.3%

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

      5. unpow2 65.3%

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

      6. times-frac 70.7%

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

      7. *-commutative 70.7%

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

      8. associate-*l* 75.6%

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

      9. associate-*l* 75.6%

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

      10. associate-/l* 76.7%

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

      11. associate-/r/ 75.7%

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

   8. Simplified 75.7%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 8 \cdot 10^{-111}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \frac{D}{\frac{\frac{\frac{\ell}{h} \cdot \frac{d}{M}}{M}}{0.25}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot h\right) \cdot \frac{M}{\ell}\right)\right)\right)}\\ \end{array} \]

Alternative 4: 80.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.7 \cdot 10^{-99}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \frac{D}{\frac{\frac{\frac{\ell}{h} \cdot \frac{d}{M}}{M}}{0.25}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \frac{\frac{D}{d} \cdot \left(M \cdot \left(0.25 \cdot \left(M \cdot h\right)\right)\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 1.7e-99)
   (* w0 (sqrt (- 1.0 (* (/ D d) (/ D (/ (/ (* (/ l h) (/ d M)) M) 0.25))))))
   (* w0 (sqrt (- 1.0 (* (/ D d) (/ (* (/ D d) (* M (* 0.25 (* M h)))) l)))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.7e-99) {
		tmp = w0 * sqrt((1.0 - ((D / d) * (D / ((((l / h) * (d / M)) / M) / 0.25)))));
	} else {
		tmp = w0 * sqrt((1.0 - ((D / d) * (((D / d) * (M * (0.25 * (M * h)))) / l))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (d_1 <= 1.7d-99) then
        tmp = w0 * sqrt((1.0d0 - ((d / d_1) * (d / ((((l / h) * (d_1 / m)) / m) / 0.25d0)))))
    else
        tmp = w0 * sqrt((1.0d0 - ((d / d_1) * (((d / d_1) * (m * (0.25d0 * (m * h)))) / l))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.7e-99) {
		tmp = w0 * Math.sqrt((1.0 - ((D / d) * (D / ((((l / h) * (d / M)) / M) / 0.25)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((D / d) * (((D / d) * (M * (0.25 * (M * h)))) / l))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 1.7e-99:
		tmp = w0 * math.sqrt((1.0 - ((D / d) * (D / ((((l / h) * (d / M)) / M) / 0.25)))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((D / d) * (((D / d) * (M * (0.25 * (M * h)))) / l))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 1.7e-99)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D / d) * Float64(D / Float64(Float64(Float64(Float64(l / h) * Float64(d / M)) / M) / 0.25))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D / d) * Float64(Float64(Float64(D / d) * Float64(M * Float64(0.25 * Float64(M * h)))) / l)))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 1.7e-99)
		tmp = w0 * sqrt((1.0 - ((D / d) * (D / ((((l / h) * (d / M)) / M) / 0.25)))));
	else
		tmp = w0 * sqrt((1.0 - ((D / d) * (((D / d) * (M * (0.25 * (M * h)))) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 1.7e-99], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D / d), $MachinePrecision] * N[(D / N[(N[(N[(N[(l / h), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision] / M), $MachinePrecision] / 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D / d), $MachinePrecision] * N[(N[(N[(D / d), $MachinePrecision] * N[(M * N[(0.25 * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.70000000000000003e-99

   1. Initial program 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in D around 0 48.2%

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

      1. *-commutative 48.2%

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

      2. times-frac 48.6%

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

      3. unpow2 48.6%

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

      4. unpow2 48.6%

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

      5. unpow2 48.6%

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

   6. Simplified 48.6%

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

      1. *-un-lft-identity 48.6%

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

      2. associate-*l* 48.6%

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

      3. times-frac 65.0%

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

      4. associate-*l* 66.3%

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

   8. Applied egg-rr 66.3%

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

      1. *-lft-identity 66.3%

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

      2. associate-*l* 70.7%

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

      3. associate-*l/ 70.7%

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

   10. Simplified 70.7%

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

   11. Taylor expanded in D around 0 67.0%

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

       1. *-commutative 67.0%

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

       2. times-frac 68.9%

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

       3. associate-*l* 68.9%

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

       4. unpow2 68.9%

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

       5. associate-*r* 70.7%

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

       6. associate-*l/ 70.7%

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

       7. associate-*r* 70.7%

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

       8. times-frac 68.9%

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

       9. *-commutative 68.9%

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

       10. associate-/l* 70.0%

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

       11. associate-*r* 70.0%

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

       12. associate-/r* 70.0%

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

       13. *-commutative 70.0%

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

       14. associate-/r* 74.2%

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

       15. *-commutative 74.2%

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

       16. times-frac 75.2%

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

   13. Simplified 75.2%

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

   if 1.70000000000000003e-99 < d

   1. Initial program 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in D around 0 63.3%

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

      1. *-commutative 63.3%

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

      2. times-frac 63.4%

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

      3. unpow2 63.4%

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

      4. unpow2 63.4%

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

      5. unpow2 63.4%

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

   6. Simplified 63.4%

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

      1. *-un-lft-identity 63.4%

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

      2. associate-*l* 63.4%

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

      3. times-frac 69.0%

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

      4. associate-*l* 72.4%

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

   8. Applied egg-rr 72.4%

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

      1. *-lft-identity 72.4%

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

      2. associate-*l* 76.9%

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

      3. associate-*l/ 76.9%

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

   10. Simplified 76.9%

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

       1. associate-*r/ 82.3%

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

       2. associate-*l* 82.3%

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

   12. Applied egg-rr 82.3%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.7 \cdot 10^{-99}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \frac{D}{\frac{\frac{\frac{\ell}{h} \cdot \frac{d}{M}}{M}}{0.25}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \frac{\frac{D}{d} \cdot \left(M \cdot \left(0.25 \cdot \left(M \cdot h\right)\right)\right)}{\ell}}\\ \end{array} \]

Alternative 5: 85.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot h\right)\right)\right)}}} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (*
  w0
  (sqrt
   (+ 1.0 (/ -1.0 (/ l (* 0.25 (* (/ D d) (* (* M (/ D d)) (* M h))))))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 + (-1.0 / (l / (0.25 * ((D / d) * ((M * (D / d)) * (M * h))))))));
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 + ((-1.0d0) / (l / (0.25d0 * ((d / d_1) * ((m * (d / d_1)) * (m * h))))))))
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 + (-1.0 / (l / (0.25 * ((D / d) * ((M * (D / d)) * (M * h))))))));
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 + (-1.0 / (l / (0.25 * ((D / d) * ((M * (D / d)) * (M * h))))))))

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 + Float64(-1.0 / Float64(l / Float64(0.25 * Float64(Float64(D / d) * Float64(Float64(M * Float64(D / d)) * Float64(M * h)))))))))
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 + (-1.0 / (l / (0.25 * ((D / d) * ((M * (D / d)) * (M * h))))))));
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 + N[(-1.0 / N[(l / N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.3%

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

3. Simplified 77.2%

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

   1. associate-*r/ 82.0%

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

   2. frac-times 82.1%

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

   3. *-commutative 82.1%

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

   4. clear-num 82.1%

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

   5. *-commutative 82.1%

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

   6. *-commutative 82.1%

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

   7. frac-times 82.0%

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

   8. div-inv 82.0%

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

   9. metadata-eval 82.0%

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

   10. associate-*l* 82.0%

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

5. Applied egg-rr 82.0%

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

6. Taylor expanded in h around 0 56.2%

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

   1. unpow2 56.2%

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

   2. associate-*r* 58.4%

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

   3. associate-/l* 57.4%

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

   4. associate-/r/ 58.3%

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

   5. unpow2 58.3%

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

   6. unpow2 58.3%

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

   7. times-frac 71.5%

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

   8. associate-*l* 76.1%

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

   9. associate-*r* 81.2%

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

8. Simplified 81.2%

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

9. Final simplification 81.2%

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

Alternative 6: 82.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \sqrt{1 - \frac{D}{d} \cdot \left(0.25 \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot h\right) \cdot \frac{M}{\ell}\right)\right)\right)} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (/ D d) (* 0.25 (* (/ D d) (* (* M h) (/ M l)))))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - ((D / d) * (0.25 * ((D / d) * ((M * h) * (M / l)))))));
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 - ((d / d_1) * (0.25d0 * ((d / d_1) * ((m * h) * (m / l)))))))
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - ((D / d) * (0.25 * ((D / d) * ((M * h) * (M / l)))))));
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - ((D / d) * (0.25 * ((D / d) * ((M * h) * (M / l)))))))

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D / d) * Float64(0.25 * Float64(Float64(D / d) * Float64(Float64(M * h) * Float64(M / l))))))))
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((D / d) * (0.25 * ((D / d) * ((M * h) * (M / l)))))));
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D / d), $MachinePrecision] * N[(0.25 * N[(N[(D / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] * N[(M / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.3%

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

3. Simplified 77.2%

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

   1. *-commutative 77.2%

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

   2. frac-times 77.3%

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

   3. *-commutative 77.3%

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

   4. associate-*l/ 82.1%

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

   5. *-commutative 82.1%

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

   6. frac-times 82.0%

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

   7. div-inv 82.0%

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

   8. metadata-eval 82.0%

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

   9. associate-*l* 82.0%

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

5. Applied egg-rr 82.0%

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

6. Taylor expanded in h around 0 53.4%

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

   1. unpow2 53.4%

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

   2. associate-*r* 55.6%

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

   3. times-frac 55.8%

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

   4. unpow2 55.8%

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

   5. unpow2 55.8%

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

   6. times-frac 68.4%

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

   7. *-commutative 68.4%

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

   8. associate-*l* 72.8%

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

   9. associate-*l* 72.8%

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

   10. associate-/l* 75.2%

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

   11. associate-/r/ 72.9%

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

8. Simplified 72.9%

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

9. Final simplification 72.9%

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

Alternative 7: 73.0% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.36 \cdot 10^{-139}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.36e-139)
   w0
   (* w0 (+ 1.0 (* -0.125 (* (* (/ D d) (/ D d)) (/ (* h (* M M)) l)))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.36e-139) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h * (M * M)) / l))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (m <= 1.36d-139) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d / d_1) * (d / d_1)) * ((h * (m * m)) / l))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.36e-139) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h * (M * M)) / l))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.36e-139:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h * (M * M)) / l))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.36e-139)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(h * Float64(M * M)) / l)))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.36e-139)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h * (M * M)) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.36e-139], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.36000000000000003e-139

   1. Initial program 78.4%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in D around 0 63.0%

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

   if 1.36000000000000003e-139 < M

   1. Initial program 75.7%

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

   3. Simplified 74.8%

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

   4. Taylor expanded in D around 0 56.4%

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

      1. times-frac 54.5%

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

      2. unpow2 54.5%

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

      3. unpow2 54.5%

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

      4. unpow2 54.5%

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

   6. Simplified 54.5%

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

   7. Taylor expanded in D around 0 54.5%

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

      1. unpow2 54.5%

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

      2. unpow2 54.5%

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

      3. times-frac 63.9%

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

   9. Simplified 63.9%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.36 \cdot 10^{-139}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)\\ \end{array} \]

Alternative 8: 69.9% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.35 \cdot 10^{+59}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \left(M \cdot \left(w0 \cdot h\right)\right)\right) \cdot \left(\frac{D}{\ell} \cdot \frac{D}{d \cdot d}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 2.35e+59)
   w0
   (* -0.125 (* (* M (* M (* w0 h))) (* (/ D l) (/ D (* d d)))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.35e+59) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((M * (M * (w0 * h))) * ((D / l) * (D / (d * d))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (m <= 2.35d+59) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((m * (m * (w0 * h))) * ((d / l) * (d / (d_1 * d_1))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.35e+59) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((M * (M * (w0 * h))) * ((D / l) * (D / (d * d))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.35e+59:
		tmp = w0
	else:
		tmp = -0.125 * ((M * (M * (w0 * h))) * ((D / l) * (D / (d * d))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.35e+59)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(M * Float64(M * Float64(w0 * h))) * Float64(Float64(D / l) * Float64(D / Float64(d * d)))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.35e+59)
		tmp = w0;
	else
		tmp = -0.125 * ((M * (M * (w0 * h))) * ((D / l) * (D / (d * d))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 2.35e+59], w0, N[(-0.125 * N[(N[(M * N[(M * N[(w0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D / l), $MachinePrecision] * N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.35e59

   1. Initial program 77.9%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in D around 0 65.1%

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

   if 2.35e59 < M

   1. Initial program 74.8%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in D around 0 48.9%

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

      1. +-commutative 48.9%

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

      2. fma-def 48.9%

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

      3. *-commutative 48.9%

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

      4. *-commutative 48.9%

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

      5. associate-*r* 48.9%

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

      6. unpow2 48.9%

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

      7. unpow2 48.9%

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

      8. swap-sqr 57.2%

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

      9. *-commutative 57.2%

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

      10. associate-*l* 55.2%

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

      11. unpow2 55.2%

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

      12. associate-*l* 57.3%

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

   6. Simplified 57.3%

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

   7. Taylor expanded in h around inf 31.1%

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

      1. associate-/l* 31.0%

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

      2. unpow2 31.0%

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

      3. unpow2 31.0%

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

      4. unpow2 31.0%

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

   9. Simplified 31.0%

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

   10. Taylor expanded in D around 0 31.1%

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

       1. unpow2 31.1%

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

       2. unpow2 31.1%

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

       3. unpow2 31.1%

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

       4. associate-*r* 31.4%

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

       5. *-commutative 31.4%

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

       6. associate-*l/ 31.7%

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

       7. *-commutative 31.7%

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

       8. times-frac 33.9%

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

   12. Simplified 33.9%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.35 \cdot 10^{+59}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(M \cdot \left(M \cdot \left(w0 \cdot h\right)\right)\right) \cdot \left(\frac{D}{\ell} \cdot \frac{D}{d \cdot d}\right)\right)\\ \end{array} \]

Alternative 9: 70.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.85 \cdot 10^{+76}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{D \cdot D}{\frac{\ell}{w0 \cdot h} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.85e+76)
   w0
   (* -0.125 (/ (* D D) (* (/ l (* w0 h)) (* (/ d M) (/ d M)))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.85e+76) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * D) / ((l / (w0 * h)) * ((d / M) * (d / M))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (m <= 1.85d+76) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((d * d) / ((l / (w0 * h)) * ((d_1 / m) * (d_1 / m))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.85e+76) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * D) / ((l / (w0 * h)) * ((d / M) * (d / M))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.85e+76:
		tmp = w0
	else:
		tmp = -0.125 * ((D * D) / ((l / (w0 * h)) * ((d / M) * (d / M))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.85e+76)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(D * D) / Float64(Float64(l / Float64(w0 * h)) * Float64(Float64(d / M) * Float64(d / M)))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.85e+76)
		tmp = w0;
	else
		tmp = -0.125 * ((D * D) / ((l / (w0 * h)) * ((d / M) * (d / M))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.85e+76], w0, N[(-0.125 * N[(N[(D * D), $MachinePrecision] / N[(N[(l / N[(w0 * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.85e76

   1. Initial program 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in D around 0 64.6%

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

   if 1.85e76 < M

   1. Initial program 76.2%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in D around 0 49.6%

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

      1. +-commutative 49.6%

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

      2. fma-def 49.6%

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

      3. *-commutative 49.6%

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

      4. *-commutative 49.6%

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

      5. associate-*r* 49.6%

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

      6. unpow2 49.6%

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

      7. unpow2 49.6%

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

      8. swap-sqr 58.9%

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

      9. *-commutative 58.9%

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

      10. associate-*l* 56.7%

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

      11. unpow2 56.7%

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

      12. associate-*l* 59.0%

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

   6. Simplified 59.0%

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

   7. Taylor expanded in h around inf 32.1%

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

      1. associate-/l* 32.0%

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

      2. unpow2 32.0%

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

      3. unpow2 32.0%

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

      4. unpow2 32.0%

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

   9. Simplified 32.0%

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

   10. Taylor expanded in d around 0 32.0%

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

       1. unpow2 32.0%

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

       2. times-frac 32.0%

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

       3. unpow2 32.0%

          \[\leadsto -0.125 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{M \cdot M}} \cdot \frac{\ell}{h \cdot w0}} \]

       4. times-frac 36.8%

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

   12. Simplified 36.8%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.85 \cdot 10^{+76}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{D \cdot D}{\frac{\ell}{w0 \cdot h} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}\\ \end{array} \]

Alternative 10: 70.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 7.2 \cdot 10^{+34}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{D \cdot D}{\frac{d \cdot \frac{\ell}{w0 \cdot h}}{\frac{M}{\frac{d}{M}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 7.2e+34)
   w0
   (* -0.125 (/ (* D D) (/ (* d (/ l (* w0 h))) (/ M (/ d M)))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 7.2e+34) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * D) / ((d * (l / (w0 * h))) / (M / (d / M))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (m <= 7.2d+34) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((d * d) / ((d_1 * (l / (w0 * h))) / (m / (d_1 / m))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 7.2e+34) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * D) / ((d * (l / (w0 * h))) / (M / (d / M))));
	}
	return tmp;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 7.2e+34:
		tmp = w0
	else:
		tmp = -0.125 * ((D * D) / ((d * (l / (w0 * h))) / (M / (d / M))))
	return tmp

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 7.2e+34)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(D * D) / Float64(Float64(d * Float64(l / Float64(w0 * h))) / Float64(M / Float64(d / M)))));
	end
	return tmp
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 7.2e+34)
		tmp = w0;
	else
		tmp = -0.125 * ((D * D) / ((d * (l / (w0 * h))) / (M / (d / M))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 7.2e+34], w0, N[(-0.125 * N[(N[(D * D), $MachinePrecision] / N[(N[(d * N[(l / N[(w0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.2000000000000001e34

   1. Initial program 78.2%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in D around 0 65.5%

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

   if 7.2000000000000001e34 < M

   1. Initial program 74.1%

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

   3. Simplified 72.2%

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

   4. Taylor expanded in D around 0 49.1%

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

      1. +-commutative 49.1%

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

      2. fma-def 49.1%

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

      3. *-commutative 49.1%

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

      4. *-commutative 49.1%

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

      5. associate-*r* 49.1%

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

      6. unpow2 49.1%

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

      7. unpow2 49.1%

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

      8. swap-sqr 56.6%

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

      9. *-commutative 56.6%

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

      10. associate-*l* 54.8%

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

      11. unpow2 54.8%

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

      12. associate-*l* 56.7%

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

   6. Simplified 56.7%

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

   7. Taylor expanded in h around inf 29.9%

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

      1. associate-/l* 29.8%

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

      2. unpow2 29.8%

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

      3. unpow2 29.8%

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

      4. unpow2 29.8%

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

   9. Simplified 29.8%

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

   10. Taylor expanded in d around 0 29.8%

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

       1. unpow2 29.8%

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

       2. times-frac 29.8%

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

       3. associate-/l* 33.4%

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

       4. associate-*l/ 33.5%

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

       5. unpow2 33.5%

          \[\leadsto -0.125 \cdot \frac{D \cdot D}{\frac{d \cdot \frac{\ell}{h \cdot w0}}{\frac{\color{blue}{M \cdot M}}{d}}} \]

       6. associate-/l* 33.7%

          \[\leadsto -0.125 \cdot \frac{D \cdot D}{\frac{d \cdot \frac{\ell}{h \cdot w0}}{\color{blue}{\frac{M}{\frac{d}{M}}}}} \]

   12. Simplified 33.7%

       \[\leadsto -0.125 \cdot \frac{D \cdot D}{\color{blue}{\frac{d \cdot \frac{\ell}{h \cdot w0}}{\frac{M}{\frac{d}{M}}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.2 \cdot 10^{+34}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{D \cdot D}{\frac{d \cdot \frac{\ell}{w0 \cdot h}}{\frac{M}{\frac{d}{M}}}}\\ \end{array} \]

Alternative 11: 76.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot h\right) \cdot \frac{M}{\ell}\right)\right)\right)\right) \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (+ 1.0 (* -0.125 (* (/ D d) (* (/ D d) (* (* M h) (/ M l))))))))

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * ((M * h) * (M / l))))));
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 * (1.0d0 + ((-0.125d0) * ((d / d_1) * ((d / d_1) * ((m * h) * (m / l))))))
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * ((M * h) * (M / l))))));
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * ((M * h) * (M / l))))))

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D / d) * Float64(Float64(D / d) * Float64(Float64(M * h) * Float64(M / l)))))))
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * (1.0 + (-0.125 * ((D / d) * ((D / d) * ((M * h) * (M / l))))));
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D / d), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] * N[(M / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.3%

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

3. Simplified 77.2%

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

4. Taylor expanded in D around 0 51.6%

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

   1. times-frac 52.7%

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

   2. unpow2 52.7%

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

   3. unpow2 52.7%

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

   4. unpow2 52.7%

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

6. Simplified 52.7%

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

7. Taylor expanded in D around 0 51.6%

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

   1. unpow2 51.6%

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

   2. associate-*r* 53.3%

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

   3. times-frac 54.3%

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

   4. unpow2 54.3%

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

   5. unpow2 54.3%

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

   6. times-frac 66.3%

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

   7. unpow1 66.3%

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

   8. sqr-pow 66.3%

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

   9. associate-*l* 66.3%

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

9. Simplified 69.4%

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

10. Final simplification 69.4%

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

Alternative 12: 68.0% accurate, 216.0× speedup

Math

\[\begin{array}{l} M = |M|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d) :precision binary64 w0)

C

M = abs(M);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
end function

Java

M = Math.abs(M);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Python

M = abs(M)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0

Julia

M = abs(M)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return w0
end

MATLAB

M = abs(M)
M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 77.3%

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

3. Simplified 77.2%

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

4. Taylor expanded in D around 0 58.9%

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

5. Final simplification 58.9%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023293 
(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))))))