Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 86.5%

Time: 14.3s

Alternatives: 11

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+271}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{\frac{2}{\frac{M}{d}}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{M \cdot h}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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 (<= (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))) 5e+271)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ D (/ 2.0 (/ M d))) 2.0)))))
   (* w0 (sqrt (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (* M (/ (* M h) l)))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= 5e+271) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((D / (2.0 / (M / d))), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (M * ((M * h) / l))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 ((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))) <= 5d+271) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d / (2.0d0 / (m / d_1))) ** 2.0d0))))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d / d_1) * (d / d_1)) * (m * ((m * h) / l))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= 5e+271) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((D / (2.0 / (M / d))), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (M * ((M * h) / l))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= 5e+271:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((D / (2.0 / (M / d))), 2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (M * ((M * h) / l))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))) <= 5e+271)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D / Float64(2.0 / Float64(M / d))) ^ 2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(M * Float64(Float64(M * h) / l)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))) <= 5e+271)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D / (2.0 / (M / d))) ^ 2.0))));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (M * ((M * h) / l))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D 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[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], 5e+271], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D / N[(2.0 / N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(M * N[(N[(M * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 5.0000000000000003e271

   1. Initial program 99.4%

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

   3. Simplified 96.5%

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

      1. associate-*l/ 96.5%

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

      2. associate-/l* 96.5%

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

   5. Applied egg-rr 96.5%

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

   if 5.0000000000000003e271 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 43.6%

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

   3. Simplified 44.8%

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

   4. Taylor expanded in D around 0 45.6%

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

      1. times-frac 49.1%

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

      2. unpow2 49.1%

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

      3. unpow2 49.1%

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

      4. unpow2 49.1%

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

   6. Simplified 49.1%

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

      1. times-frac 58.8%

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

   8. Applied egg-rr 58.8%

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

   9. Taylor expanded in M around 0 58.8%

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

       1. associate-/l* 48.9%

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

       2. unpow2 48.9%

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

       3. associate-/l* 51.8%

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

       4. associate-/r/ 51.8%

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

   11. Simplified 51.8%

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

   12. Taylor expanded in M around 0 64.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+271}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{\frac{2}{\frac{M}{d}}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{M \cdot h}{\ell}\right)\right)}\\ \end{array} \]

Alternative 2: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.7 \cdot 10^{-29}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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-29)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
   (* w0 (sqrt (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (/ (* M (* M h)) l))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.7e-29) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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-29) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / d_1) * (d / 2.0d0)) ** 2.0d0))))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d / d_1) * (d / d_1)) * ((m * (m * h)) / l)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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-29) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 1.7e-29:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 1.7e-29)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * Float64(M * h)) / l))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 1.7e-29)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D 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-29], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.69999999999999986e-29

   1. Initial program 80.0%

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

   3. Simplified 77.8%

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

   if 1.69999999999999986e-29 < d

   1. Initial program 83.6%

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

   3. Simplified 83.7%

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

   4. Taylor expanded in D around 0 58.9%

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

      1. times-frac 58.8%

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

      2. unpow2 58.8%

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

      3. unpow2 58.8%

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

      4. unpow2 58.8%

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

   6. Simplified 58.8%

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

      1. times-frac 68.1%

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

   8. Applied egg-rr 68.1%

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

   9. Taylor expanded in M around 0 68.1%

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

       1. unpow2 64.5%

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

       2. associate-*l* 67.3%

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

   11. Simplified 74.5%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.7 \cdot 10^{-29}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}\right)}\\ \end{array} \]

Alternative 3: 81.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 6.8 \cdot 10^{-151}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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 6.8e-151)
   w0
   (* w0 (sqrt (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (* h (* M (/ M l))))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 6.8e-151) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (h * (M * (M / l)))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 <= 6.8d-151) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d / d_1) * (d / d_1)) * (h * (m * (m / l)))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 6.8e-151) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (h * (M * (M / l)))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 6.8e-151:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (h * (M * (M / l)))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 6.8e-151)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(h * Float64(M * Float64(M / l))))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 6.8e-151)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (h * (M * (M / l)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D 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, 6.8e-151], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M * N[(M / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 6.8000000000000005e-151

   1. Initial program 82.2%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in D around 0 69.9%

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

   if 6.8000000000000005e-151 < M

   1. Initial program 79.2%

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

   3. Simplified 78.1%

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

   4. Taylor expanded in D around 0 48.6%

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

      1. times-frac 50.7%

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

      2. unpow2 50.7%

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

      3. unpow2 50.7%

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

      4. unpow2 50.7%

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

   6. Simplified 50.7%

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

      1. times-frac 61.8%

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

   8. Applied egg-rr 61.8%

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

   9. Taylor expanded in M around 0 61.8%

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

       1. unpow2 61.8%

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

       2. associate-*l* 64.0%

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

   11. Simplified 65.0%

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

   12. Taylor expanded in M around 0 61.8%

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

       1. unpow2 61.8%

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

       2. associate-/l* 62.7%

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

       3. associate-*l/ 67.2%

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

       4. associate-/r/ 69.3%

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

       5. *-commutative 69.3%

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

       6. associate-*l* 67.1%

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

   14. Simplified 67.1%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.8 \cdot 10^{-151}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right)\right)}\\ \end{array} \]

Alternative 4: 81.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 9.2 \cdot 10^{-151}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{M \cdot h}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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 9.2e-151)
   w0
   (* w0 (sqrt (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (* M (/ (* M h) l)))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 9.2e-151) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (M * ((M * h) / l))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 <= 9.2d-151) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d / d_1) * (d / d_1)) * (m * ((m * h) / l))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 9.2e-151) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (M * ((M * h) / l))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 9.2e-151:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (M * ((M * h) / l))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 9.2e-151)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(M * Float64(Float64(M * h) / l)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 9.2e-151)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (M * ((M * h) / l))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D 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, 9.2e-151], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(M * N[(N[(M * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 9.19999999999999984e-151

   1. Initial program 82.2%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in D around 0 69.9%

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

   if 9.19999999999999984e-151 < M

   1. Initial program 79.2%

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

   3. Simplified 78.1%

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

   4. Taylor expanded in D around 0 48.6%

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

      1. times-frac 50.7%

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

      2. unpow2 50.7%

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

      3. unpow2 50.7%

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

      4. unpow2 50.7%

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

   6. Simplified 50.7%

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

      1. times-frac 61.8%

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

   8. Applied egg-rr 61.8%

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

   9. Taylor expanded in M around 0 61.8%

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

       1. associate-/l* 62.7%

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

       2. unpow2 62.7%

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

       3. associate-/l* 67.2%

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

       4. associate-/r/ 67.2%

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

   11. Simplified 67.2%

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

   12. Taylor expanded in M around 0 67.2%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9.2 \cdot 10^{-151}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{M \cdot h}{\ell}\right)\right)}\\ \end{array} \]

Alternative 5: 81.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4.3 \cdot 10^{-150}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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 4.3e-150)
   w0
   (* w0 (sqrt (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (/ (* M (* M h)) l))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.3e-150) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 <= 4.3d-150) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d / d_1) * (d / d_1)) * ((m * (m * h)) / l)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.3e-150) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 4.3e-150:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 4.3e-150)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * Float64(M * h)) / l))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 4.3e-150)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D 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, 4.3e-150], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.30000000000000004e-150

   1. Initial program 82.2%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in D around 0 69.9%

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

   if 4.30000000000000004e-150 < M

   1. Initial program 79.2%

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

   3. Simplified 78.1%

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

   4. Taylor expanded in D around 0 48.6%

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

      1. times-frac 50.7%

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

      2. unpow2 50.7%

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

      3. unpow2 50.7%

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

      4. unpow2 50.7%

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

   6. Simplified 50.7%

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

      1. times-frac 61.8%

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

   8. Applied egg-rr 61.8%

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

   9. Taylor expanded in M around 0 61.8%

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

       1. unpow2 61.8%

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

       2. associate-*l* 64.0%

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

   11. Simplified 65.0%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.3 \cdot 10^{-150}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}\right)}\\ \end{array} \]

Alternative 6: 75.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 76:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \left(\left(M \cdot \frac{M}{d}\right) \cdot \frac{D \cdot D}{d \cdot \ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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 76.0)
   w0
   (* w0 (+ 1.0 (* -0.125 (* h (* (* M (/ M d)) (/ (* D D) (* d l)))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 76.0) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (h * ((M * (M / d)) * ((D * D) / (d * l))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 <= 76.0d0) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (h * ((m * (m / d_1)) * ((d * d) / (d_1 * l))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 76.0) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (h * ((M * (M / d)) * ((D * D) / (d * l))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 76.0:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (h * ((M * (M / d)) * ((D * D) / (d * l))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 76.0)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(h * Float64(Float64(M * Float64(M / d)) * Float64(Float64(D * D) / Float64(d * l)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 76.0)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (h * ((M * (M / d)) * ((D * D) / (d * l))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D 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, 76.0], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(h * N[(N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 76

   1. Initial program 82.8%

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

   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 70.9%

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

   if 76 < M

   1. Initial program 75.8%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in D around 0 40.5%

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

      1. *-commutative 40.5%

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

      2. times-frac 43.6%

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

      3. unpow2 43.6%

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

      4. unpow2 43.6%

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

      5. unpow2 43.6%

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

   6. Simplified 43.6%

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

   7. Taylor expanded in D around 0 40.5%

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

      1. times-frac 43.6%

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

      2. unpow2 43.6%

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

      3. unpow2 43.6%

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

      4. times-frac 50.3%

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

      5. associate-/l* 51.8%

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

      6. associate-/r/ 50.4%

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

      7. associate-*r* 50.4%

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

      8. times-frac 43.7%

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

      9. unpow2 43.7%

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

      10. unpow2 43.7%

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

      11. times-frac 42.3%

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

      12. *-commutative 42.3%

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

      13. unpow2 42.3%

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

      14. associate-*l* 44.1%

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

      15. times-frac 48.9%

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

      16. unpow2 48.9%

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

      17. unpow2 48.9%

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

   9. Simplified 48.9%

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

   10. Taylor expanded in M around 0 48.9%

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

       1. unpow2 48.9%

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

       2. associate-*l/ 63.1%

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

       3. *-commutative 63.1%

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

   12. Simplified 63.1%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 76:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \left(\left(M \cdot \frac{M}{d}\right) \cdot \frac{D \cdot D}{d \cdot \ell}\right)\right)\right)\\ \end{array} \]

Alternative 7: 77.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 10^{-125}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M}{\frac{\frac{\ell}{h}}{M}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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 1e-125)
   w0
   (* w0 (+ 1.0 (* -0.125 (* (* (/ D d) (/ D d)) (/ M (/ (/ l h) M))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1e-125) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * (M / ((l / h) / M)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 <= 1d-125) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d / d_1) * (d / d_1)) * (m / ((l / h) / m)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1e-125) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * (M / ((l / h) / M)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1e-125:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * (M / ((l / h) / M)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1e-125)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(M / Float64(Float64(l / h) / M))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1e-125)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * (M / ((l / h) / M)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D 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, 1e-125], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(M / N[(N[(l / h), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.00000000000000001e-125

   1. Initial program 81.9%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in D around 0 70.4%

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

   if 1.00000000000000001e-125 < M

   1. Initial program 79.6%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in D around 0 46.8%

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

      1. *-commutative 46.8%

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

      2. times-frac 49.0%

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

      3. unpow2 49.0%

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

      4. unpow2 49.0%

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

      5. unpow2 49.0%

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

   6. Simplified 49.0%

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

      1. times-frac 60.5%

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

   8. Applied egg-rr 60.5%

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

   9. Taylor expanded in M around 0 60.5%

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

       1. associate-/l* 61.6%

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

       2. unpow2 61.6%

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

       3. associate-/l* 66.1%

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

   11. Simplified 66.1%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 10^{-125}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M}{\frac{\frac{\ell}{h}}{M}}\right)\right)\\ \end{array} \]

Alternative 8: 78.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 6.2 \cdot 10^{-141}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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 6.2e-141)
   w0
   (* w0 (+ 1.0 (* (* (* (/ D d) (/ D d)) (/ (* M (* M h)) l)) -0.125)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 6.2e-141) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((((D / d) * (D / d)) * ((M * (M * h)) / l)) * -0.125));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 <= 6.2d-141) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((((d / d_1) * (d / d_1)) * ((m * (m * h)) / l)) * (-0.125d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 6.2e-141) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((((D / d) * (D / d)) * ((M * (M * h)) / l)) * -0.125));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 6.2e-141:
		tmp = w0
	else:
		tmp = w0 * (1.0 + ((((D / d) * (D / d)) * ((M * (M * h)) / l)) * -0.125))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 6.2e-141)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * Float64(M * h)) / l)) * -0.125)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 6.2e-141)
		tmp = w0;
	else
		tmp = w0 * (1.0 + ((((D / d) * (D / d)) * ((M * (M * h)) / l)) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D 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, 6.2e-141], w0, N[(w0 * N[(1.0 + N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 6.20000000000000055e-141

   1. Initial program 81.7%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in D around 0 70.0%

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

   if 6.20000000000000055e-141 < M

   1. Initial program 80.1%

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

   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 48.0%

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

      1. *-commutative 48.0%

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

      2. times-frac 50.1%

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

      3. unpow2 50.1%

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

      4. unpow2 50.1%

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

      5. unpow2 50.1%

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

   6. Simplified 50.1%

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

      1. times-frac 61.4%

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

   8. Applied egg-rr 61.4%

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

   9. Taylor expanded in M around 0 61.4%

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

       1. unpow2 61.4%

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

       2. associate-*l* 63.6%

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

   11. Simplified 63.6%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.2 \cdot 10^{-141}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 9: 72.6% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.1 \cdot 10^{+124}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125}{d \cdot d} \cdot \frac{\left(D \cdot D\right) \cdot \left(M \cdot \left(M \cdot \left(h \cdot w0\right)\right)\right)}{\ell}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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 2.1e+124)
   w0
   (* (/ -0.125 (* d d)) (/ (* (* D D) (* M (* M (* h w0)))) l))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.1e+124) {
		tmp = w0;
	} else {
		tmp = (-0.125 / (d * d)) * (((D * D) * (M * (M * (h * w0)))) / l);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 <= 2.1d+124) then
        tmp = w0
    else
        tmp = ((-0.125d0) / (d_1 * d_1)) * (((d * d) * (m * (m * (h * w0)))) / l)
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.1e+124) {
		tmp = w0;
	} else {
		tmp = (-0.125 / (d * d)) * (((D * D) * (M * (M * (h * w0)))) / l);
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.1e+124:
		tmp = w0
	else:
		tmp = (-0.125 / (d * d)) * (((D * D) * (M * (M * (h * w0)))) / l)
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.1e+124)
		tmp = w0;
	else
		tmp = Float64(Float64(-0.125 / Float64(d * d)) * Float64(Float64(Float64(D * D) * Float64(M * Float64(M * Float64(h * w0)))) / l));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.1e+124)
		tmp = w0;
	else
		tmp = (-0.125 / (d * d)) * (((D * D) * (M * (M * (h * w0)))) / l);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D 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, 2.1e+124], w0, N[(N[(-0.125 / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * N[(M * N[(M * N[(h * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.10000000000000011e124

   1. Initial program 82.7%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in D around 0 70.5%

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

   if 2.10000000000000011e124 < M

   1. Initial program 71.7%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in D around 0 30.7%

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

      1. *-commutative 30.7%

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

      2. times-frac 30.6%

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

      3. unpow2 30.6%

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

      4. unpow2 30.6%

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

      5. unpow2 30.6%

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

   6. Simplified 30.6%

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

      1. times-frac 33.8%

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

   8. Applied egg-rr 33.8%

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

   9. Taylor expanded in D around inf 22.8%

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

       1. associate-*r/ 22.8%

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

       2. associate-*r* 22.3%

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

       3. unpow2 22.3%

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

       4. unpow2 22.3%

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

       5. unpow2 22.3%

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

   11. Simplified 22.3%

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

       1. times-frac 22.3%

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

       2. associate-*l* 22.7%

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

       3. associate-*l* 23.6%

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

   13. Applied egg-rr 23.6%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.1 \cdot 10^{+124}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125}{d \cdot d} \cdot \frac{\left(D \cdot D\right) \cdot \left(M \cdot \left(M \cdot \left(h \cdot w0\right)\right)\right)}{\ell}\\ \end{array} \]

Alternative 10: 72.7% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.2 \cdot 10^{+124}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \left(h \cdot w0\right)\right)}{d \cdot \left(d \cdot \ell\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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 2.2e+124)
   w0
   (/ (* -0.125 (* (* (* D D) (* M M)) (* h w0))) (* d (* d l)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.2e+124) {
		tmp = w0;
	} else {
		tmp = (-0.125 * (((D * D) * (M * M)) * (h * w0))) / (d * (d * l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 <= 2.2d+124) then
        tmp = w0
    else
        tmp = ((-0.125d0) * (((d * d) * (m * m)) * (h * w0))) / (d_1 * (d_1 * l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.2e+124) {
		tmp = w0;
	} else {
		tmp = (-0.125 * (((D * D) * (M * M)) * (h * w0))) / (d * (d * l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.2e+124:
		tmp = w0
	else:
		tmp = (-0.125 * (((D * D) * (M * M)) * (h * w0))) / (d * (d * l))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.2e+124)
		tmp = w0;
	else
		tmp = Float64(Float64(-0.125 * Float64(Float64(Float64(D * D) * Float64(M * M)) * Float64(h * w0))) / Float64(d * Float64(d * l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.2e+124)
		tmp = w0;
	else
		tmp = (-0.125 * (((D * D) * (M * M)) * (h * w0))) / (d * (d * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D 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, 2.2e+124], w0, N[(N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[(h * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.2000000000000001e124

   1. Initial program 82.7%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in D around 0 70.5%

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

   if 2.2000000000000001e124 < M

   1. Initial program 71.7%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in D around 0 30.7%

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

      1. *-commutative 30.7%

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

      2. times-frac 30.6%

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

      3. unpow2 30.6%

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

      4. unpow2 30.6%

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

      5. unpow2 30.6%

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

   6. Simplified 30.6%

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

      1. times-frac 33.8%

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

   8. Applied egg-rr 33.8%

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

   9. Taylor expanded in D around inf 22.8%

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

       1. associate-*r/ 22.8%

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

       2. associate-*r* 22.3%

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

       3. unpow2 22.3%

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

       4. unpow2 22.3%

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

       5. unpow2 22.3%

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

   11. Simplified 22.3%

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

   12. Taylor expanded in d around 0 22.3%

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

       1. unpow2 22.3%

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

       2. associate-*r* 25.3%

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

   14. Simplified 25.3%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.2 \cdot 10^{+124}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \left(h \cdot w0\right)\right)}{d \cdot \left(d \cdot \ell\right)}\\ \end{array} \]

Alternative 11: 70.1% accurate, 216.0× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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);
D = abs(D);
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: D 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);
D = Math.abs(D);
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)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0

Julia

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

MATLAB

M = abs(M)
D = abs(D)
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: D 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 81.1%

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

3. Simplified 79.6%

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

4. Taylor expanded in D around 0 67.6%

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

5. Final simplification 67.6%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023271 
(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))))))