Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 87.9%

Time: 16.5s

Alternatives: 14

Speedup: 1.7×

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: 81.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-110}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq 0:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= (/ h l) (- INFINITY))
   (* w0 (sqrt (- 1.0 (* 0.25 (* (/ (* D D) l) (* (/ M d) (/ (* h M) d)))))))
   (if (<= (/ h l) -1e-110)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ D d) (/ M 2.0)) 2.0)))))
     (if (<= (/ h l) 0.0)
       (*
        w0
        (sqrt (- 1.0 (* 0.25 (* D (* (/ D l) (* h (* (/ M d) (/ M d)))))))))
       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) {
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D * D) / l) * ((M / d) * ((h * M) / d))))));
	} else if ((h / l) <= -1e-110) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D / d) * (M / 2.0)), 2.0))));
	} else if ((h / l) <= 0.0) {
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

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 ((h / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D * D) / l) * ((M / d) * ((h * M) / d))))));
	} else if ((h / l) <= -1e-110) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D / d) * (M / 2.0)), 2.0))));
	} else if ((h / l) <= 0.0) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))));
	} else {
		tmp = w0;
	}
	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 (h / l) <= -math.inf:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D * D) / l) * ((M / d) * ((h * M) / d))))))
	elif (h / l) <= -1e-110:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D / d) * (M / 2.0)), 2.0))))
	elif (h / l) <= 0.0:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))))
	else:
		tmp = w0
	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(h / l) <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D * D) / l) * Float64(Float64(M / d) * Float64(Float64(h * M) / d)))))));
	elseif (Float64(h / l) <= -1e-110)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0)))));
	elseif (Float64(h / l) <= 0.0)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D * Float64(Float64(D / l) * Float64(h * Float64(Float64(M / d) * Float64(M / d)))))))));
	else
		tmp = w0;
	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 ((h / l) <= -Inf)
		tmp = w0 * sqrt((1.0 - (0.25 * (((D * D) / l) * ((M / d) * ((h * M) / d))))));
	elseif ((h / l) <= -1e-110)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D / d) * (M / 2.0)) ^ 2.0))));
	elseif ((h / l) <= 0.0)
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))));
	else
		tmp = w0;
	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[(h / l), $MachinePrecision], (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-110], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], 0.0], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(h * N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 h l) < -inf.0

   1. Initial program 48.1%

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

      1. times-frac 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in M around 0 63.6%

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

      1. associate-*r/ 63.6%

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

      2. *-commutative 63.6%

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

      3. associate-*r/ 63.6%

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

      4. times-frac 63.5%

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

      5. unpow2 63.5%

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

      6. *-commutative 63.5%

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

      7. unpow2 63.5%

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

      8. unpow2 63.5%

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

   6. Simplified 63.5%

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

   7. Taylor expanded in M around 0 63.5%

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

      1. unpow2 63.5%

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

      2. *-commutative 63.5%

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

      3. associate-*r* 63.5%

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

      4. unpow2 63.5%

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

      5. times-frac 70.3%

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

      6. *-commutative 70.3%

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

   9. Simplified 70.3%

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

   if -inf.0 < (/.f64 h l) < -1.0000000000000001e-110

   1. Initial program 84.5%

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

      1. times-frac 83.5%

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

   3. Simplified 83.5%

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

   if -1.0000000000000001e-110 < (/.f64 h l) < -0.0

   1. Initial program 72.1%

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

      1. times-frac 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in M around 0 53.1%

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

      1. associate-*r/ 53.1%

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

      2. *-commutative 53.1%

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

      3. *-commutative 53.1%

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

      4. times-frac 53.1%

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

      5. unpow2 53.1%

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

      6. *-commutative 53.1%

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

      7. associate-*l* 53.2%

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

      8. unpow2 53.2%

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

      9. unpow2 53.2%

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

      10. swap-sqr 67.5%

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

      11. *-commutative 67.5%

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

      12. associate-*l* 66.0%

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

      13. *-commutative 66.0%

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

   6. Simplified 66.0%

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

   7. Taylor expanded in d around 0 53.1%

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

      1. *-commutative 53.1%

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

      2. times-frac 49.9%

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

      3. unpow2 49.9%

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

      4. associate-*r/ 56.2%

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

      5. unpow2 56.2%

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

      6. times-frac 64.2%

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

      7. unpow2 64.2%

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

      8. associate-/l* 68.9%

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

      9. *-commutative 68.9%

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

      10. associate-*l* 75.3%

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

      11. *-commutative 75.3%

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

   9. Simplified 75.3%

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

   10. Taylor expanded in M around 0 62.6%

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

       1. unpow2 62.6%

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

       2. associate-*l* 67.3%

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

       3. *-commutative 67.3%

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

       4. unpow2 67.3%

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

       5. times-frac 76.9%

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

       6. associate-*l/ 78.4%

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

       7. *-commutative 78.4%

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

       8. associate-*l* 76.9%

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

   12. Simplified 76.9%

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

   if -0.0 < (/.f64 h l)

   1. Initial program 95.4%

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

      1. times-frac 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in M around 0 100.0%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-110}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq 0:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 10^{+264}:\\ \;\;\;\;w0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (- 1.0 (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ h l)))))
   (if (<= t_0 1e+264)
     (* w0 (sqrt t_0))
     (*
      w0
      (sqrt (- 1.0 (* 0.25 (* D (* (/ D l) (* h (* (/ M d) (/ M d))))))))))))

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 t_0 = 1.0 - (pow(((D * M) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 1e+264) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((((d * m) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))
    if (t_0 <= 1d+264) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d * ((d / l) * (h * ((m / d_1) * (m / d_1))))))))
    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 t_0 = 1.0 - (Math.pow(((D * M) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 1e+264) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = 1.0 - (math.pow(((D * M) / (2.0 * d)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 1e+264:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(1.0 - Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 1e+264)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D * Float64(Float64(D / l) * Float64(h * Float64(Float64(M / d) * Float64(M / d)))))))));
	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)
	t_0 = 1.0 - ((((D * M) / (2.0 * d)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 1e+264)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))));
	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_] := Block[{t$95$0 = N[(1.0 - N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+264], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(h * N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $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))) < 1.00000000000000004e264

   1. Initial program 99.9%

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

   if 1.00000000000000004e264 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 46.1%

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

      1. times-frac 47.2%

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

   3. Simplified 47.2%

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

   4. Taylor expanded in M around 0 44.7%

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

      1. associate-*r/ 44.7%

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

      2. *-commutative 44.7%

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

      3. *-commutative 44.7%

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

      4. times-frac 45.8%

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

      5. unpow2 45.8%

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

      6. *-commutative 45.8%

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

      7. associate-*l* 47.2%

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

      8. unpow2 47.2%

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

      9. unpow2 47.2%

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

      10. swap-sqr 54.5%

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

      11. *-commutative 54.5%

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

      12. associate-*l* 54.5%

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

      13. *-commutative 54.5%

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

   6. Simplified 54.5%

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

   7. Taylor expanded in d around 0 44.7%

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

      1. *-commutative 44.7%

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

      2. times-frac 42.2%

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

      3. unpow2 42.2%

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

      4. associate-*r/ 44.5%

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

      5. unpow2 44.5%

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

      6. times-frac 52.1%

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

      7. unpow2 52.1%

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

      8. associate-/l* 56.8%

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

      9. *-commutative 56.8%

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

      10. associate-*l* 64.1%

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

      11. *-commutative 64.1%

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

   9. Simplified 64.1%

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

   10. Taylor expanded in M around 0 48.4%

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

       1. unpow2 48.4%

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

       2. associate-*l* 48.4%

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

       3. *-commutative 48.4%

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

       4. unpow2 48.4%

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

       5. times-frac 61.7%

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

       6. associate-*l/ 65.2%

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

       7. *-commutative 65.2%

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

       8. associate-*l* 64.0%

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

   12. Simplified 64.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{+264}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 3: 87.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-110}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq 0:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= (/ h l) (- INFINITY))
   (* w0 (sqrt (- 1.0 (* 0.25 (* (/ (* D D) l) (* (/ M d) (/ (* h M) d)))))))
   (if (<= (/ h l) -1e-110)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* M (* D (/ 0.5 d))) 2.0)))))
     (if (<= (/ h l) 0.0)
       (*
        w0
        (sqrt (- 1.0 (* 0.25 (* D (* (/ D l) (* h (* (/ M d) (/ M d)))))))))
       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) {
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D * D) / l) * ((M / d) * ((h * M) / d))))));
	} else if ((h / l) <= -1e-110) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((M * (D * (0.5 / d))), 2.0))));
	} else if ((h / l) <= 0.0) {
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

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 ((h / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D * D) / l) * ((M / d) * ((h * M) / d))))));
	} else if ((h / l) <= -1e-110) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((M * (D * (0.5 / d))), 2.0))));
	} else if ((h / l) <= 0.0) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))));
	} else {
		tmp = w0;
	}
	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 (h / l) <= -math.inf:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D * D) / l) * ((M / d) * ((h * M) / d))))))
	elif (h / l) <= -1e-110:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((M * (D * (0.5 / d))), 2.0))))
	elif (h / l) <= 0.0:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))))
	else:
		tmp = w0
	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(h / l) <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D * D) / l) * Float64(Float64(M / d) * Float64(Float64(h * M) / d)))))));
	elseif (Float64(h / l) <= -1e-110)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0)))));
	elseif (Float64(h / l) <= 0.0)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D * Float64(Float64(D / l) * Float64(h * Float64(Float64(M / d) * Float64(M / d)))))))));
	else
		tmp = w0;
	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 ((h / l) <= -Inf)
		tmp = w0 * sqrt((1.0 - (0.25 * (((D * D) / l) * ((M / d) * ((h * M) / d))))));
	elseif ((h / l) <= -1e-110)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M * (D * (0.5 / d))) ^ 2.0))));
	elseif ((h / l) <= 0.0)
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))));
	else
		tmp = w0;
	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[(h / l), $MachinePrecision], (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-110], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], 0.0], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(h * N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 h l) < -inf.0

   1. Initial program 48.1%

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

      1. times-frac 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in M around 0 63.6%

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

      1. associate-*r/ 63.6%

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

      2. *-commutative 63.6%

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

      3. associate-*r/ 63.6%

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

      4. times-frac 63.5%

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

      5. unpow2 63.5%

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

      6. *-commutative 63.5%

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

      7. unpow2 63.5%

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

      8. unpow2 63.5%

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

   6. Simplified 63.5%

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

   7. Taylor expanded in M around 0 63.5%

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

      1. unpow2 63.5%

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

      2. *-commutative 63.5%

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

      3. associate-*r* 63.5%

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

      4. unpow2 63.5%

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

      5. times-frac 70.3%

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

      6. *-commutative 70.3%

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

   9. Simplified 70.3%

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

   if -inf.0 < (/.f64 h l) < -1.0000000000000001e-110

   1. Initial program 84.5%

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

      1. associate-/l* 83.5%

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

   3. Simplified 83.5%

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

      1. clear-num 83.5%

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

      2. associate-/r/ 83.5%

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

      3. clear-num 83.5%

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

      4. div-inv 83.5%

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

      5. associate-/r* 83.5%

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

      6. metadata-eval 83.5%

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

   5. Applied egg-rr 83.5%

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

   if -1.0000000000000001e-110 < (/.f64 h l) < -0.0

   1. Initial program 72.1%

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

      1. times-frac 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in M around 0 53.1%

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

      1. associate-*r/ 53.1%

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

      2. *-commutative 53.1%

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

      3. *-commutative 53.1%

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

      4. times-frac 53.1%

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

      5. unpow2 53.1%

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

      6. *-commutative 53.1%

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

      7. associate-*l* 53.2%

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

      8. unpow2 53.2%

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

      9. unpow2 53.2%

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

      10. swap-sqr 67.5%

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

      11. *-commutative 67.5%

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

      12. associate-*l* 66.0%

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

      13. *-commutative 66.0%

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

   6. Simplified 66.0%

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

   7. Taylor expanded in d around 0 53.1%

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

      1. *-commutative 53.1%

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

      2. times-frac 49.9%

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

      3. unpow2 49.9%

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

      4. associate-*r/ 56.2%

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

      5. unpow2 56.2%

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

      6. times-frac 64.2%

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

      7. unpow2 64.2%

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

      8. associate-/l* 68.9%

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

      9. *-commutative 68.9%

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

      10. associate-*l* 75.3%

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

      11. *-commutative 75.3%

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

   9. Simplified 75.3%

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

   10. Taylor expanded in M around 0 62.6%

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

       1. unpow2 62.6%

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

       2. associate-*l* 67.3%

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

       3. *-commutative 67.3%

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

       4. unpow2 67.3%

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

       5. times-frac 76.9%

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

       6. associate-*l/ 78.4%

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

       7. *-commutative 78.4%

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

       8. associate-*l* 76.9%

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

   12. Simplified 76.9%

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

   if -0.0 < (/.f64 h l)

   1. Initial program 95.4%

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

      1. times-frac 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in M around 0 100.0%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-110}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq 0:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;2 \cdot d \leq 5 \cdot 10^{-113}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

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 (<= (* 2.0 d) 5e-113)
   (* w0 (sqrt (- 1.0 (* 0.25 (* D (* (/ D l) (* h (* (/ M d) (/ M d)))))))))
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* (* M 0.5) (/ D d)) 2.0)) 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 ((2.0 * d) <= 5e-113) {
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))));
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow(((M * 0.5) * (D / d)), 2.0)) / 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 ((2.0d0 * d_1) <= 5d-113) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d * ((d / l) * (h * ((m / d_1) * (m / d_1))))))))
    else
        tmp = w0 * sqrt((1.0d0 - ((h * (((m * 0.5d0) * (d / d_1)) ** 2.0d0)) / 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 ((2.0 * d) <= 5e-113) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow(((M * 0.5) * (D / d)), 2.0)) / 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 (2.0 * d) <= 5e-113:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow(((M * 0.5) * (D / d)), 2.0)) / 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(2.0 * d) <= 5e-113)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D * Float64(Float64(D / l) * Float64(h * Float64(Float64(M / d) * Float64(M / d)))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0)) / 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 ((2.0 * d) <= 5e-113)
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))));
	else
		tmp = w0 * sqrt((1.0 - ((h * (((M * 0.5) * (D / d)) ^ 2.0)) / 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[(2.0 * d), $MachinePrecision], 5e-113], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(h * N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 2 d) < 4.9999999999999997e-113

   1. Initial program 80.8%

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

      1. times-frac 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in M around 0 53.1%

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

      1. associate-*r/ 53.1%

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

      2. *-commutative 53.1%

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

      3. *-commutative 53.1%

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

      4. times-frac 53.9%

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

      5. unpow2 53.9%

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

      6. *-commutative 53.9%

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

      7. associate-*l* 55.2%

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

      8. unpow2 55.2%

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

      9. unpow2 55.2%

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

      10. swap-sqr 68.5%

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

      11. *-commutative 68.5%

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

      12. associate-*l* 66.5%

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

      13. *-commutative 66.5%

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

   6. Simplified 66.5%

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

   7. Taylor expanded in d around 0 53.1%

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

      1. *-commutative 53.1%

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

      2. times-frac 51.9%

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

      3. unpow2 51.9%

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

      4. associate-*r/ 55.1%

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

      5. unpow2 55.1%

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

      6. times-frac 62.5%

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

      7. unpow2 62.5%

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

      8. associate-/l* 68.5%

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

      9. *-commutative 68.5%

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

      10. associate-*l* 75.2%

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

      11. *-commutative 75.2%

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

   9. Simplified 75.2%

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

   10. Taylor expanded in M around 0 60.5%

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

       1. unpow2 60.5%

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

       2. associate-*l* 62.0%

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

       3. *-commutative 62.0%

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

       4. unpow2 62.0%

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

       5. times-frac 76.4%

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

       6. associate-*l/ 79.6%

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

       7. *-commutative 79.6%

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

       8. associate-*l* 78.3%

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

   12. Simplified 78.3%

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

   if 4.9999999999999997e-113 < (*.f64 2 d)

   1. Initial program 84.9%

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

      1. times-frac 85.0%

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

   3. Simplified 85.0%

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

      1. associate-*r/ 91.0%

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

      2. frac-times 90.9%

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

      3. frac-times 91.0%

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

      4. unpow2 91.0%

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

      5. unpow2 91.0%

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

      6. div-inv 91.0%

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

      7. metadata-eval 91.0%

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

   5. Applied egg-rr 91.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot d \leq 5 \cdot 10^{-113}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 5: 83.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq 0:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= (/ h l) 0.0)
   (* w0 (sqrt (- 1.0 (* 0.25 (* D (* (/ D l) (* h (* (/ M d) (/ M d)))))))))
   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) {
	double tmp;
	if ((h / l) <= 0.0) {
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))));
	} else {
		tmp = w0;
	}
	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 ((h / l) <= 0.0d0) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d * ((d / l) * (h * ((m / d_1) * (m / d_1))))))))
    else
        tmp = w0
    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 ((h / l) <= 0.0) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))));
	} else {
		tmp = w0;
	}
	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 (h / l) <= 0.0:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))))
	else:
		tmp = w0
	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(h / l) <= 0.0)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D * Float64(Float64(D / l) * Float64(h * Float64(Float64(M / d) * Float64(M / d)))))))));
	else
		tmp = w0;
	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 ((h / l) <= 0.0)
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D / l) * (h * ((M / d) * (M / d))))))));
	else
		tmp = w0;
	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[(h / l), $MachinePrecision], 0.0], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(h * N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -0.0

   1. Initial program 75.8%

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

      1. times-frac 74.7%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in M around 0 51.7%

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

      1. associate-*r/ 51.7%

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

      2. *-commutative 51.7%

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

      3. *-commutative 51.7%

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

      4. times-frac 52.8%

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

      5. unpow2 52.8%

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

      6. *-commutative 52.8%

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

      7. associate-*l* 52.9%

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

      8. unpow2 52.9%

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

      9. unpow2 52.9%

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

      10. swap-sqr 67.6%

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

      11. *-commutative 67.6%

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

      12. associate-*l* 66.5%

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

      13. *-commutative 66.5%

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

   6. Simplified 66.5%

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

   7. Taylor expanded in d around 0 51.7%

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

      1. *-commutative 51.7%

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

      2. times-frac 50.4%

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

      3. unpow2 50.4%

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

      4. associate-*r/ 54.6%

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

      5. unpow2 54.6%

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

      6. times-frac 61.2%

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

      7. unpow2 61.2%

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

      8. associate-/l* 68.9%

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

      9. *-commutative 68.9%

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

      10. associate-*l* 76.1%

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

      11. *-commutative 76.1%

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

   9. Simplified 76.1%

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

   10. Taylor expanded in M around 0 60.1%

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

       1. unpow2 60.1%

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

       2. associate-*l* 62.5%

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

       3. *-commutative 62.5%

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

       4. unpow2 62.5%

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

       5. times-frac 75.4%

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

       6. associate-*l/ 78.8%

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

       7. *-commutative 78.8%

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

       8. associate-*l* 77.1%

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

   12. Simplified 77.1%

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

   if -0.0 < (/.f64 h l)

   1. Initial program 95.4%

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

      1. times-frac 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in M around 0 100.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq 0:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 6: 77.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.05 \cdot 10^{-201}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \frac{M \cdot \frac{h}{d}}{d}\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 1.05e-201)
   w0
   (* w0 (+ 1.0 (* -0.125 (* D (* (/ D l) (* M (/ (* M (/ h d)) d)))))))))

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 <= 1.05e-201) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M * (h / d)) / d))))));
	}
	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 <= 1.05d-201) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (d * ((d / l) * (m * ((m * (h / d_1)) / d_1))))))
    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 <= 1.05e-201) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M * (h / d)) / d))))));
	}
	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 <= 1.05e-201:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M * (h / d)) / d))))))
	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 <= 1.05e-201)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(D / l) * Float64(M * Float64(Float64(M * Float64(h / d)) / d)))))));
	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 <= 1.05e-201)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * (M * ((M * (h / d)) / d))))));
	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, 1.05e-201], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(M * N[(N[(M * N[(h / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.05000000000000006e-201

   1. Initial program 83.2%

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

      1. times-frac 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in M around 0 69.9%

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

   if 1.05000000000000006e-201 < M

   1. Initial program 81.4%

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

      1. times-frac 81.4%

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

   3. Simplified 81.4%

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

   4. Taylor expanded in M around 0 56.8%

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

      1. associate-*r/ 56.8%

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

      2. *-commutative 56.8%

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

      3. associate-*r/ 56.8%

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

      4. *-commutative 56.8%

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

      5. times-frac 54.9%

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

      6. unpow2 54.9%

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

      7. *-commutative 54.9%

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

      8. unpow2 54.9%

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

      9. unpow2 54.9%

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

   6. Simplified 54.9%

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

   7. Taylor expanded in D around 0 56.8%

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

      1. *-commutative 56.8%

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

      2. times-frac 54.9%

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

      3. unpow2 54.9%

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

      4. associate-*l/ 57.8%

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

      5. *-commutative 57.8%

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

      6. unpow2 57.8%

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

      7. *-commutative 57.8%

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

      8. associate-*r* 61.7%

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

      9. unpow2 61.7%

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

      10. times-frac 70.4%

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

      11. *-commutative 70.4%

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

   9. Simplified 70.4%

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

   10. Taylor expanded in D around 0 56.8%

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

       1. *-commutative 56.8%

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

       2. times-frac 54.9%

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

       3. unpow2 54.9%

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

       4. associate-*r/ 57.8%

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

       5. unpow2 57.8%

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

       6. times-frac 63.6%

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

       7. unpow2 63.6%

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

       8. associate-/l* 73.3%

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

       9. associate-*l* 79.1%

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

       10. associate-*r/ 80.1%

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

       11. *-commutative 80.1%

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

       12. associate-/r/ 79.1%

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

       13. *-commutative 79.1%

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

   12. Simplified 79.1%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.05 \cdot 10^{-201}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(M \cdot \frac{M \cdot \frac{h}{d}}{d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 77.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.3 \cdot 10^{-201}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 1.3e-201)
   w0
   (* w0 (+ 1.0 (* -0.125 (* D (* (/ D l) (* (/ M (/ d M)) (/ h d)))))))))

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 <= 1.3e-201) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * ((M / (d / M)) * (h / d))))));
	}
	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 <= 1.3d-201) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (d * ((d / l) * ((m / (d_1 / m)) * (h / d_1))))))
    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 <= 1.3e-201) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * ((M / (d / M)) * (h / d))))));
	}
	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 <= 1.3e-201:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * ((M / (d / M)) * (h / d))))))
	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 <= 1.3e-201)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(D / l) * Float64(Float64(M / Float64(d / M)) * Float64(h / d)))))));
	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 <= 1.3e-201)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * ((M / (d / M)) * (h / d))))));
	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, 1.3e-201], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.29999999999999991e-201

   1. Initial program 83.2%

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

      1. times-frac 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in M around 0 69.9%

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

   if 1.29999999999999991e-201 < M

   1. Initial program 81.4%

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

      1. times-frac 81.4%

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

   3. Simplified 81.4%

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

   4. Taylor expanded in M around 0 56.8%

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

      1. associate-*r/ 56.8%

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

      2. *-commutative 56.8%

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

      3. associate-*r/ 56.8%

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

      4. *-commutative 56.8%

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

      5. times-frac 54.9%

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

      6. unpow2 54.9%

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

      7. *-commutative 54.9%

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

      8. unpow2 54.9%

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

      9. unpow2 54.9%

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

   6. Simplified 54.9%

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

   7. Taylor expanded in D around 0 56.8%

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

      1. *-commutative 56.8%

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

      2. times-frac 54.9%

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

      3. unpow2 54.9%

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

      4. associate-*l/ 57.8%

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

      5. *-commutative 57.8%

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

      6. unpow2 57.8%

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

      7. *-commutative 57.8%

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

      8. associate-*r* 61.7%

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

      9. unpow2 61.7%

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

      10. times-frac 70.4%

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

      11. *-commutative 70.4%

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

   9. Simplified 70.4%

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

   10. Taylor expanded in w0 around 0 56.8%

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

   12. Simplified 79.1%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.3 \cdot 10^{-201}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 8: 75.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.26 \cdot 10^{+150}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \frac{h \cdot M}{d \cdot \frac{d}{M}}\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 1.26e+150)
   (* w0 (+ 1.0 (* (* D (* (/ D l) (/ (* h M) (* d (/ d M))))) -0.125)))
   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) {
	double tmp;
	if (d <= 1.26e+150) {
		tmp = w0 * (1.0 + ((D * ((D / l) * ((h * M) / (d * (d / M))))) * -0.125));
	} else {
		tmp = w0;
	}
	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 <= 1.26d+150) then
        tmp = w0 * (1.0d0 + ((d * ((d / l) * ((h * m) / (d_1 * (d_1 / m))))) * (-0.125d0)))
    else
        tmp = w0
    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 <= 1.26e+150) {
		tmp = w0 * (1.0 + ((D * ((D / l) * ((h * M) / (d * (d / M))))) * -0.125));
	} else {
		tmp = w0;
	}
	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 <= 1.26e+150:
		tmp = w0 * (1.0 + ((D * ((D / l) * ((h * M) / (d * (d / M))))) * -0.125))
	else:
		tmp = w0
	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 <= 1.26e+150)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D * Float64(Float64(D / l) * Float64(Float64(h * M) / Float64(d * Float64(d / M))))) * -0.125)));
	else
		tmp = w0;
	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 <= 1.26e+150)
		tmp = w0 * (1.0 + ((D * ((D / l) * ((h * M) / (d * (d / M))))) * -0.125));
	else
		tmp = w0;
	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, 1.26e+150], N[(w0 * N[(1.0 + N[(N[(D * N[(N[(D / l), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / N[(d * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 1.26e150

   1. Initial program 82.5%

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

      1. times-frac 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in M around 0 57.0%

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

      1. associate-*r/ 57.0%

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

      2. *-commutative 57.0%

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

      3. associate-*r/ 57.0%

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

      4. *-commutative 57.0%

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

      5. times-frac 57.5%

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

      6. unpow2 57.5%

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

      7. *-commutative 57.5%

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

      8. unpow2 57.5%

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

      9. unpow2 57.5%

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

   6. Simplified 57.5%

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

   7. Taylor expanded in D around 0 57.0%

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

      1. *-commutative 57.0%

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

      2. times-frac 57.5%

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

      3. unpow2 57.5%

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

      4. associate-*l/ 59.4%

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

      5. *-commutative 59.4%

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

      6. unpow2 59.4%

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

      7. *-commutative 59.4%

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

      8. associate-*r* 61.0%

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

      9. unpow2 61.0%

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

      10. times-frac 69.0%

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

      11. *-commutative 69.0%

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

   9. Simplified 69.0%

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

       1. add-cbrt-cube 33.5%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\left(w0 \cdot \left(1 + \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{M \cdot h}{d} \cdot \frac{M}{d}\right)\right) \cdot -0.125\right)\right) \cdot \left(w0 \cdot \left(1 + \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{M \cdot h}{d} \cdot \frac{M}{d}\right)\right) \cdot -0.125\right)\right)\right) \cdot \left(w0 \cdot \left(1 + \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{M \cdot h}{d} \cdot \frac{M}{d}\right)\right) \cdot -0.125\right)\right)}} \]

   11. Applied egg-rr 33.5%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\left(w0 \cdot \left(1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{h}} \cdot \frac{M}{d}\right) \cdot -0.125\right)\right)\right) \cdot \left(w0 \cdot \left(1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{h}} \cdot \frac{M}{d}\right) \cdot -0.125\right)\right)\right)\right) \cdot \left(w0 \cdot \left(1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{h}} \cdot \frac{M}{d}\right) \cdot -0.125\right)\right)\right)}} \]
   12. Step-by-step derivation

       1. associate-*l* 33.5%

          \[\leadsto \sqrt[3]{\color{blue}{\left(w0 \cdot \left(1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{h}} \cdot \frac{M}{d}\right) \cdot -0.125\right)\right)\right) \cdot \left(\left(w0 \cdot \left(1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{h}} \cdot \frac{M}{d}\right) \cdot -0.125\right)\right)\right) \cdot \left(w0 \cdot \left(1 + \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(\frac{M}{\frac{d}{h}} \cdot \frac{M}{d}\right) \cdot -0.125\right)\right)\right)\right)}} \]

   13. Simplified 36.2%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.125\right) \cdot w0\right) \cdot \left(w0 \cdot \left(\left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.125\right) \cdot \left(\left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.125\right) \cdot w0\right)\right)\right)}} \]
   14. Step-by-step derivation

       1. *-commutative 36.2%

          \[\leadsto \sqrt[3]{\color{blue}{\left(w0 \cdot \left(\left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.125\right) \cdot \left(\left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.125\right) \cdot w0\right)\right)\right) \cdot \left(\left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.125\right) \cdot w0\right)}} \]

       2. associate-*r* 36.2%

          \[\leadsto \sqrt[3]{\color{blue}{\left(\left(w0 \cdot \left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.125\right)\right) \cdot \left(\left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.125\right) \cdot w0\right)\right)} \cdot \left(\left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.125\right) \cdot w0\right)} \]

       3. *-commutative 36.2%

          \[\leadsto \sqrt[3]{\left(\color{blue}{\left(\left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.125\right) \cdot w0\right)} \cdot \left(\left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.125\right) \cdot w0\right)\right) \cdot \left(\left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right) \cdot -0.125\right) \cdot w0\right)} \]

   15. Applied egg-rr 74.4%

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

   if 1.26e150 < d

   1. Initial program 82.2%

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

      1. times-frac 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in M around 0 86.9%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.26 \cdot 10^{+150}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \left(\frac{D}{\ell} \cdot \frac{h \cdot M}{d \cdot \frac{d}{M}}\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 9: 71.7% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 9.8 \cdot 10^{+45}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{h \cdot \left(M \cdot \left(w0 \cdot M\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

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 9.8e+45)
   w0
   (* -0.125 (* (/ (* D D) (* d d)) (/ (* h (* M (* w0 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 <= 9.8e+45) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) / (d * d)) * ((h * (M * (w0 * 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 <= 9.8d+45) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d * d) / (d_1 * d_1)) * ((h * (m * (w0 * 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 <= 9.8e+45) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) / (d * d)) * ((h * (M * (w0 * 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 <= 9.8e+45:
		tmp = w0
	else:
		tmp = -0.125 * (((D * D) / (d * d)) * ((h * (M * (w0 * 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 <= 9.8e+45)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) / Float64(d * d)) * Float64(Float64(h * Float64(M * Float64(w0 * 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 <= 9.8e+45)
		tmp = w0;
	else
		tmp = -0.125 * (((D * D) / (d * d)) * ((h * (M * (w0 * 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, 9.8e+45], w0, N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * N[(w0 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 9.8000000000000004e45

   1. Initial program 83.9%

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

      1. times-frac 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in M around 0 74.7%

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

   if 9.8000000000000004e45 < M

   1. Initial program 76.7%

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

      1. times-frac 76.7%

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

   3. Simplified 76.7%

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

   4. Taylor expanded in M around 0 37.0%

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

      1. associate-*r/ 37.0%

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

      2. *-commutative 37.0%

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

      3. associate-*r/ 37.0%

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

      4. *-commutative 37.0%

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

      5. times-frac 36.9%

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

      6. unpow2 36.9%

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

      7. *-commutative 36.9%

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

      8. unpow2 36.9%

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

      9. unpow2 36.9%

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

   6. Simplified 36.9%

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

   7. Taylor expanded in D around 0 37.0%

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

      1. *-commutative 37.0%

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

      2. times-frac 36.9%

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

      3. unpow2 36.9%

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

      4. associate-*l/ 37.0%

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

      5. *-commutative 37.0%

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

      6. unpow2 37.0%

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

      7. *-commutative 37.0%

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

      8. associate-*r* 45.1%

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

      9. unpow2 45.1%

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

      10. times-frac 61.3%

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

      11. *-commutative 61.3%

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

   9. Simplified 61.3%

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

   10. Taylor expanded in D around inf 27.3%

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

       1. associate-/r* 27.4%

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

       2. associate-/r* 27.3%

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

       3. times-frac 25.2%

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

       4. *-commutative 25.2%

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

       5. unpow2 25.2%

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

       6. unpow2 25.2%

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

       7. *-commutative 25.2%

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

       8. *-commutative 25.2%

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

       9. *-commutative 25.2%

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

       10. unpow2 25.2%

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

   12. Simplified 25.2%

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

   13. Taylor expanded in D around 0 27.3%

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

       1. *-commutative 27.3%

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

       2. times-frac 25.2%

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

       3. unpow2 25.2%

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

       4. unpow2 25.2%

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

       5. *-commutative 25.2%

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

       6. associate-*r* 25.3%

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

       7. unpow2 25.3%

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

       8. associate-*l* 25.3%

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

   15. Simplified 25.3%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9.8 \cdot 10^{+45}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{h \cdot \left(M \cdot \left(w0 \cdot M\right)\right)}{\ell}\right)\\ \end{array} \]

Alternative 10: 71.7% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.05 \cdot 10^{+45}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

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 1.05e+45)
   w0
   (* -0.125 (* (/ (* D D) (* d d)) (/ (* w0 (* 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 <= 1.05e+45) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) / (d * d)) * ((w0 * (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 <= 1.05d+45) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d * d) / (d_1 * d_1)) * ((w0 * (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 <= 1.05e+45) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) / (d * d)) * ((w0 * (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 <= 1.05e+45:
		tmp = w0
	else:
		tmp = -0.125 * (((D * D) / (d * d)) * ((w0 * (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 <= 1.05e+45)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) / Float64(d * d)) * Float64(Float64(w0 * Float64(h * Float64(M * 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 <= 1.05e+45)
		tmp = w0;
	else
		tmp = -0.125 * (((D * D) / (d * d)) * ((w0 * (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, 1.05e+45], w0, N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.04999999999999997e45

   1. Initial program 83.9%

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

      1. times-frac 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in M around 0 74.7%

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

   if 1.04999999999999997e45 < M

   1. Initial program 76.7%

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

      1. times-frac 76.7%

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

   3. Simplified 76.7%

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

   4. Taylor expanded in M around 0 37.0%

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

      1. associate-*r/ 37.0%

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

      2. *-commutative 37.0%

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

      3. associate-*r/ 37.0%

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

      4. *-commutative 37.0%

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

      5. times-frac 36.9%

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

      6. unpow2 36.9%

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

      7. *-commutative 36.9%

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

      8. unpow2 36.9%

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

      9. unpow2 36.9%

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

   6. Simplified 36.9%

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

   7. Taylor expanded in D around 0 37.0%

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

      1. *-commutative 37.0%

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

      2. times-frac 36.9%

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

      3. unpow2 36.9%

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

      4. associate-*l/ 37.0%

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

      5. *-commutative 37.0%

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

      6. unpow2 37.0%

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

      7. *-commutative 37.0%

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

      8. associate-*r* 45.1%

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

      9. unpow2 45.1%

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

      10. times-frac 61.3%

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

      11. *-commutative 61.3%

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

   9. Simplified 61.3%

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

   10. Taylor expanded in D around inf 27.3%

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

       1. associate-/r* 27.4%

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

       2. associate-/r* 27.3%

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

       3. times-frac 25.2%

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

       4. *-commutative 25.2%

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

       5. unpow2 25.2%

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

       6. unpow2 25.2%

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

       7. *-commutative 25.2%

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

       8. *-commutative 25.2%

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

       9. *-commutative 25.2%

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

       10. unpow2 25.2%

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

   12. Simplified 25.2%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.05 \cdot 10^{+45}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell}\right)\\ \end{array} \]

Alternative 11: 71.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 10^{+46}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \end{array} \end{array} \]

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+46)
   w0
   (* -0.125 (/ (* (* D D) (* h (* w0 (* M M)))) (* l (* d d))))))

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+46) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) * (h * (w0 * (M * M)))) / (l * (d * d)));
	}
	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+46) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d * d) * (h * (w0 * (m * m)))) / (l * (d_1 * d_1)))
    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+46) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) * (h * (w0 * (M * M)))) / (l * (d * d)));
	}
	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+46:
		tmp = w0
	else:
		tmp = -0.125 * (((D * D) * (h * (w0 * (M * M)))) / (l * (d * d)))
	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+46)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) * Float64(h * Float64(w0 * Float64(M * M)))) / Float64(l * Float64(d * d))));
	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+46)
		tmp = w0;
	else
		tmp = -0.125 * (((D * D) * (h * (w0 * (M * M)))) / (l * (d * d)));
	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+46], w0, N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] * N[(h * N[(w0 * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 9.9999999999999999e45

   1. Initial program 83.9%

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

      1. times-frac 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in M around 0 74.7%

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

   if 9.9999999999999999e45 < M

   1. Initial program 76.7%

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

      1. times-frac 76.7%

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

   3. Simplified 76.7%

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

   4. Taylor expanded in M around 0 37.0%

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

      1. associate-*r/ 37.0%

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

      2. *-commutative 37.0%

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

      3. associate-*r/ 37.0%

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

      4. *-commutative 37.0%

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

      5. times-frac 36.9%

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

      6. unpow2 36.9%

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

      7. *-commutative 36.9%

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

      8. unpow2 36.9%

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

      9. unpow2 36.9%

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

   6. Simplified 36.9%

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

   7. Taylor expanded in D around 0 37.0%

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

      1. *-commutative 37.0%

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

      2. times-frac 36.9%

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

      3. unpow2 36.9%

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

      4. associate-*l/ 37.0%

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

      5. *-commutative 37.0%

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

      6. unpow2 37.0%

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

      7. *-commutative 37.0%

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

      8. associate-*r* 45.1%

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

      9. unpow2 45.1%

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

      10. times-frac 61.3%

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

      11. *-commutative 61.3%

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

   9. Simplified 61.3%

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

   10. Taylor expanded in D around inf 27.3%

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

       1. associate-/r* 27.4%

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

       2. associate-/r* 27.3%

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

       3. times-frac 25.2%

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

       4. *-commutative 25.2%

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

       5. unpow2 25.2%

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

       6. unpow2 25.2%

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

       7. *-commutative 25.2%

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

       8. *-commutative 25.2%

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

       9. *-commutative 25.2%

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

       10. unpow2 25.2%

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

   12. Simplified 25.2%

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

       1. frac-times 27.3%

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

       2. associate-*l* 27.5%

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

   14. Applied egg-rr 27.5%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 10^{+46}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \end{array} \]

Alternative 12: 72.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \left(w0 \cdot \frac{h}{d}\right)}{\frac{d}{M} \cdot \frac{\ell}{M}}\\ \end{array} \end{array} \]

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 6.5e+15)
   w0
   (* -0.125 (/ (* (* D D) (* w0 (/ h d))) (* (/ d M) (/ l 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 <= 6.5e+15) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) * (w0 * (h / d))) / ((d / M) * (l / 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 <= 6.5d+15) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d * d) * (w0 * (h / d_1))) / ((d_1 / m) * (l / 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 <= 6.5e+15) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) * (w0 * (h / d))) / ((d / M) * (l / 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 <= 6.5e+15:
		tmp = w0
	else:
		tmp = -0.125 * (((D * D) * (w0 * (h / d))) / ((d / M) * (l / 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 <= 6.5e+15)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) * Float64(w0 * Float64(h / d))) / Float64(Float64(d / M) * Float64(l / 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 <= 6.5e+15)
		tmp = w0;
	else
		tmp = -0.125 * (((D * D) * (w0 * (h / d))) / ((d / M) * (l / 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, 6.5e+15], w0, N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] * N[(w0 * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] * N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 6.5e15

   1. Initial program 84.1%

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

      1. times-frac 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in M around 0 75.4%

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

   if 6.5e15 < M

   1. Initial program 76.9%

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

      1. times-frac 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in M around 0 39.9%

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

      1. associate-*r/ 39.9%

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

      2. *-commutative 39.9%

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

      3. associate-*r/ 39.9%

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

      4. *-commutative 39.9%

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

      5. times-frac 39.8%

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

      6. unpow2 39.8%

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

      7. *-commutative 39.8%

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

      8. unpow2 39.8%

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

      9. unpow2 39.8%

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

   6. Simplified 39.8%

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

   7. Taylor expanded in D around 0 39.9%

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

      1. *-commutative 39.9%

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

      2. times-frac 39.8%

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

      3. unpow2 39.8%

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

      4. associate-*l/ 40.0%

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

      5. *-commutative 40.0%

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

      6. unpow2 40.0%

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

      7. *-commutative 40.0%

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

      8. associate-*r* 46.8%

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

      9. unpow2 46.8%

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

      10. times-frac 60.7%

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

      11. *-commutative 60.7%

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

   9. Simplified 60.7%

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

   10. Taylor expanded in D around inf 25.2%

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

       1. associate-*r/ 25.2%

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

       2. *-commutative 25.2%

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

       3. associate-*r/ 25.2%

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

       4. times-frac 25.1%

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

       5. *-commutative 25.1%

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

       6. associate-*r* 25.1%

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

       7. unpow2 25.1%

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

       8. times-frac 28.9%

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

       9. associate-*r/ 28.9%

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

       10. unpow2 28.9%

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

       11. associate-/l* 29.6%

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

       12. *-commutative 29.6%

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

       13. unpow2 29.6%

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

       14. associate-*r/ 29.8%

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

       15. associate-*l* 29.7%

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

   12. Simplified 29.2%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \left(w0 \cdot \frac{h}{d}\right)}{\frac{d}{M} \cdot \frac{\ell}{M}}\\ \end{array} \]

Alternative 13: 72.4% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\ell}\\ \end{array} \end{array} \]

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 6.5e+15)
   w0
   (* -0.125 (/ (* (* (/ D d) (/ D d)) (* h (* w0 (* 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 <= 6.5e+15) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((((D / d) * (D / d)) * (h * (w0 * (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 <= 6.5d+15) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((((d / d_1) * (d / d_1)) * (h * (w0 * (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 <= 6.5e+15) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((((D / d) * (D / d)) * (h * (w0 * (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 <= 6.5e+15:
		tmp = w0
	else:
		tmp = -0.125 * ((((D / d) * (D / d)) * (h * (w0 * (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 <= 6.5e+15)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(h * Float64(w0 * Float64(M * 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 <= 6.5e+15)
		tmp = w0;
	else
		tmp = -0.125 * ((((D / d) * (D / d)) * (h * (w0 * (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, 6.5e+15], w0, N[(-0.125 * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(w0 * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 6.5e15

   1. Initial program 84.1%

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

      1. times-frac 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in M around 0 75.4%

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

   if 6.5e15 < M

   1. Initial program 76.9%

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

      1. times-frac 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in M around 0 39.9%

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

      1. associate-*r/ 39.9%

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

      2. *-commutative 39.9%

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

      3. associate-*r/ 39.9%

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

      4. *-commutative 39.9%

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

      5. times-frac 39.8%

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

      6. unpow2 39.8%

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

      7. *-commutative 39.8%

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

      8. unpow2 39.8%

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

      9. unpow2 39.8%

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

   6. Simplified 39.8%

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

   7. Taylor expanded in D around 0 39.9%

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

      1. *-commutative 39.9%

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

      2. times-frac 39.8%

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

      3. unpow2 39.8%

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

      4. associate-*l/ 40.0%

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

      5. *-commutative 40.0%

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

      6. unpow2 40.0%

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

      7. *-commutative 40.0%

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

      8. associate-*r* 46.8%

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

      9. unpow2 46.8%

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

      10. times-frac 60.7%

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

      11. *-commutative 60.7%

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

   9. Simplified 60.7%

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

   10. Taylor expanded in D around inf 25.2%

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

       1. associate-/r* 25.2%

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

       2. associate-/r* 25.2%

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

       3. times-frac 23.4%

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

       4. *-commutative 23.4%

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

       5. unpow2 23.4%

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

       6. unpow2 23.4%

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

       7. *-commutative 23.4%

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

       8. *-commutative 23.4%

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

       9. *-commutative 23.4%

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

       10. unpow2 23.4%

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

   12. Simplified 23.4%

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

       1. associate-*r/ 23.4%

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

       2. times-frac 28.8%

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

       3. associate-*l* 28.9%

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

   14. Applied egg-rr 28.9%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\ell}\\ \end{array} \]

Alternative 14: 68.2% 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 82.5%

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

   1. times-frac 81.7%

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

3. Simplified 81.7%

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

4. Taylor expanded in M around 0 68.1%

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

5. Final simplification 68.1%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023261 
(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))))))