Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 86.7%

Time: 19.9s

Alternatives: 10

Speedup: 1.0×

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: 86.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}} \leq \infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell} \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<=
      (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ h l)))))
      INFINITY)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ D 2.0) (/ M d)) 2.0)))))
   (* w0 (+ 1.0 (* (/ (* (* (/ D d) (/ D d)) (* M (* h M))) l) -0.125)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((w0 * sqrt((1.0 - (pow(((M * D) / (d * 2.0)), 2.0) * (h / l))))) <= ((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D / 2.0) * (M / d)), 2.0))));
	} else {
		tmp = w0 * (1.0 + (((((D / d) * (D / d)) * (M * (h * M))) / l) * -0.125));
	}
	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 ((w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l))))) <= Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D / 2.0) * (M / d)), 2.0))));
	} else {
		tmp = w0 * (1.0 + (((((D / d) * (D / d)) * (M * (h * M))) / l) * -0.125));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (w0 * math.sqrt((1.0 - (math.pow(((M * D) / (d * 2.0)), 2.0) * (h / l))))) <= math.inf:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D / 2.0) * (M / d)), 2.0))))
	else:
		tmp = w0 * (1.0 + (((((D / d) * (D / d)) * (M * (h * M))) / l) * -0.125))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l))))) <= Inf)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D / 2.0) * Float64(M / d)) ^ 2.0)))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(M * Float64(h * M))) / l) * -0.125)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((w0 * sqrt((1.0 - ((((M * D) / (d * 2.0)) ^ 2.0) * (h / l))))) <= Inf)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D / 2.0) * (M / d)) ^ 2.0))));
	else
		tmp = w0 * (1.0 + (((((D / d) * (D / d)) * (M * (h * M))) / l) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], Infinity], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / 2.0), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < +inf.0

   1. Initial program 84.7%

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

   3. Simplified 85.2%

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

   if +inf.0 < (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in D around 0 50.0%

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

      1. *-commutative 50.0%

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

      2. times-frac 41.7%

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

      3. unpow2 41.7%

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

      4. unpow2 41.7%

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

      5. unpow2 41.7%

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

   6. Simplified 41.7%

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

      1. associate-*r/ 50.6%

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

      2. times-frac 67.1%

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

      3. *-commutative 67.1%

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

   8. Applied egg-rr 67.1%

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

   9. Taylor expanded in h around 0 67.1%

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

       1. unpow2 67.1%

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

       2. associate-*l* 75.1%

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

   11. Simplified 75.1%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}} \leq \infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell} \cdot -0.125\right)\\ \end{array} \]

Alternative 2: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}} \end{array} \]

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 (sqrt (- 1.0 (/ (* h (pow (* M (* D (/ 0.5 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) {
	return w0 * sqrt((1.0 - ((h * pow((M * (D * (0.5 / d))), 2.0)) / l)));
}

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 * sqrt((1.0d0 - ((h * ((m * (d * (0.5d0 / d_1))) ** 2.0d0)) / l)))
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 * Math.sqrt((1.0 - ((h * Math.pow((M * (D * (0.5 / d))), 2.0)) / l)));
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - ((h * math.pow((M * (D * (0.5 / d))), 2.0)) / l)))

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0)) / l))))
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 * sqrt((1.0 - ((h * ((M * (D * (0.5 / d))) ^ 2.0)) / l)));
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_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

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

3. Simplified 81.2%

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

   1. *-commutative 81.2%

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

   2. frac-times 80.7%

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

   3. *-commutative 80.7%

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

   4. associate-*l/ 84.3%

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

   5. div-inv 84.3%

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

   6. associate-*l* 84.4%

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

   7. associate-/r* 84.4%

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

   8. metadata-eval 84.4%

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

5. Applied egg-rr 84.4%

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

6. Final simplification 84.4%

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

Alternative 3: 78.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.7 \cdot 10^{-184}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.25\right) \cdot \frac{M}{\frac{\frac{\ell}{h}}{M}}}\\ \end{array} \end{array} \]

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 1.7e-184)
   w0
   (*
    w0
    (sqrt (+ 1.0 (* (* (* (/ D d) (/ D d)) -0.25) (/ M (/ (/ l h) M))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.7e-184) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 + ((((D / d) * (D / d)) * -0.25) * (M / ((l / h) / M)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 1.7d-184) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 + ((((d / d_1) * (d / d_1)) * (-0.25d0)) * (m / ((l / h) / m)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.7e-184) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((((D / d) * (D / d)) * -0.25) * (M / ((l / h) / M)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.7e-184:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 + ((((D / d) * (D / d)) * -0.25) * (M / ((l / h) / M)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.7e-184)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) * -0.25) * Float64(M / Float64(Float64(l / h) / M))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.7e-184)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 + ((((D / d) * (D / d)) * -0.25) * (M / ((l / h) / M)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.7e-184], w0, N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] * N[(M / N[(N[(l / h), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.70000000000000002e-184

   1. Initial program 80.9%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in D around 0 66.9%

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

   if 1.70000000000000002e-184 < M

   1. Initial program 80.4%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in D around 0 57.7%

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

      1. times-frac 57.1%

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

      2. unpow2 57.1%

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

      3. unpow2 57.1%

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

      4. unpow2 57.1%

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

   6. Simplified 57.1%

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

      1. *-un-lft-identity 57.1%

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

      2. cancel-sign-sub-inv 57.1%

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

      3. metadata-eval 57.1%

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

      4. times-frac 66.4%

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

      5. associate-/l* 66.7%

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

   8. Applied egg-rr 66.7%

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

   10. Simplified 72.1%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.7 \cdot 10^{-184}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.25\right) \cdot \frac{M}{\frac{\frac{\ell}{h}}{M}}}\\ \end{array} \]

Alternative 4: 79.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.25 \cdot 10^{-166}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 2.25e-166)
   w0
   (* w0 (sqrt (- 1.0 (* 0.25 (/ (* (* (/ D d) (/ D d)) (* h (* M M))) l)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.25e-166) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * ((((D / d) * (D / d)) * (h * (M * M))) / l))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 2.25d-166) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * ((((d / d_1) * (d / d_1)) * (h * (m * m))) / l))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.25e-166) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * ((((D / d) * (D / d)) * (h * (M * M))) / l))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.25e-166:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * ((((D / d) * (D / d)) * (h * (M * M))) / l))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.25e-166)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) * 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 <= 2.25e-166)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - (0.25 * ((((D / d) * (D / d)) * (h * (M * M))) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 2.25e-166], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.2499999999999999e-166

   1. Initial program 80.9%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in D around 0 66.8%

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

   if 2.2499999999999999e-166 < M

   1. Initial program 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in D around 0 57.5%

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

      1. times-frac 56.9%

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

      2. unpow2 56.9%

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

      3. unpow2 56.9%

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

      4. unpow2 56.9%

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

   6. Simplified 56.9%

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

      1. associate-*r/ 57.6%

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

      2. times-frac 66.3%

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

      3. *-commutative 66.3%

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

   8. Applied egg-rr 70.4%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.25 \cdot 10^{-166}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell}}\\ \end{array} \]

Alternative 5: 77.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 5.8 \cdot 10^{-168}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \left(M \cdot \frac{M}{\ell}\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 5.8e-168)
   w0
   (* w0 (+ 1.0 (* -0.125 (* h (* (pow (/ D d) 2.0) (* 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 <= 5.8e-168) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (h * (pow((D / d), 2.0) * (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 <= 5.8d-168) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (h * (((d / d_1) ** 2.0d0) * (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 <= 5.8e-168) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (h * (Math.pow((D / d), 2.0) * (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 <= 5.8e-168:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (h * (math.pow((D / d), 2.0) * (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 <= 5.8e-168)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(h * Float64((Float64(D / d) ^ 2.0) * Float64(M * Float64(M / l)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 5.8e-168)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (h * (((D / d) ^ 2.0) * (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, 5.8e-168], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(h * N[(N[Power[N[(D / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(M * N[(M / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.7999999999999997e-168

   1. Initial program 80.9%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in D around 0 66.8%

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

   if 5.7999999999999997e-168 < M

   1. Initial program 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in D around 0 56.1%

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

      1. *-commutative 56.1%

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

      2. times-frac 53.9%

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

      3. unpow2 53.9%

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

      4. unpow2 53.9%

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

      5. unpow2 53.9%

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

   6. Simplified 53.9%

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

      1. associate-*r/ 57.6%

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

      2. times-frac 66.3%

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

      3. *-commutative 66.3%

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

   8. Applied egg-rr 66.3%

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

   9. Taylor expanded in D around 0 56.1%

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

       1. associate-*r* 58.5%

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

       2. unpow2 58.5%

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

       3. associate-*r* 58.6%

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

       4. associate-/l* 59.6%

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

       5. associate-*r/ 62.1%

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

       6. *-commutative 62.1%

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

       7. associate-*r/ 62.1%

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

       8. associate-*r* 62.1%

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

       9. associate-/r* 62.5%

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

       10. associate-*l/ 62.5%

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

       11. unpow2 62.5%

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

       12. associate-/r/ 62.6%

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

   11. Simplified 72.3%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5.8 \cdot 10^{-168}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \left(M \cdot \frac{M}{\ell}\right)\right)\right)\right)\\ \end{array} \]

Alternative 6: 73.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 2.7 \cdot 10^{+35}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d \cdot d\right)}\right)\right)\\ \end{array} \end{array} \]

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 (<= D 2.7e+35)
   w0
   (* w0 (+ 1.0 (* -0.125 (* h (/ (* (* M D) (* M D)) (* 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 (D <= 2.7e+35) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (h * (((M * D) * (M * D)) / (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 (d <= 2.7d+35) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (h * (((m * d) * (m * d)) / (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 (D <= 2.7e+35) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (h * (((M * D) * (M * D)) / (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 D <= 2.7e+35:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (h * (((M * D) * (M * D)) / (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 (D <= 2.7e+35)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(h * Float64(Float64(Float64(M * D) * Float64(M * D)) / 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 (D <= 2.7e+35)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (h * (((M * D) * (M * D)) / (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[D, 2.7e+35], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(h * N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.70000000000000003e35

   1. Initial program 83.3%

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

   3. Simplified 84.0%

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

   4. Taylor expanded in D around 0 68.0%

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

   if 2.70000000000000003e35 < D

   1. Initial program 70.4%

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

   3. Simplified 70.4%

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

   4. Taylor expanded in D around 0 44.8%

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

      1. *-commutative 44.8%

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

      2. times-frac 43.0%

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

      3. unpow2 43.0%

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

      4. unpow2 43.0%

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

      5. unpow2 43.0%

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

   6. Simplified 43.0%

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

   7. Taylor expanded in D around 0 44.8%

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

      1. associate-*r* 46.8%

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

      2. unpow2 46.8%

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

      3. unpow2 46.8%

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

      4. unpow2 46.8%

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

      5. associate-*r* 46.9%

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

      6. associate-*l/ 46.9%

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

      7. *-commutative 46.9%

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

      8. unswap-sqr 62.3%

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

      9. associate-*r* 62.2%

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

   9. Simplified 62.2%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.7 \cdot 10^{+35}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\ell \cdot \left(d \cdot d\right)}\right)\right)\\ \end{array} \]

Alternative 7: 75.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.16 \cdot 10^{-182}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d \cdot d} \cdot \left(D \cdot \frac{h \cdot M}{\frac{\ell}{M}}\right)\right)\right)\\ \end{array} \end{array} \]

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 1.16e-182)
   w0
   (* w0 (+ 1.0 (* -0.125 (* (/ D (* d d)) (* D (/ (* h 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 <= 1.16e-182) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D / (d * d)) * (D * ((h * 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 <= 1.16d-182) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d / (d_1 * d_1)) * (d * ((h * 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 <= 1.16e-182) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D / (d * d)) * (D * ((h * 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 <= 1.16e-182:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((D / (d * d)) * (D * ((h * 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 <= 1.16e-182)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D / Float64(d * d)) * Float64(D * Float64(Float64(h * 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 <= 1.16e-182)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((D / (d * d)) * (D * ((h * 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, 1.16e-182], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(D * N[(N[(h * M), $MachinePrecision] / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.16e-182

   1. Initial program 80.4%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in D around 0 66.5%

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

   if 1.16e-182 < M

   1. Initial program 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in D around 0 57.0%

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

      1. *-commutative 57.0%

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

      2. times-frac 54.9%

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

      3. unpow2 54.9%

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

      4. unpow2 54.9%

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

      5. unpow2 54.9%

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

   6. Simplified 54.9%

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

   7. Taylor expanded in D around 0 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%

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

      2. unpow2 57.0%

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

      3. unpow2 57.0%

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

      4. associate-*l* 59.5%

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

      5. times-frac 64.2%

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

      6. unpow2 64.2%

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

      7. associate-*r/ 60.8%

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

      8. associate-*r* 64.9%

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

      9. associate-/l* 68.2%

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

      10. *-commutative 68.2%

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

   9. Simplified 68.2%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.16 \cdot 10^{-182}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d \cdot d} \cdot \left(D \cdot \frac{h \cdot M}{\frac{\ell}{M}}\right)\right)\right)\\ \end{array} \]

Alternative 8: 76.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.75 \cdot 10^{-200}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell} \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 2.75e-200)
   w0
   (* w0 (+ 1.0 (* (/ (* (* (/ D d) (/ D d)) (* M (* h M))) l) -0.125)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.75e-200) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (((((D / d) * (D / d)) * (M * (h * M))) / l) * -0.125));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 2.75d-200) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + (((((d / d_1) * (d / d_1)) * (m * (h * m))) / l) * (-0.125d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.75e-200) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (((((D / d) * (D / d)) * (M * (h * M))) / l) * -0.125));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.75e-200:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (((((D / d) * (D / d)) * (M * (h * M))) / l) * -0.125))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.75e-200)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(M * Float64(h * M))) / l) * -0.125)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.75e-200)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (((((D / d) * (D / d)) * (M * (h * M))) / l) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 2.75e-200], w0, N[(w0 * N[(1.0 + N[(N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.7499999999999998e-200

   1. Initial program 80.9%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in D around 0 65.9%

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

   if 2.7499999999999998e-200 < M

   1. Initial program 80.4%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in D around 0 54.4%

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

      1. *-commutative 54.4%

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

      2. times-frac 52.4%

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

      3. unpow2 52.4%

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

      4. unpow2 52.4%

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

      5. unpow2 52.4%

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

   6. Simplified 52.4%

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

      1. associate-*r/ 55.7%

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

      2. times-frac 67.8%

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

      3. *-commutative 67.8%

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

   8. Applied egg-rr 67.8%

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

   9. Taylor expanded in h around 0 67.8%

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

       1. unpow2 67.8%

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

       2. associate-*l* 73.2%

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

   11. Simplified 73.2%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.75 \cdot 10^{-200}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell} \cdot -0.125\right)\\ \end{array} \]

Alternative 9: 71.7% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 6.5 \cdot 10^{+30}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d \cdot \ell}\right) \cdot \left(\left(h \cdot M\right) \cdot \left(w0 \cdot M\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 6.5e+30)
   w0
   (* -0.125 (* (* (/ D d) (/ D (* d l))) (* (* h M) (* w0 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+30) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (D / (d * l))) * ((h * M) * (w0 * 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+30) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d / d_1) * (d / (d_1 * l))) * ((h * m) * (w0 * 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+30) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (D / (d * l))) * ((h * M) * (w0 * 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+30:
		tmp = w0
	else:
		tmp = -0.125 * (((D / d) * (D / (d * l))) * ((h * M) * (w0 * 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+30)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / Float64(d * l))) * Float64(Float64(h * M) * Float64(w0 * 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+30)
		tmp = w0;
	else
		tmp = -0.125 * (((D / d) * (D / (d * l))) * ((h * M) * (w0 * 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+30], w0, N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] * N[(w0 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 6.5e30

   1. Initial program 81.6%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in D around 0 69.6%

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

   if 6.5e30 < M

   1. Initial program 76.5%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in D around 0 42.9%

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

      1. +-commutative 42.9%

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

      2. *-commutative 42.9%

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

      3. fma-def 42.9%

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

      4. associate-*r* 43.0%

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

      5. unpow2 43.0%

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

      6. unpow2 43.0%

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

      7. swap-sqr 60.8%

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

      8. associate-/l* 60.4%

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

      9. swap-sqr 42.5%

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

      10. unpow2 42.5%

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

      11. associate-*l* 42.6%

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

   6. Simplified 42.6%

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

   7. Taylor expanded in D around inf 30.0%

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

      1. unpow2 30.0%

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

      2. associate-*r* 30.2%

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

      3. associate-/l* 30.1%

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

      4. unpow2 30.1%

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

      5. associate-*r* 29.8%

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

      6. associate-*r* 30.0%

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

      7. *-commutative 30.0%

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

      8. unpow2 30.0%

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

      9. associate-*r* 30.1%

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

   9. Simplified 30.1%

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

       1. associate-*r* 30.3%

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

       2. associate-/r/ 30.3%

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

       3. associate-*l* 30.6%

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

   11. Applied egg-rr 30.6%

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

       1. times-frac 33.7%

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

   13. Applied egg-rr 33.7%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.5 \cdot 10^{+30}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d \cdot \ell}\right) \cdot \left(\left(h \cdot M\right) \cdot \left(w0 \cdot M\right)\right)\right)\\ \end{array} \]

Alternative 10: 67.0% 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 80.7%

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

3. Simplified 81.2%

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

4. Taylor expanded in D around 0 64.1%

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

5. Final simplification 64.1%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023282 
(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))))))