Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.6% → 87.3%

Time: 16.3s

Alternatives: 11

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.6% 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: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;2 \cdot d \leq 5 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{1 + -0.25 \cdot \frac{D \cdot h}{\frac{\ell}{D} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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-128)
   (* (sqrt (+ 1.0 (* -0.25 (/ (* D h) (* (/ l D) (* (/ d M) (/ d M))))))) w0)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* (/ M d) (* D 0.5)) 2.0)) l))))))

C

M = abs(M);
D = abs(D);
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-128) {
		tmp = sqrt((1.0 + (-0.25 * ((D * h) / ((l / D) * ((d / M) * (d / M))))))) * w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow(((M / d) * (D * 0.5)), 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: 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-128) then
        tmp = sqrt((1.0d0 + ((-0.25d0) * ((d * h) / ((l / d) * ((d_1 / m) * (d_1 / m))))))) * w0
    else
        tmp = w0 * sqrt((1.0d0 - ((h * (((m / d_1) * (d * 0.5d0)) ** 2.0d0)) / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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-128) {
		tmp = Math.sqrt((1.0 + (-0.25 * ((D * h) / ((l / D) * ((d / M) * (d / M))))))) * w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow(((M / d) * (D * 0.5)), 2.0)) / l)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (2.0 * d) <= 5e-128:
		tmp = math.sqrt((1.0 + (-0.25 * ((D * h) / ((l / D) * ((d / M) * (d / M))))))) * w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow(((M / d) * (D * 0.5)), 2.0)) / l)))
	return tmp

Julia

M = abs(M)
D = abs(D)
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-128)
		tmp = Float64(sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D * h) / Float64(Float64(l / D) * Float64(Float64(d / M) * Float64(d / M))))))) * w0);
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(M / d) * Float64(D * 0.5)) ^ 2.0)) / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
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-128)
		tmp = sqrt((1.0 + (-0.25 * ((D * h) / ((l / D) * ((d / M) * (d / M))))))) * w0;
	else
		tmp = w0 * sqrt((1.0 - ((h * (((M / d) * (D * 0.5)) ^ 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: 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-128], N[(N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D * h), $MachinePrecision] / N[(N[(l / D), $MachinePrecision] * N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 2 d) < 5.0000000000000001e-128

   1. Initial program 80.7%

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

      1. times-frac 78.6%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in w0 around 0 56.5%

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

   6. Simplified 59.6%

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

      1. frac-times 62.0%

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

      2. times-frac 76.9%

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

   8. Applied egg-rr 76.9%

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

   if 5.0000000000000001e-128 < (*.f64 2 d)

   1. Initial program 87.8%

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

      1. times-frac 89.0%

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

   3. Simplified 89.0%

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

      1. associate-*r/ 93.6%

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

      2. frac-times 92.4%

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

      3. *-commutative 92.4%

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

      4. frac-times 92.5%

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

      5. div-inv 92.5%

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

      6. metadata-eval 92.5%

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

   5. Applied egg-rr 92.5%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot d \leq 5 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{1 + -0.25 \cdot \frac{D \cdot h}{\frac{\ell}{D} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 2: 87.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+291}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + -0.25 \cdot \frac{D \cdot h}{\frac{\ell}{D} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}} \cdot w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (- 1.0 (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ h l))) 5e+291)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
   (*
    (sqrt (+ 1.0 (* -0.25 (/ (* D h) (* (/ l D) (* (/ d M) (/ d M)))))))
    w0)))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((1.0 - (pow(((D * M) / (2.0 * d)), 2.0) * (h / l))) <= 5e+291) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = sqrt((1.0 + (-0.25 * ((D * h) / ((l / D) * ((d / M) * (d / M))))))) * w0;
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((1.0d0 - ((((d * m) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))) <= 5d+291) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / d_1) * (d / 2.0d0)) ** 2.0d0))))
    else
        tmp = sqrt((1.0d0 + ((-0.25d0) * ((d * h) / ((l / d) * ((d_1 / m) * (d_1 / m))))))) * w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((1.0 - (Math.pow(((D * M) / (2.0 * d)), 2.0) * (h / l))) <= 5e+291) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = Math.sqrt((1.0 + (-0.25 * ((D * h) / ((l / D) * ((d / M) * (d / M))))))) * w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (1.0 - (math.pow(((D * M) / (2.0 * d)), 2.0) * (h / l))) <= 5e+291:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	else:
		tmp = math.sqrt((1.0 + (-0.25 * ((D * h) / ((l / D) * ((d / M) * (d / M))))))) * w0
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(1.0 - Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))) <= 5e+291)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0)))));
	else
		tmp = Float64(sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D * h) / Float64(Float64(l / D) * Float64(Float64(d / M) * Float64(d / M))))))) * w0);
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((1.0 - ((((D * M) / (2.0 * d)) ^ 2.0) * (h / l))) <= 5e+291)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	else
		tmp = sqrt((1.0 + (-0.25 * ((D * h) / ((l / D) * ((d / M) * (d / M))))))) * 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: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(1.0 - N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+291], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D * h), $MachinePrecision] / N[(N[(l / D), $MachinePrecision] * N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 5.0000000000000001e291

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. times-frac 98.8%

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

   3. Simplified 98.8%

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

   if 5.0000000000000001e291 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 48.9%

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

      1. times-frac 50.1%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in w0 around 0 51.4%

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

   6. Simplified 52.2%

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

      1. frac-times 53.2%

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

      2. times-frac 61.9%

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

   8. Applied egg-rr 61.9%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+291}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + -0.25 \cdot \frac{D \cdot h}{\frac{\ell}{D} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}} \cdot w0\\ \end{array} \]

Alternative 3: 84.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{-132}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}\right)}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+23}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(D \cdot \left(M \cdot \frac{D \cdot h}{d \cdot \frac{d}{M}}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{d \cdot d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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 2e-132)
   (* w0 (sqrt (+ 1.0 (* -0.25 (* (/ D (/ l D)) (/ h (* (/ d M) (/ d M))))))))
   (if (<= d 5e+23)
     (* w0 (sqrt (- 1.0 (/ (* 0.25 (* D (* M (/ (* D h) (* d (/ d M)))))) l))))
     (*
      w0
      (sqrt (- 1.0 (* 0.25 (* D (* (/ D (* d d)) (/ (* h (* M M)) l))))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 2e-132) {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / (l / D)) * (h / ((d / M) * (d / M)))))));
	} else if (d <= 5e+23) {
		tmp = w0 * sqrt((1.0 - ((0.25 * (D * (M * ((D * h) / (d * (d / M)))))) / l)));
	} 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: 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 <= 2d-132) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((d / (l / d)) * (h / ((d_1 / m) * (d_1 / m)))))))
    else if (d_1 <= 5d+23) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * (d * (m * ((d * h) / (d_1 * (d_1 / m)))))) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d * ((d / (d_1 * d_1)) * ((h * (m * m)) / l))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 2e-132) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((D / (l / D)) * (h / ((d / M) * (d / M)))))));
	} else if (d <= 5e+23) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * (D * (M * ((D * h) / (d * (d / M)))))) / l)));
	} 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)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 2e-132:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((D / (l / D)) * (h / ((d / M) * (d / M)))))))
	elif d <= 5e+23:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * (D * (M * ((D * h) / (d * (d / M)))))) / l)))
	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)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 2e-132)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D / Float64(l / D)) * Float64(h / Float64(Float64(d / M) * Float64(d / M))))))));
	elseif (d <= 5e+23)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(D * Float64(M * Float64(Float64(D * h) / Float64(d * Float64(d / M)))))) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D * Float64(Float64(D / Float64(d * d)) * Float64(Float64(h * Float64(M * M)) / l)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 2e-132)
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / (l / D)) * (h / ((d / M) * (d / M)))))));
	elseif (d <= 5e+23)
		tmp = w0 * sqrt((1.0 - ((0.25 * (D * (M * ((D * h) / (d * (d / M)))))) / l)));
	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: 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, 2e-132], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(h / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5e+23], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(D * N[(M * N[(N[(D * h), $MachinePrecision] / N[(d * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D * N[(N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < 2e-132

   1. Initial program 80.5%

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

      1. times-frac 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in w0 around 0 56.6%

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

   6. Simplified 59.1%

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

   7. Taylor expanded in d around 0 59.1%

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

      1. unpow2 59.1%

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

      2. unpow2 59.1%

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

      3. times-frac 71.7%

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

   9. Simplified 71.7%

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

   if 2e-132 < d < 4.9999999999999999e23

   1. Initial program 86.5%

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

      1. times-frac 86.5%

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

   3. Simplified 86.5%

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

      1. associate-*r/ 88.7%

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

      2. frac-times 88.7%

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

      3. *-commutative 88.7%

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

      4. frac-times 85.5%

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

      5. div-inv 85.5%

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

      6. metadata-eval 85.5%

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

   5. Applied egg-rr 85.5%

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

   6. Taylor expanded in M around 0 78.4%

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

      1. unpow2 78.4%

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

      2. associate-*r* 78.4%

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

      3. unpow2 78.4%

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

      4. times-frac 78.3%

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

      5. associate-*r/ 74.9%

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

      6. associate-*r* 71.4%

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

      7. unpow2 71.4%

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

      8. associate-*l* 73.6%

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

      9. *-commutative 73.6%

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

      10. associate-/l* 73.6%

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

      11. associate-*r/ 76.9%

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

      12. associate-/r/ 76.9%

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

      13. associate-/r* 70.3%

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

      14. associate-*r/ 72.8%

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

      15. associate-*l/ 72.8%

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

      16. associate-/r/ 83.2%

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

   8. Simplified 83.2%

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

   if 4.9999999999999999e23 < d

   1. Initial program 89.0%

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

      1. times-frac 90.8%

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

   3. Simplified 90.8%

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

      1. associate-*r/ 96.5%

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

      2. frac-times 94.7%

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

      3. *-commutative 94.7%

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

      4. frac-times 96.5%

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

      5. div-inv 96.5%

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

      6. metadata-eval 96.5%

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

   5. Applied egg-rr 96.5%

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

   6. Taylor expanded in M around 0 65.3%

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

      1. *-commutative 65.3%

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

   8. Simplified 85.4%

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

   9. Taylor expanded in D around 0 83.9%

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

       1. *-commutative 83.9%

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

       2. times-frac 83.5%

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

       3. unpow2 83.5%

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

       4. *-commutative 83.5%

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

       5. unpow2 83.5%

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

   11. Simplified 83.5%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{-132}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}\right)}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+23}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(D \cdot \left(M \cdot \frac{D \cdot h}{d \cdot \frac{d}{M}}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{d \cdot d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)}\\ \end{array} \]

Alternative 4: 84.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 3.1 \cdot 10^{-132}:\\ \;\;\;\;\sqrt{1 + -0.25 \cdot \frac{D \cdot h}{\frac{\ell}{D} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}} \cdot w0\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+23}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(D \cdot \left(M \cdot \frac{D \cdot h}{d \cdot \frac{d}{M}}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{d \cdot d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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 3.1e-132)
   (* (sqrt (+ 1.0 (* -0.25 (/ (* D h) (* (/ l D) (* (/ d M) (/ d M))))))) w0)
   (if (<= d 5e+23)
     (* w0 (sqrt (- 1.0 (/ (* 0.25 (* D (* M (/ (* D h) (* d (/ d M)))))) l))))
     (*
      w0
      (sqrt (- 1.0 (* 0.25 (* D (* (/ D (* d d)) (/ (* h (* M M)) l))))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 3.1e-132) {
		tmp = sqrt((1.0 + (-0.25 * ((D * h) / ((l / D) * ((d / M) * (d / M))))))) * w0;
	} else if (d <= 5e+23) {
		tmp = w0 * sqrt((1.0 - ((0.25 * (D * (M * ((D * h) / (d * (d / M)))))) / l)));
	} 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: 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 <= 3.1d-132) then
        tmp = sqrt((1.0d0 + ((-0.25d0) * ((d * h) / ((l / d) * ((d_1 / m) * (d_1 / m))))))) * w0
    else if (d_1 <= 5d+23) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * (d * (m * ((d * h) / (d_1 * (d_1 / m)))))) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d * ((d / (d_1 * d_1)) * ((h * (m * m)) / l))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 3.1e-132) {
		tmp = Math.sqrt((1.0 + (-0.25 * ((D * h) / ((l / D) * ((d / M) * (d / M))))))) * w0;
	} else if (d <= 5e+23) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * (D * (M * ((D * h) / (d * (d / M)))))) / l)));
	} 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)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 3.1e-132:
		tmp = math.sqrt((1.0 + (-0.25 * ((D * h) / ((l / D) * ((d / M) * (d / M))))))) * w0
	elif d <= 5e+23:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * (D * (M * ((D * h) / (d * (d / M)))))) / l)))
	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)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 3.1e-132)
		tmp = Float64(sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D * h) / Float64(Float64(l / D) * Float64(Float64(d / M) * Float64(d / M))))))) * w0);
	elseif (d <= 5e+23)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(D * Float64(M * Float64(Float64(D * h) / Float64(d * Float64(d / M)))))) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D * Float64(Float64(D / Float64(d * d)) * Float64(Float64(h * Float64(M * M)) / l)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 3.1e-132)
		tmp = sqrt((1.0 + (-0.25 * ((D * h) / ((l / D) * ((d / M) * (d / M))))))) * w0;
	elseif (d <= 5e+23)
		tmp = w0 * sqrt((1.0 - ((0.25 * (D * (M * ((D * h) / (d * (d / M)))))) / l)));
	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: 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, 3.1e-132], N[(N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D * h), $MachinePrecision] / N[(N[(l / D), $MachinePrecision] * N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * w0), $MachinePrecision], If[LessEqual[d, 5e+23], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(D * N[(M * N[(N[(D * h), $MachinePrecision] / N[(d * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D * N[(N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < 3.10000000000000008e-132

   1. Initial program 80.5%

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

      1. times-frac 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in w0 around 0 56.6%

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

   6. Simplified 59.1%

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

      1. frac-times 61.5%

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

      2. times-frac 76.6%

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

   8. Applied egg-rr 76.6%

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

   if 3.10000000000000008e-132 < d < 4.9999999999999999e23

   1. Initial program 86.5%

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

      1. times-frac 86.5%

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

   3. Simplified 86.5%

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

      1. associate-*r/ 88.7%

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

      2. frac-times 88.7%

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

      3. *-commutative 88.7%

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

      4. frac-times 85.5%

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

      5. div-inv 85.5%

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

      6. metadata-eval 85.5%

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

   5. Applied egg-rr 85.5%

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

   6. Taylor expanded in M around 0 78.4%

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

      1. unpow2 78.4%

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

      2. associate-*r* 78.4%

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

      3. unpow2 78.4%

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

      4. times-frac 78.3%

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

      5. associate-*r/ 74.9%

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

      6. associate-*r* 71.4%

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

      7. unpow2 71.4%

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

      8. associate-*l* 73.6%

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

      9. *-commutative 73.6%

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

      10. associate-/l* 73.6%

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

      11. associate-*r/ 76.9%

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

      12. associate-/r/ 76.9%

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

      13. associate-/r* 70.3%

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

      14. associate-*r/ 72.8%

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

      15. associate-*l/ 72.8%

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

      16. associate-/r/ 83.2%

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

   8. Simplified 83.2%

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

   if 4.9999999999999999e23 < d

   1. Initial program 89.0%

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

      1. times-frac 90.8%

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

   3. Simplified 90.8%

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

      1. associate-*r/ 96.5%

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

      2. frac-times 94.7%

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

      3. *-commutative 94.7%

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

      4. frac-times 96.5%

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

      5. div-inv 96.5%

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

      6. metadata-eval 96.5%

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

   5. Applied egg-rr 96.5%

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

   6. Taylor expanded in M around 0 65.3%

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

      1. *-commutative 65.3%

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

   8. Simplified 85.4%

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

   9. Taylor expanded in D around 0 83.9%

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

       1. *-commutative 83.9%

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

       2. times-frac 83.5%

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

       3. unpow2 83.5%

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

       4. *-commutative 83.5%

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

       5. unpow2 83.5%

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

   11. Simplified 83.5%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 3.1 \cdot 10^{-132}:\\ \;\;\;\;\sqrt{1 + -0.25 \cdot \frac{D \cdot h}{\frac{\ell}{D} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}} \cdot w0\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+23}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(D \cdot \left(M \cdot \frac{D \cdot h}{d \cdot \frac{d}{M}}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{d \cdot d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)}\\ \end{array} \]

Alternative 5: 81.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4.2 \cdot 10^{-67}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{D \cdot h}{\frac{\ell}{D} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{d \cdot d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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 4.2e-67)
   (* w0 (fma -0.125 (/ (* D h) (* (/ l D) (* (/ d M) (/ d M)))) 1.0))
   (* w0 (sqrt (- 1.0 (* 0.25 (* D (* (/ D (* d d)) (/ (* h (* M M)) l)))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 4.2e-67) {
		tmp = w0 * fma(-0.125, ((D * h) / ((l / D) * ((d / M) * (d / M)))), 1.0);
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D / (d * d)) * ((h * (M * M)) / l))))));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 4.2e-67)
		tmp = Float64(w0 * fma(-0.125, Float64(Float64(D * h) / Float64(Float64(l / D) * Float64(Float64(d / M) * Float64(d / M)))), 1.0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D * Float64(Float64(D / Float64(d * d)) * Float64(Float64(h * Float64(M * M)) / l)))))));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D 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, 4.2e-67], N[(w0 * N[(-0.125 * N[(N[(D * h), $MachinePrecision] / N[(N[(l / D), $MachinePrecision] * N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D * N[(N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.2000000000000003e-67

   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 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in M around 0 54.6%

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

      1. +-commutative 54.6%

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

      2. *-commutative 54.6%

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

      3. *-commutative 54.6%

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

      4. fma-def 54.6%

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

      5. *-commutative 54.6%

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

      6. times-frac 54.6%

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

      7. unpow2 54.6%

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

      8. associate-/l* 54.8%

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

      9. associate-/l* 57.0%

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

      10. unpow2 57.0%

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

      11. unpow2 57.0%

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

   6. Simplified 57.0%

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

      1. frac-times 61.0%

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

      2. times-frac 75.1%

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

   8. Applied egg-rr 69.0%

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

   if 4.2000000000000003e-67 < d

   1. Initial program 88.7%

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

      1. times-frac 90.1%

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

   3. Simplified 90.1%

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

      1. associate-*r/ 95.4%

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

      2. frac-times 94.0%

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

      3. *-commutative 94.0%

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

      4. frac-times 95.4%

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

      5. div-inv 95.4%

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

      6. metadata-eval 95.4%

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

   5. Applied egg-rr 95.4%

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

   6. Taylor expanded in M around 0 69.9%

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

      1. *-commutative 69.9%

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

   8. Simplified 85.9%

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

   9. Taylor expanded in D around 0 84.8%

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

       1. *-commutative 84.8%

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

       2. times-frac 83.2%

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

       3. unpow2 83.2%

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

       4. *-commutative 83.2%

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

       5. unpow2 83.2%

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

   11. Simplified 83.2%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.2 \cdot 10^{-67}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{D \cdot h}{\frac{\ell}{D} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{d \cdot d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)}\\ \end{array} \]

Alternative 6: 84.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 6 \cdot 10^{-67}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \frac{D \cdot h}{\ell \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{d \cdot d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D 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 6e-67)
   (* w0 (sqrt (- 1.0 (* 0.25 (* D (/ (* D h) (* l (* (/ d M) (/ d M)))))))))
   (* w0 (sqrt (- 1.0 (* 0.25 (* D (* (/ D (* d d)) (/ (* h (* M M)) l)))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 6e-67) {
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D * h) / (l * ((d / M) * (d / M))))))));
	} 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: 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 <= 6d-67) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d * ((d * h) / (l * ((d_1 / m) * (d_1 / m))))))))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (d * ((d / (d_1 * d_1)) * ((h * (m * m)) / l))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 6e-67) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (D * ((D * h) / (l * ((d / M) * (d / M))))))));
	} 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)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 6e-67:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (D * ((D * h) / (l * ((d / M) * (d / M))))))))
	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)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 6e-67)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D * Float64(Float64(D * h) / Float64(l * Float64(Float64(d / M) * Float64(d / M)))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(D * Float64(Float64(D / Float64(d * d)) * Float64(Float64(h * Float64(M * M)) / l)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 6e-67)
		tmp = w0 * sqrt((1.0 - (0.25 * (D * ((D * h) / (l * ((d / M) * (d / M))))))));
	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: 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, 6e-67], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D * N[(N[(D * h), $MachinePrecision] / N[(l * N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(D * N[(N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 6.00000000000000065e-67

   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 78.8%

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

   3. Simplified 78.8%

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

      1. associate-*r/ 82.1%

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

      2. frac-times 83.1%

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

      3. *-commutative 83.1%

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

      4. frac-times 82.6%

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

      5. div-inv 82.6%

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

      6. metadata-eval 82.6%

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

   5. Applied egg-rr 82.6%

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

   6. Taylor expanded in M around 0 55.9%

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

      1. *-commutative 55.9%

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

   8. Simplified 74.0%

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

   if 6.00000000000000065e-67 < d

   1. Initial program 88.7%

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

      1. times-frac 90.1%

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

   3. Simplified 90.1%

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

      1. associate-*r/ 95.4%

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

      2. frac-times 94.0%

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

      3. *-commutative 94.0%

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

      4. frac-times 95.4%

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

      5. div-inv 95.4%

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

      6. metadata-eval 95.4%

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

   5. Applied egg-rr 95.4%

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

   6. Taylor expanded in M around 0 69.9%

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

      1. *-commutative 69.9%

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

   8. Simplified 85.9%

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

   9. Taylor expanded in D around 0 84.8%

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

       1. *-commutative 84.8%

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

       2. times-frac 83.2%

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

       3. unpow2 83.2%

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

       4. *-commutative 83.2%

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

       5. unpow2 83.2%

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

   11. Simplified 83.2%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6 \cdot 10^{-67}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \frac{D \cdot h}{\ell \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(D \cdot \left(\frac{D}{d \cdot d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)}\\ \end{array} \]

Alternative 7: 80.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 4.3 \cdot 10^{+51}:\\ \;\;\;\;w0 + w0 \cdot \left(\left(D \cdot -0.125\right) \cdot \left(\frac{D}{\ell} \cdot \frac{h}{{\left(\frac{d}{M}\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, D \cdot \frac{D \cdot \frac{w0 \cdot \left(h \cdot \frac{M}{d}\right)}{\frac{d}{M}}}{\ell}, w0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 4.3e+51)
   (+ w0 (* w0 (* (* D -0.125) (* (/ D l) (/ h (pow (/ d M) 2.0))))))
   (fma -0.125 (* D (/ (* D (/ (* w0 (* h (/ M d))) (/ d M))) l)) w0)))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 4.3e+51) {
		tmp = w0 + (w0 * ((D * -0.125) * ((D / l) * (h / pow((d / M), 2.0)))));
	} else {
		tmp = fma(-0.125, (D * ((D * ((w0 * (h * (M / d))) / (d / M))) / l)), w0);
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 4.3e+51)
		tmp = Float64(w0 + Float64(w0 * Float64(Float64(D * -0.125) * Float64(Float64(D / l) * Float64(h / (Float64(d / M) ^ 2.0))))));
	else
		tmp = fma(-0.125, Float64(D * Float64(Float64(D * Float64(Float64(w0 * Float64(h * Float64(M / d))) / Float64(d / M))) / l)), w0);
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D 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, 4.3e+51], N[(w0 + N[(w0 * N[(N[(D * -0.125), $MachinePrecision] * N[(N[(D / l), $MachinePrecision] * N[(h / N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.125 * N[(D * N[(N[(D * N[(N[(w0 * N[(h * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 4.2999999999999997e51

   1. Initial program 83.8%

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

      1. times-frac 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in M around 0 60.9%

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

      1. associate-*r/ 60.9%

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

      2. *-commutative 60.9%

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

      3. associate-*r/ 60.9%

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

      4. times-frac 60.1%

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

      5. unpow2 60.1%

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

      6. *-commutative 60.1%

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

      7. unpow2 60.1%

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

      8. unpow2 60.1%

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

   6. Simplified 60.1%

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

   7. Taylor expanded in D around 0 60.9%

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

      1. *-commutative 60.9%

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

      2. times-frac 60.1%

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

      3. unpow2 60.1%

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

      4. *-commutative 60.1%

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

      5. unpow2 60.1%

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

      6. unpow2 60.1%

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

      7. associate-/l* 61.7%

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

      8. unpow2 61.7%

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

      9. associate-/l* 63.6%

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

      10. unpow2 63.6%

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

      11. times-frac 71.6%

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

      12. associate-*l/ 74.7%

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

      13. associate-/r/ 76.7%

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

      14. *-commutative 76.7%

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

      15. associate-*r/ 76.4%

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

      16. associate-/l/ 75.5%

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

   9. Simplified 75.5%

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

       1. distribute-rgt-in 75.5%

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

       2. *-un-lft-identity 75.5%

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

       3. associate-*r* 75.5%

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

       4. times-frac 74.7%

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

       5. pow2 74.7%

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

   11. Applied egg-rr 74.7%

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

   if 4.2999999999999997e51 < D

   1. Initial program 78.8%

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

      1. times-frac 74.2%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in M around 0 50.0%

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

      1. fma-def 50.0%

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

      2. times-frac 44.7%

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

      3. unpow2 44.7%

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

      4. associate-/l* 44.7%

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

      5. *-commutative 44.7%

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

      6. associate-*r* 44.5%

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

      7. unpow2 44.5%

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

      8. times-frac 44.6%

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

      9. unpow2 44.6%

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

      10. associate-*r* 47.1%

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

      11. *-commutative 47.1%

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

   6. Simplified 47.1%

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

      1. associate-*l/ 59.7%

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

      2. associate-/l* 62.3%

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

   8. Applied egg-rr 62.3%

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

   9. Taylor expanded in M around 0 57.2%

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

       1. *-commutative 57.2%

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

       2. associate-*r* 57.1%

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

       3. unpow2 57.1%

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

       4. times-frac 57.2%

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

       5. unpow2 57.2%

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

       6. associate-*r* 59.7%

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

       7. *-commutative 59.7%

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

       8. associate-/l* 62.3%

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

       9. times-frac 62.3%

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

       10. *-commutative 62.3%

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

       11. times-frac 62.3%

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

       12. *-commutative 62.3%

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

       13. associate-/l* 62.3%

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

       14. *-rgt-identity 62.3%

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

       15. associate-*r/ 62.3%

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

       16. associate-/r/ 62.3%

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

       17. associate-*l/ 62.3%

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

       18. *-lft-identity 62.3%

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

   11. Simplified 62.3%

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

       1. associate-/r/ 67.7%

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

       2. associate-*l/ 67.7%

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

   13. Applied egg-rr 67.7%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 4.3 \cdot 10^{+51}:\\ \;\;\;\;w0 + w0 \cdot \left(\left(D \cdot -0.125\right) \cdot \left(\frac{D}{\ell} \cdot \frac{h}{{\left(\frac{d}{M}\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, D \cdot \frac{D \cdot \frac{w0 \cdot \left(h \cdot \frac{M}{d}\right)}{\frac{d}{M}}}{\ell}, w0\right)\\ \end{array} \]

Alternative 8: 78.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 8.6 \cdot 10^{+108}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{D \cdot h}{\ell \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 8.6e+108)
   (* w0 (+ 1.0 (* (* D (/ (* D h) (* l (* (/ d M) (/ d M))))) -0.125)))
   w0))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 8.6e+108) {
		tmp = w0 * (1.0 + ((D * ((D * h) / (l * ((d / M) * (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: 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 <= 8.6d+108) then
        tmp = w0 * (1.0d0 + ((d * ((d * h) / (l * ((d_1 / m) * (d_1 / m))))) * (-0.125d0)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 8.6e+108) {
		tmp = w0 * (1.0 + ((D * ((D * h) / (l * ((d / M) * (d / M))))) * -0.125));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 8.6e+108:
		tmp = w0 * (1.0 + ((D * ((D * h) / (l * ((d / M) * (d / M))))) * -0.125))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 8.6e+108)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D * Float64(Float64(D * h) / Float64(l * Float64(Float64(d / M) * Float64(d / M))))) * -0.125)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 8.6e+108)
		tmp = w0 * (1.0 + ((D * ((D * h) / (l * ((d / M) * (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: 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, 8.6e+108], N[(w0 * N[(1.0 + N[(N[(D * N[(N[(D * h), $MachinePrecision] / N[(l * N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 8.59999999999999993e108

   1. Initial program 81.9%

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

      1. times-frac 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in M around 0 57.7%

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

      1. associate-*r/ 57.7%

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

      2. *-commutative 57.7%

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

      3. associate-*r/ 57.7%

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

      4. times-frac 57.3%

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

      5. unpow2 57.3%

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

      6. *-commutative 57.3%

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

      7. unpow2 57.3%

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

      8. unpow2 57.3%

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

   6. Simplified 57.3%

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

   7. Taylor expanded in D around 0 57.7%

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

      1. *-commutative 57.7%

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

      2. times-frac 57.3%

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

      3. unpow2 57.3%

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

      4. *-commutative 57.3%

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

      5. unpow2 57.3%

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

      6. unpow2 57.3%

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

      7. associate-/l* 57.5%

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

      8. unpow2 57.5%

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

      9. associate-/l* 59.4%

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

      10. unpow2 59.4%

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

      11. times-frac 67.3%

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

      12. associate-*l/ 70.7%

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

      13. associate-/r/ 72.8%

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

      14. *-commutative 72.8%

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

      15. associate-*r/ 72.5%

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

      16. associate-/l/ 71.6%

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

   9. Simplified 71.6%

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

   if 8.59999999999999993e108 < d

   1. Initial program 89.2%

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

      1. times-frac 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in M around 0 90.1%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 8.6 \cdot 10^{+108}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot \frac{D \cdot h}{\ell \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 9: 70.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.65 \cdot 10^{-28}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot \left(D \cdot w0\right)}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 2.65e-28)
   w0
   (* -0.125 (* (/ (* D (* D w0)) l) (/ (* h (* M M)) (* d d))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.65e-28) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * (D * w0)) / l) * ((h * (M * M)) / (d * d)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 <= 2.65d-28) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d * (d * w0)) / l) * ((h * (m * m)) / (d_1 * d_1)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 2.65e-28) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * (D * w0)) / l) * ((h * (M * M)) / (d * d)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.65e-28:
		tmp = w0
	else:
		tmp = -0.125 * (((D * (D * w0)) / l) * ((h * (M * M)) / (d * d)))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.65e-28)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * Float64(D * w0)) / l) * Float64(Float64(h * Float64(M * M)) / Float64(d * d))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.65e-28)
		tmp = w0;
	else
		tmp = -0.125 * (((D * (D * w0)) / l) * ((h * (M * M)) / (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: 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, 2.65e-28], w0, N[(-0.125 * N[(N[(N[(D * N[(D * w0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.64999999999999994e-28

   1. Initial program 84.3%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 68.9%

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

   if 2.64999999999999994e-28 < M

   1. Initial program 79.2%

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

      1. times-frac 80.7%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in M around 0 52.5%

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

      1. associate-*r/ 52.5%

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

      2. *-commutative 52.5%

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

      3. associate-*r/ 52.5%

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

      4. times-frac 48.2%

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

      5. unpow2 48.2%

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

      6. *-commutative 48.2%

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

      7. unpow2 48.2%

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

      8. unpow2 48.2%

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

   6. Simplified 48.2%

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

   7. Taylor expanded in D around 0 52.5%

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

      1. *-commutative 52.5%

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

      2. times-frac 48.2%

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

      3. unpow2 48.2%

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

      4. *-commutative 48.2%

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

      5. unpow2 48.2%

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

      6. unpow2 48.2%

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

      7. associate-/l* 48.9%

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

      8. unpow2 48.9%

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

      9. associate-/l* 53.5%

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

      10. unpow2 53.5%

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

      11. times-frac 61.3%

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

      12. associate-*l/ 66.3%

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

      13. associate-/r/ 66.6%

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

      14. *-commutative 66.6%

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

      15. associate-*r/ 69.1%

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

      16. associate-/l/ 66.1%

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

   9. Simplified 66.1%

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

   10. Taylor expanded in D around inf 23.7%

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

       1. associate-*r* 23.8%

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

       2. unpow2 23.8%

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

       3. *-commutative 23.8%

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

       4. unpow2 23.8%

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

       5. *-commutative 23.8%

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

       6. unpow2 23.8%

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

   12. Simplified 23.8%

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

       1. times-frac 23.7%

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

       2. associate-*l* 24.2%

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

       3. *-commutative 24.2%

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

   14. Applied egg-rr 24.2%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.65 \cdot 10^{-28}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot \left(D \cdot w0\right)}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right)\\ \end{array} \]

Alternative 10: 69.9% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 8.5 \cdot 10^{-31}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{D \cdot \left(\left(D \cdot w0\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 8.5e-31)
   w0
   (* -0.125 (/ (* D (* (* D w0) (* h (* M M)))) (* l (* d d))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 8.5e-31) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * ((D * w0) * (h * (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: 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 <= 8.5d-31) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((d * ((d * w0) * (h * (m * m)))) / (l * (d_1 * d_1)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 <= 8.5e-31) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * ((D * w0) * (h * (M * M)))) / (l * (d * d)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 8.5e-31:
		tmp = w0
	else:
		tmp = -0.125 * ((D * ((D * w0) * (h * (M * M)))) / (l * (d * d)))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 8.5e-31)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(D * Float64(Float64(D * w0) * Float64(h * Float64(M * M)))) / Float64(l * Float64(d * d))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 8.5e-31)
		tmp = w0;
	else
		tmp = -0.125 * ((D * ((D * w0) * (h * (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: 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, 8.5e-31], w0, N[(-0.125 * N[(N[(D * N[(N[(D * w0), $MachinePrecision] * N[(h * 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 < 8.5000000000000007e-31

   1. Initial program 84.3%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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 68.9%

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

   if 8.5000000000000007e-31 < M

   1. Initial program 79.2%

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

      1. times-frac 80.7%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in M around 0 52.5%

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

      1. associate-*r/ 52.5%

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

      2. *-commutative 52.5%

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

      3. associate-*r/ 52.5%

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

      4. times-frac 48.2%

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

      5. unpow2 48.2%

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

      6. *-commutative 48.2%

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

      7. unpow2 48.2%

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

      8. unpow2 48.2%

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

   6. Simplified 48.2%

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

   7. Taylor expanded in D around 0 52.5%

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

      1. *-commutative 52.5%

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

      2. times-frac 48.2%

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

      3. unpow2 48.2%

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

      4. *-commutative 48.2%

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

      5. unpow2 48.2%

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

      6. unpow2 48.2%

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

      7. associate-/l* 48.9%

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

      8. unpow2 48.9%

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

      9. associate-/l* 53.5%

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

      10. unpow2 53.5%

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

      11. times-frac 61.3%

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

      12. associate-*l/ 66.3%

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

      13. associate-/r/ 66.6%

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

      14. *-commutative 66.6%

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

      15. associate-*r/ 69.1%

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

      16. associate-/l/ 66.1%

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

   9. Simplified 66.1%

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

   10. Taylor expanded in D around inf 23.7%

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

       1. associate-*r* 23.8%

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

       2. unpow2 23.8%

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

       3. *-commutative 23.8%

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

       4. unpow2 23.8%

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

       5. *-commutative 23.8%

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

       6. unpow2 23.8%

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

   12. Simplified 23.8%

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

   13. Taylor expanded in D around 0 23.7%

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

       1. *-commutative 23.7%

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

       2. unpow2 23.7%

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

       3. associate-*r* 24.6%

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

       4. associate-*r* 24.7%

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

       5. unpow2 24.7%

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

   15. Simplified 24.7%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 8.5 \cdot 10^{-31}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{D \cdot \left(\left(D \cdot w0\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \end{array} \]

Alternative 11: 67.7% accurate, 216.0× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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);
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: 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);
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)
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)
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)
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: 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 83.0%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
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 64.7%

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

5. Final simplification 64.7%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023200 
(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))))))