Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.9% → 88.2%

Time: 15.3s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+120}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(h \cdot {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}\right) \cdot \frac{-1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{\ell} \cdot \left(D \cdot h\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 D) (* 2.0 d)) 5e+120)
   (* w0 (sqrt (+ 1.0 (* (* h (pow (* 0.5 (* D (/ M d))) 2.0)) (/ -1.0 l)))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (* (* (/ M d) (/ M d)) (* (/ D l) (* D h)))))))))

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 * D) / (2.0 * d)) <= 5e+120) {
		tmp = w0 * sqrt((1.0 + ((h * pow((0.5 * (D * (M / d))), 2.0)) * (-1.0 / l))));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * (((M / d) * (M / d)) * ((D / l) * (D * h))))));
	}
	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 * d) / (2.0d0 * d_1)) <= 5d+120) then
        tmp = w0 * sqrt((1.0d0 + ((h * ((0.5d0 * (d * (m / d_1))) ** 2.0d0)) * ((-1.0d0) / l))))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (((m / d_1) * (m / d_1)) * ((d / l) * (d * h))))))
    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 * D) / (2.0 * d)) <= 5e+120) {
		tmp = w0 * Math.sqrt((1.0 + ((h * Math.pow((0.5 * (D * (M / d))), 2.0)) * (-1.0 / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (((M / d) * (M / d)) * ((D / l) * (D * h))))));
	}
	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 * D) / (2.0 * d)) <= 5e+120:
		tmp = w0 * math.sqrt((1.0 + ((h * math.pow((0.5 * (D * (M / d))), 2.0)) * (-1.0 / l))))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (((M / d) * (M / d)) * ((D / l) * (D * h))))))
	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(Float64(M * D) / Float64(2.0 * d)) <= 5e+120)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(h * (Float64(0.5 * Float64(D * Float64(M / d))) ^ 2.0)) * Float64(-1.0 / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(Float64(M / d) * Float64(M / d)) * Float64(Float64(D / l) * Float64(D * h)))))));
	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 * D) / (2.0 * d)) <= 5e+120)
		tmp = w0 * sqrt((1.0 + ((h * ((0.5 * (D * (M / d))) ^ 2.0)) * (-1.0 / l))));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * (((M / d) * (M / d)) * ((D / l) * (D * h))))));
	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[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 5e+120], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(h * N[Power[N[(0.5 * N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(N[(D / l), $MachinePrecision] * N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 5.00000000000000019e120

   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 83.9%

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

   3. Simplified 83.9%

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

      1. associate-*r/ 90.5%

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

      2. clear-num 90.5%

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

      3. div-inv 90.5%

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

      4. metadata-eval 90.5%

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

   5. Applied egg-rr 90.5%

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

      1. associate-/r/ 90.5%

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

      2. *-commutative 90.5%

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

      3. *-commutative 90.5%

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

      4. associate-*r* 90.5%

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

      5. associate-*r/ 90.9%

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

      6. associate-*l/ 90.1%

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

      7. *-commutative 90.1%

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

   7. Simplified 90.1%

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

   if 5.00000000000000019e120 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 49.4%

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

      1. times-frac 49.4%

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

   3. Simplified 49.4%

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

   4. Taylor expanded in w0 around 0 26.8%

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

      1. cancel-sign-sub-inv 26.8%

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

      2. *-commutative 26.8%

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

      3. cancel-sign-sub-inv 26.8%

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

   6. Simplified 21.8%

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

      1. *-un-lft-identity 21.8%

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

      2. *-commutative 21.8%

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

      3. *-commutative 21.8%

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

      4. times-frac 30.0%

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

      5. associate-/l* 33.4%

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

      6. *-commutative 33.4%

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

   8. Applied egg-rr 33.4%

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

      1. *-lft-identity 33.4%

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

      2. *-commutative 33.4%

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

      3. associate-/r/ 33.9%

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

      4. *-commutative 33.9%

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

   10. Simplified 33.9%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+120}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(h \cdot {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}\right) \cdot \frac{-1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{\ell} \cdot \left(D \cdot h\right)\right)\right)}\\ \end{array} \]

Alternative 2: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{+288}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \left(D \cdot \left(D \cdot h\right)\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))) 2e+288)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (/ (* (* (/ M d) (/ M d)) (* D (* D h))) 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 ((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= 2e+288) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((((M / d) * (M / d)) * (D * (D * h))) / 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 ((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))) <= 2d+288) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / d_1) * (d / 2.0d0)) ** 2.0d0))))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((((m / d_1) * (m / d_1)) * (d * (d * h))) / 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 ((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= 2e+288) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((((M / d) * (M / d)) * (D * (D * h))) / 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 (1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= 2e+288:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((((M / d) * (M / d)) * (D * (D * h))) / 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(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))) <= 2e+288)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(Float64(Float64(M / d) * Float64(M / d)) * Float64(D * Float64(D * h))) / 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 ((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))) <= 2e+288)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * ((((M / d) * (M / d)) * (D * (D * h))) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+288], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 2e288

   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 97.7%

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

   3. Simplified 97.7%

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

   if 2e288 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 34.1%

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

      1. times-frac 34.1%

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

   3. Simplified 34.1%

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

   4. Taylor expanded in w0 around 0 40.8%

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

      1. cancel-sign-sub-inv 40.8%

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

      2. *-commutative 40.8%

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

      3. cancel-sign-sub-inv 40.8%

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

   6. Simplified 45.8%

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

      1. associate-*l/ 47.3%

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

      2. *-commutative 47.3%

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

      3. times-frac 57.0%

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

   8. Applied egg-rr 57.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{+288}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \left(D \cdot \left(D \cdot h\right)\right)}{\ell}}\\ \end{array} \]

Alternative 3: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(h \cdot \frac{M}{\frac{\ell}{M}}\right) \cdot \frac{D \cdot D}{d \cdot d}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-296}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= (/ h l) (- INFINITY))
   (* w0 (sqrt (+ 1.0 (* -0.25 (* (* h (/ M (/ l M))) (/ (* D D) (* d d)))))))
   (if (<= (/ h l) -1e-296)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ D d) (/ M 2.0)) 2.0)))))
     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 ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((h * (M / (l / M))) * ((D * D) / (d * d))))));
	} else if ((h / l) <= -1e-296) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D / d) * (M / 2.0)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

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 ((h / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((h * (M / (l / M))) * ((D * D) / (d * d))))));
	} else if ((h / l) <= -1e-296) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D / d) * (M / 2.0)), 2.0))));
	} 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 (h / l) <= -math.inf:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((h * (M / (l / M))) * ((D * D) / (d * d))))))
	elif (h / l) <= -1e-296:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D / d) * (M / 2.0)), 2.0))))
	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 (Float64(h / l) <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(h * Float64(M / Float64(l / M))) * Float64(Float64(D * D) / Float64(d * d)))))));
	elseif (Float64(h / l) <= -1e-296)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0)))));
	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 ((h / l) <= -Inf)
		tmp = w0 * sqrt((1.0 + (-0.25 * ((h * (M / (l / M))) * ((D * D) / (d * d))))));
	elseif ((h / l) <= -1e-296)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D / d) * (M / 2.0)) ^ 2.0))));
	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[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(h * N[(M / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-296], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 26.2%

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

      1. times-frac 26.2%

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

   3. Simplified 26.2%

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

   4. Taylor expanded in w0 around 0 38.3%

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

      1. cancel-sign-sub-inv 38.3%

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

      2. *-commutative 38.3%

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

      3. cancel-sign-sub-inv 38.3%

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

   6. Simplified 54.9%

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

      1. associate-*l/ 59.5%

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

      2. *-commutative 59.5%

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

      3. times-frac 80.3%

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

   8. Applied egg-rr 80.3%

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

   9. Taylor expanded in D around 0 46.9%

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

       1. unpow2 46.9%

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

       2. unpow2 46.9%

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

       3. associate-/l* 46.9%

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

       4. unpow2 46.9%

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

       5. associate-/r/ 46.5%

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

       6. times-frac 51.3%

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

       7. unpow2 51.3%

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

   11. Simplified 51.3%

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

   12. Taylor expanded in D around 0 38.3%

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

       1. *-commutative 38.3%

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

       2. unpow2 38.3%

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

       3. times-frac 42.4%

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

       4. associate-*r/ 42.4%

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

       5. associate-/l* 54.5%

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

       6. unpow2 54.5%

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

       7. unpow2 54.5%

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

   14. Simplified 54.5%

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

   if -inf.0 < (/.f64 h l) < -1e-296

   1. Initial program 84.5%

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

      1. times-frac 84.6%

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

   3. Simplified 84.6%

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

   if -1e-296 < (/.f64 h l)

   1. Initial program 88.9%

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

      1. times-frac 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in M around 0 94.7%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(h \cdot \frac{M}{\frac{\ell}{M}}\right) \cdot \frac{D \cdot D}{d \cdot d}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-296}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4: 88.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+120}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{\ell} \cdot \left(D \cdot h\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 D) (* 2.0 d)) 5e+120)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* (/ M d) (* D 0.5)) 2.0)) l))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (* (* (/ M d) (/ M d)) (* (/ D l) (* D h)))))))))

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 * D) / (2.0 * d)) <= 5e+120) {
		tmp = w0 * sqrt((1.0 - ((h * pow(((M / d) * (D * 0.5)), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * (((M / d) * (M / d)) * ((D / l) * (D * h))))));
	}
	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 * d) / (2.0d0 * d_1)) <= 5d+120) then
        tmp = w0 * sqrt((1.0d0 - ((h * (((m / d_1) * (d * 0.5d0)) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (((m / d_1) * (m / d_1)) * ((d / l) * (d * h))))))
    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 * D) / (2.0 * d)) <= 5e+120) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow(((M / d) * (D * 0.5)), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (((M / d) * (M / d)) * ((D / l) * (D * h))))));
	}
	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 * D) / (2.0 * d)) <= 5e+120:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow(((M / d) * (D * 0.5)), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (((M / d) * (M / d)) * ((D / l) * (D * h))))))
	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(Float64(M * D) / Float64(2.0 * d)) <= 5e+120)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(M / d) * Float64(D * 0.5)) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(Float64(M / d) * Float64(M / d)) * Float64(Float64(D / l) * Float64(D * h)))))));
	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 * D) / (2.0 * d)) <= 5e+120)
		tmp = w0 * sqrt((1.0 - ((h * (((M / d) * (D * 0.5)) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * (((M / d) * (M / d)) * ((D / l) * (D * h))))));
	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[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 5e+120], 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], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(N[(D / l), $MachinePrecision] * N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 5.00000000000000019e120

   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. *-commutative 84.3%

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

      2. times-frac 83.4%

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

   3. Simplified 83.4%

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

      1. associate-*r/ 90.1%

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

      2. div-inv 90.1%

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

      3. metadata-eval 90.1%

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

   5. Applied egg-rr 90.1%

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

   if 5.00000000000000019e120 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 49.4%

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

      1. times-frac 49.4%

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

   3. Simplified 49.4%

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

   4. Taylor expanded in w0 around 0 26.8%

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

      1. cancel-sign-sub-inv 26.8%

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

      2. *-commutative 26.8%

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

      3. cancel-sign-sub-inv 26.8%

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

   6. Simplified 21.8%

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

      1. *-un-lft-identity 21.8%

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

      2. *-commutative 21.8%

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

      3. *-commutative 21.8%

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

      4. times-frac 30.0%

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

      5. associate-/l* 33.4%

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

      6. *-commutative 33.4%

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

   8. Applied egg-rr 33.4%

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

      1. *-lft-identity 33.4%

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

      2. *-commutative 33.4%

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

      3. associate-/r/ 33.9%

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

      4. *-commutative 33.9%

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

   10. Simplified 33.9%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+120}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{\ell} \cdot \left(D \cdot h\right)\right)\right)}\\ \end{array} \]

Alternative 5: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;2 \cdot d \leq 3.5 \cdot 10^{-37}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{\ell} \cdot \left(D \cdot h\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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) 3.5e-37)
   (* w0 (sqrt (+ 1.0 (* -0.25 (* (* (/ M d) (/ M d)) (* (/ D l) (* D h)))))))
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* (* M 0.5) (/ D d)) 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) <= 3.5e-37) {
		tmp = w0 * sqrt((1.0 + (-0.25 * (((M / d) * (M / d)) * ((D / l) * (D * h))))));
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow(((M * 0.5) * (D / d)), 2.0)) / l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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) <= 3.5d-37) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (((m / d_1) * (m / d_1)) * ((d / l) * (d * h))))))
    else
        tmp = w0 * sqrt((1.0d0 - ((h * (((m * 0.5d0) * (d / d_1)) ** 2.0d0)) / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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) <= 3.5e-37) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (((M / d) * (M / d)) * ((D / l) * (D * h))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow(((M * 0.5) * (D / d)), 2.0)) / l)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (2.0 * d) <= 3.5e-37:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (((M / d) * (M / d)) * ((D / l) * (D * h))))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow(((M * 0.5) * (D / d)), 2.0)) / l)))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(2.0 * d) <= 3.5e-37)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(Float64(M / d) * Float64(M / d)) * Float64(Float64(D / l) * Float64(D * h)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0)) / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
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) <= 3.5e-37)
		tmp = w0 * sqrt((1.0 + (-0.25 * (((M / d) * (M / d)) * ((D / l) * (D * h))))));
	else
		tmp = w0 * sqrt((1.0 - ((h * (((M * 0.5) * (D / d)) ^ 2.0)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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], 3.5e-37], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(N[(D / l), $MachinePrecision] * N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 2 d) < 3.5000000000000001e-37

   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 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 45.3%

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

      1. cancel-sign-sub-inv 45.3%

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

      2. *-commutative 45.3%

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

      3. cancel-sign-sub-inv 45.3%

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

   6. Simplified 50.1%

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

      1. *-un-lft-identity 50.1%

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

      2. *-commutative 50.1%

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

      3. *-commutative 50.1%

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

      4. times-frac 69.7%

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

      5. associate-/l* 70.2%

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

      6. *-commutative 70.2%

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

   8. Applied egg-rr 70.2%

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

      1. *-lft-identity 70.2%

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

      2. *-commutative 70.2%

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

      3. associate-/r/ 70.8%

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

      4. *-commutative 70.8%

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

   10. Simplified 70.8%

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

   if 3.5000000000000001e-37 < (*.f64 2 d)

   1. Initial program 86.3%

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

      1. times-frac 86.3%

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

   3. Simplified 86.3%

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

      1. associate-*r/ 92.1%

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

      2. div-inv 92.1%

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

      3. metadata-eval 92.1%

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

   5. Applied egg-rr 92.1%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot d \leq 3.5 \cdot 10^{-37}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{\ell} \cdot \left(D \cdot h\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 6: 80.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 3.3 \cdot 10^{+125}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{\ell} \cdot \left(D \cdot h\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 3.3e+125)
   (* w0 (sqrt (+ 1.0 (* -0.25 (* (* (/ M d) (/ M d)) (* (/ D l) (* D h)))))))
   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 <= 3.3e+125) {
		tmp = w0 * sqrt((1.0 + (-0.25 * (((M / d) * (M / d)) * ((D / l) * (D * h))))));
	} 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 <= 3.3d+125) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (((m / d_1) * (m / d_1)) * ((d / l) * (d * h))))))
    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 <= 3.3e+125) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (((M / d) * (M / d)) * ((D / l) * (D * h))))));
	} 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 <= 3.3e+125:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (((M / d) * (M / d)) * ((D / l) * (D * h))))))
	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 <= 3.3e+125)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(Float64(M / d) * Float64(M / d)) * Float64(Float64(D / l) * Float64(D * h)))))));
	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 <= 3.3e+125)
		tmp = w0 * sqrt((1.0 + (-0.25 * (((M / d) * (M / d)) * ((D / l) * (D * h))))));
	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, 3.3e+125], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(N[(D / l), $MachinePrecision] * N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 3.30000000000000005e125

   1. Initial program 79.9%

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

      1. times-frac 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in w0 around 0 50.5%

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

      1. cancel-sign-sub-inv 50.5%

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

      2. *-commutative 50.5%

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

      3. cancel-sign-sub-inv 50.5%

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

   6. Simplified 55.4%

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

      1. *-un-lft-identity 55.4%

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

      2. *-commutative 55.4%

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

      3. *-commutative 55.4%

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

      4. times-frac 71.9%

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

      5. associate-/l* 72.3%

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

      6. *-commutative 72.3%

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

   8. Applied egg-rr 72.3%

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

      1. *-lft-identity 72.3%

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

      2. *-commutative 72.3%

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

      3. associate-/r/ 72.8%

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

      4. *-commutative 72.8%

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

   10. Simplified 72.8%

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

   if 3.30000000000000005e125 < d

   1. Initial program 86.7%

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

      1. times-frac 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in M around 0 89.7%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 3.3 \cdot 10^{+125}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(\frac{M}{d} \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{\ell} \cdot \left(D \cdot h\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 7: 77.0% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.5 \cdot 10^{-120}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot M\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 2.5e-120)
   w0
   (* w0 (+ 1.0 (* -0.125 (* (* (/ D d) (/ D d)) (* (/ h l) (* M M))))))))

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 <= 2.5e-120) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h / l) * (M * M)))));
	}
	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 <= 2.5d-120) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d / d_1) * (d / d_1)) * ((h / l) * (m * m)))))
    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 <= 2.5e-120) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h / l) * (M * M)))));
	}
	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 <= 2.5e-120:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h / l) * (M * M)))))
	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 <= 2.5e-120)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(h / l) * Float64(M * M))))));
	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 <= 2.5e-120)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * ((h / l) * (M * M)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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, 2.5e-120], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.50000000000000003e-120

   1. Initial program 82.7%

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

      1. times-frac 82.2%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in M around 0 76.4%

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

   if 2.50000000000000003e-120 < M

   1. Initial program 76.9%

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

      1. times-frac 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in M around 0 48.6%

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

      1. associate-*r/ 48.6%

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

      2. *-commutative 48.6%

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

      3. associate-*r/ 48.6%

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

      4. associate-/l* 48.7%

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

      5. unpow2 48.7%

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

      6. unpow2 48.7%

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

      7. *-commutative 48.7%

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

      8. unpow2 48.7%

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

   6. Simplified 48.7%

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

   7. Taylor expanded in D around 0 48.6%

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

      1. unpow2 48.6%

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

      2. *-commutative 48.6%

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

      3. times-frac 50.1%

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

      4. unpow2 50.1%

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

      5. unpow2 50.1%

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

      6. times-frac 55.8%

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

      7. unpow2 55.8%

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

      8. *-commutative 55.8%

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

      9. associate-/l* 55.6%

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

   9. Simplified 55.6%

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

       1. unpow2 55.6%

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

   11. Applied egg-rr 55.6%

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

   12. Taylor expanded in M around 0 55.8%

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

       1. unpow2 55.8%

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

       2. *-commutative 55.8%

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

       3. associate-/l* 57.1%

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

       4. associate-/r/ 55.6%

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

   14. Simplified 55.6%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.5 \cdot 10^{-120}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot M\right)\right)\right)\right)\\ \end{array} \]

Alternative 8: 77.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.75 \cdot 10^{-120}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.75e-120)
   w0
   (* w0 (+ 1.0 (* -0.125 (* (* (/ 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 (M <= 1.75e-120) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * (h * ((M * M) / l)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: 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 <= 1.75d-120) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d / d_1) * (d / d_1)) * (h * ((m * m) / l)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
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 <= 1.75e-120) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((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 M <= 1.75e-120:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (((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 (M <= 1.75e-120)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(h * Float64(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 (M <= 1.75e-120)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * (h * ((M * M) / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: 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, 1.75e-120], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.75e-120

   1. Initial program 82.7%

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

      1. times-frac 82.2%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in M around 0 76.4%

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

   if 1.75e-120 < M

   1. Initial program 76.9%

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

      1. times-frac 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in M around 0 48.6%

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

      1. associate-*r/ 48.6%

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

      2. *-commutative 48.6%

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

      3. associate-*r/ 48.6%

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

      4. associate-/l* 48.7%

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

      5. unpow2 48.7%

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

      6. unpow2 48.7%

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

      7. *-commutative 48.7%

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

      8. unpow2 48.7%

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

   6. Simplified 48.7%

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

   7. Taylor expanded in D around 0 48.6%

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

      1. unpow2 48.6%

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

      2. *-commutative 48.6%

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

      3. times-frac 50.1%

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

      4. unpow2 50.1%

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

      5. unpow2 50.1%

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

      6. times-frac 55.8%

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

      7. unpow2 55.8%

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

      8. *-commutative 55.8%

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

      9. associate-/l* 55.6%

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

   9. Simplified 55.6%

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

       1. unpow2 55.6%

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

   11. Applied egg-rr 55.6%

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

       1. associate-/r/ 57.1%

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

   13. Applied egg-rr 57.1%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.75 \cdot 10^{-120}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right)\right)\\ \end{array} \]

Alternative 9: 76.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 3.3 \cdot 10^{-102}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{\ell} \cdot \frac{D}{d \cdot d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 3.3e-102)
   w0
   (* w0 (+ 1.0 (* -0.125 (* (* (/ D l) (/ D (* d d))) (* h (* M M))))))))

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 <= 3.3e-102) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((D / l) * (D / (d * d))) * (h * (M * M)))));
	}
	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 <= 3.3d-102) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d / l) * (d / (d_1 * d_1))) * (h * (m * m)))))
    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 <= 3.3e-102) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((D / l) * (D / (d * d))) * (h * (M * M)))));
	}
	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 <= 3.3e-102:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (((D / l) * (D / (d * d))) * (h * (M * M)))))
	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 <= 3.3e-102)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D / l) * Float64(D / Float64(d * d))) * Float64(h * Float64(M * M))))));
	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 <= 3.3e-102)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (((D / l) * (D / (d * d))) * (h * (M * M)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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, 3.3e-102], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D / l), $MachinePrecision] * N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.3e-102

   1. Initial program 82.6%

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

      1. times-frac 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in M around 0 76.4%

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

   if 3.3e-102 < M

   1. Initial program 76.9%

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

      1. times-frac 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in M around 0 47.1%

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

      1. associate-*r/ 47.1%

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

      2. *-commutative 47.1%

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

      3. associate-*r/ 47.1%

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

      4. associate-/l* 47.2%

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

      5. unpow2 47.2%

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

      6. unpow2 47.2%

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

      7. *-commutative 47.2%

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

      8. unpow2 47.2%

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

   6. Simplified 47.2%

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

      1. associate-/r/ 47.2%

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

      2. *-commutative 47.2%

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

   8. Applied egg-rr 47.2%

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

      1. times-frac 53.0%

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

   10. Applied egg-rr 53.0%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.3 \cdot 10^{-102}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{\ell} \cdot \frac{D}{d \cdot d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)\\ \end{array} \]

Alternative 10: 78.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4500000000000:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{M}{d} \cdot \left(\frac{M}{d} \cdot \frac{D}{\frac{\ell}{D \cdot h}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 4500000000000.0)
   (* w0 (+ 1.0 (* -0.125 (* (/ M d) (* (/ M d) (/ D (/ l (* D h))))))))
   (* w0 (+ 1.0 (* -0.125 (* (* (/ 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 <= 4500000000000.0) {
		tmp = w0 * (1.0 + (-0.125 * ((M / d) * ((M / d) * (D / (l / (D * h)))))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * (h * ((M * M) / l)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: 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 <= 4500000000000.0d0) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((m / d_1) * ((m / d_1) * (d / (l / (d * h)))))))
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d / d_1) * (d / d_1)) * (h * ((m * m) / l)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
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 <= 4500000000000.0) {
		tmp = w0 * (1.0 + (-0.125 * ((M / d) * ((M / d) * (D / (l / (D * h)))))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((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 <= 4500000000000.0:
		tmp = w0 * (1.0 + (-0.125 * ((M / d) * ((M / d) * (D / (l / (D * h)))))))
	else:
		tmp = w0 * (1.0 + (-0.125 * (((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 <= 4500000000000.0)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(M / d) * Float64(Float64(M / d) * Float64(D / Float64(l / Float64(D * h))))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(h * Float64(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 <= 4500000000000.0)
		tmp = w0 * (1.0 + (-0.125 * ((M / d) * ((M / d) * (D / (l / (D * h)))))));
	else
		tmp = w0 * (1.0 + (-0.125 * (((D / d) * (D / d)) * (h * ((M * M) / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: 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, 4500000000000.0], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(M / d), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(D / N[(l / N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M * M), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.5e12

   1. Initial program 79.5%

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

      1. times-frac 79.0%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in M around 0 44.6%

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

      1. associate-*r/ 44.6%

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

      2. *-commutative 44.6%

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

      3. associate-*r/ 44.6%

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

      4. associate-/l* 44.7%

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

      5. unpow2 44.7%

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

      6. unpow2 44.7%

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

      7. *-commutative 44.7%

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

      8. unpow2 44.7%

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

   6. Simplified 44.7%

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

      1. associate-/r/ 44.1%

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

      2. *-commutative 44.1%

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

   8. Applied egg-rr 44.1%

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

      1. associate-*l/ 44.6%

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

   10. Applied egg-rr 44.6%

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

       1. associate-*r* 43.4%

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

       2. associate-*r* 47.1%

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

       3. frac-times 49.3%

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

       4. frac-times 66.3%

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

       5. associate-*r* 70.2%

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

       6. associate-/l* 70.7%

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

   12. Applied egg-rr 70.7%

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

   if 4.5e12 < d

   1. Initial program 85.0%

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

      1. times-frac 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in M around 0 62.5%

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

      1. associate-*r/ 62.5%

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

      2. *-commutative 62.5%

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

      3. associate-*r/ 62.5%

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

      4. associate-/l* 64.0%

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

      5. unpow2 64.0%

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

      6. unpow2 64.0%

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

      7. *-commutative 64.0%

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

      8. unpow2 64.0%

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

   6. Simplified 64.0%

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

   7. Taylor expanded in D around 0 62.5%

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

      1. unpow2 62.5%

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

      2. *-commutative 62.5%

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

      3. times-frac 64.0%

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

      4. unpow2 64.0%

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

      5. unpow2 64.0%

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

      6. times-frac 73.2%

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

      7. unpow2 73.2%

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

      8. *-commutative 73.2%

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

      9. associate-/l* 67.0%

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

   9. Simplified 67.0%

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

       1. unpow2 67.0%

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

   11. Applied egg-rr 67.0%

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

       1. associate-/r/ 73.2%

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

   13. Applied egg-rr 73.2%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4500000000000:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{M}{d} \cdot \left(\frac{M}{d} \cdot \frac{D}{\frac{\ell}{D \cdot h}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M \cdot M}{\ell}\right)\right)\right)\\ \end{array} \]

Alternative 11: 71.7% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 7.5 \cdot 10^{+102}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right)\\ \end{array} \end{array} \]

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 7.5e+102)
   w0
   (* -0.125 (* (/ D (/ l D)) (/ (* w0 (* 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 <= 7.5e+102) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D / (l / D)) * ((w0 * (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 <= 7.5d+102) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((d / (l / d)) * ((w0 * (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 <= 7.5e+102) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D / (l / D)) * ((w0 * (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 <= 7.5e+102:
		tmp = w0
	else:
		tmp = -0.125 * ((D / (l / D)) * ((w0 * (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 <= 7.5e+102)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(D / Float64(l / D)) * Float64(Float64(w0 * 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 <= 7.5e+102)
		tmp = w0;
	else
		tmp = -0.125 * ((D / (l / D)) * ((w0 * (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, 7.5e+102], w0, N[(-0.125 * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.5e102

   1. Initial program 83.3%

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

      1. times-frac 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in M around 0 74.7%

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

   if 7.5e102 < M

   1. Initial program 64.5%

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

      1. times-frac 64.5%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in M around 0 31.7%

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

      1. associate-*r/ 31.7%

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

      2. *-commutative 31.7%

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

      3. associate-*r/ 31.7%

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

      4. associate-/l* 31.7%

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

      5. unpow2 31.7%

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

      6. unpow2 31.7%

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

      7. *-commutative 31.7%

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

      8. unpow2 31.7%

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

   6. Simplified 31.7%

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

   7. Taylor expanded in D around 0 31.7%

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

      1. unpow2 31.7%

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

      2. *-commutative 31.7%

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

      3. times-frac 31.8%

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

      4. unpow2 31.8%

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

      5. unpow2 31.8%

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

      6. times-frac 35.1%

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

      7. unpow2 35.1%

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

      8. *-commutative 35.1%

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

      9. associate-/l* 34.5%

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

   9. Simplified 34.5%

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

       1. unpow2 34.5%

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

   11. Applied egg-rr 34.5%

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

   12. Taylor expanded in D around inf 25.8%

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

       1. times-frac 19.5%

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

       2. unpow2 19.5%

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

       3. associate-/l* 22.6%

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

       4. unpow2 22.6%

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

       5. *-commutative 22.6%

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

       6. unpow2 22.6%

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

   14. Simplified 22.6%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.5 \cdot 10^{+102}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right)\\ \end{array} \]

Alternative 12: 68.2% 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 80.9%

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

   1. times-frac 80.5%

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

3. Simplified 80.5%

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

4. Taylor expanded in M around 0 69.8%

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

5. Final simplification 69.8%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023257 
(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))))))