Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 87.3%

Time: 14.7s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{1 - {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+139}:\\ \;\;\;\;w0 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (sqrt (- 1.0 (* (pow (/ (* D M) (* d 2.0)) 2.0) (/ h l))))))
   (if (<= t_0 2e+139)
     (* w0 t_0)
     (* w0 (sqrt (- 1.0 (* h (/ (pow (* M (* D (/ 0.5 d))) 2.0) l))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = sqrt((1.0 - (pow(((D * M) / (d * 2.0)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 2e+139) {
		tmp = w0 * t_0;
	} else {
		tmp = w0 * sqrt((1.0 - (h * (pow((M * (D * (0.5 / d))), 2.0) / l))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 - ((((d * m) / (d_1 * 2.0d0)) ** 2.0d0) * (h / l))))
    if (t_0 <= 2d+139) then
        tmp = w0 * t_0
    else
        tmp = w0 * sqrt((1.0d0 - (h * (((m * (d * (0.5d0 / d_1))) ** 2.0d0) / l))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = Math.sqrt((1.0 - (Math.pow(((D * M) / (d * 2.0)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 2e+139) {
		tmp = w0 * t_0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow((M * (D * (0.5 / d))), 2.0) / l))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = math.sqrt((1.0 - (math.pow(((D * M) / (d * 2.0)), 2.0) * (h / l))))
	tmp = 0
	if t_0 <= 2e+139:
		tmp = w0 * t_0
	else:
		tmp = w0 * math.sqrt((1.0 - (h * (math.pow((M * (D * (0.5 / d))), 2.0) / l))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = sqrt(Float64(1.0 - Float64((Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 2e+139)
		tmp = Float64(w0 * t_0);
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0) / l)))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = sqrt((1.0 - ((((D * M) / (d * 2.0)) ^ 2.0) * (h / l))));
	tmp = 0.0;
	if (t_0 <= 2e+139)
		tmp = w0 * t_0;
	else
		tmp = w0 * sqrt((1.0 - (h * (((M * (D * (0.5 / d))) ^ 2.0) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 2e+139], N[(w0 * t$95$0), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))) < 2.00000000000000007e139

   1. Initial program 99.9%

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

   if 2.00000000000000007e139 < (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))

   1. Initial program 46.7%

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

      1. associate-/l* 49.0%

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

   3. Simplified 49.0%

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

      1. associate-/l* 46.7%

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

      2. clear-num 46.7%

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

      3. un-div-inv 49.2%

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

      4. associate-*l/ 51.6%

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

      5. *-commutative 51.6%

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

      6. associate-/r* 51.6%

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

      7. div-inv 51.6%

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

      8. metadata-eval 51.6%

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

   5. Applied egg-rr 51.6%

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

      1. associate-/r/ 70.1%

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

      2. *-commutative 70.1%

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

      3. *-commutative 70.1%

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

      4. associate-*l/ 70.1%

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

      5. associate-*l* 69.3%

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

      6. *-commutative 69.3%

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

   7. Simplified 69.3%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 2: 85.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (+ 1.0 (* (* h (pow (* D (/ (* M 0.5) d)) 2.0)) (/ -1.0 l))))))

C

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

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 + ((h * ((d * ((m * 0.5d0) / d_1)) ** 2.0d0)) * ((-1.0d0) / l))))
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 + ((h * Math.pow((D * ((M * 0.5) / d)), 2.0)) * (-1.0 / l))));
}

Python

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

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(h * (Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0)) * Float64(-1.0 / l)))))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 + ((h * ((D * ((M * 0.5) / d)) ^ 2.0)) * (-1.0 / l))));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 + N[(N[(h * N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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. associate-/l* 82.1%

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

3. Simplified 82.1%

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

   1. associate-*r/ 87.8%

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

   2. associate-/l* 88.7%

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

   3. clear-num 88.7%

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

   4. *-commutative 88.7%

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

   5. associate-*l/ 89.1%

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

   6. *-commutative 89.1%

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

   7. associate-/r* 89.1%

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

   8. div-inv 89.1%

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

   9. metadata-eval 89.1%

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

5. Applied egg-rr 89.1%

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

   1. associate-/r/ 89.1%

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

7. Simplified 89.1%

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

8. Final simplification 89.1%

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

Alternative 3: 86.0% accurate, 1.0× speedup

Math

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

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* h (/ (pow (* M (* D (/ 0.5 d))) 2.0) l))))))

C

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

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - (h * (((m * (d * (0.5d0 / d_1))) ** 2.0d0) / l))))
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (h * (Math.pow((M * (D * (0.5 / d))), 2.0) / l))));
}

Python

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

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0) / l)))))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - (h * (((M * (D * (0.5 / d))) ^ 2.0) / l))));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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. associate-/l* 82.1%

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

3. Simplified 82.1%

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

   1. associate-/l* 83.3%

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

   2. clear-num 83.3%

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

   3. un-div-inv 84.0%

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

   4. associate-*l/ 84.4%

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

   5. *-commutative 84.4%

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

   6. associate-/r* 84.4%

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

   7. div-inv 84.4%

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

   8. metadata-eval 84.4%

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

5. Applied egg-rr 84.4%

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

   1. associate-/r/ 88.7%

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

   2. *-commutative 88.7%

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

   3. *-commutative 88.7%

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

   4. associate-*l/ 88.8%

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

   5. associate-*l* 87.0%

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

   6. *-commutative 87.0%

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

7. Simplified 87.0%

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

8. Final simplification 87.0%

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

Alternative 4: 72.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \cdot M \leq 10^{-148}:\\ \;\;\;\;w0\\ \mathbf{elif}\;D \cdot M \leq 2 \cdot 10^{+119}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(d \cdot \frac{\ell}{h}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(\frac{D}{\ell} \cdot \frac{D}{d \cdot d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

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 M) 1e-148)
   w0
   (if (<= (* D M) 2e+119)
     (* w0 (+ 1.0 (* -0.125 (/ (* (* D M) (* D M)) (* d (* d (/ l h)))))))
     (*
      w0
      (sqrt (+ 1.0 (* -0.25 (* (* (/ D l) (/ D (* d d))) (* h (* M M))))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((D * M) <= 1e-148) {
		tmp = w0;
	} else if ((D * M) <= 2e+119) {
		tmp = w0 * (1.0 + (-0.125 * (((D * M) * (D * M)) / (d * (d * (l / h))))));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * (((D / l) * (D / (d * d))) * (h * (M * M))))));
	}
	return tmp;
}

Fortran

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 * m) <= 1d-148) then
        tmp = w0
    else if ((d * m) <= 2d+119) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d * m) * (d * m)) / (d_1 * (d_1 * (l / h))))))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (((d / l) * (d / (d_1 * d_1))) * (h * (m * m))))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((D * M) <= 1e-148) {
		tmp = w0;
	} else if ((D * M) <= 2e+119) {
		tmp = w0 * (1.0 + (-0.125 * (((D * M) * (D * M)) / (d * (d * (l / h))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (((D / l) * (D / (d * d))) * (h * (M * M))))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (D * M) <= 1e-148:
		tmp = w0
	elif (D * M) <= 2e+119:
		tmp = w0 * (1.0 + (-0.125 * (((D * M) * (D * M)) / (d * (d * (l / h))))))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (((D / l) * (D / (d * d))) * (h * (M * M))))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(D * M) <= 1e-148)
		tmp = w0;
	elseif (Float64(D * M) <= 2e+119)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D * M) * Float64(D * M)) / Float64(d * Float64(d * Float64(l / h)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(Float64(D / l) * Float64(D / Float64(d * d))) * Float64(h * Float64(M * M)))))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((D * M) <= 1e-148)
		tmp = w0;
	elseif ((D * M) <= 2e+119)
		tmp = w0 * (1.0 + (-0.125 * (((D * M) * (D * M)) / (d * (d * (l / h))))));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * (((D / l) * (D / (d * d))) * (h * (M * M))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(D * M), $MachinePrecision], 1e-148], w0, If[LessEqual[N[(D * M), $MachinePrecision], 2e+119], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * 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]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 9.99999999999999936e-149

   1. Initial program 87.6%

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

      1. associate-/l* 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in M around 0 79.0%

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

   if 9.99999999999999936e-149 < (*.f64 M D) < 1.99999999999999989e119

   1. Initial program 77.1%

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

      1. associate-/l* 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in M around 0 47.0%

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

      1. *-commutative 47.0%

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

      2. associate-/l* 49.0%

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

      3. unpow2 49.0%

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

      4. unpow2 49.0%

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

      5. *-commutative 49.0%

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

      6. unpow2 49.0%

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

   6. Simplified 49.0%

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

   7. Taylor expanded in D around 0 47.0%

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

      1. associate-*r* 51.0%

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

      2. unpow2 51.0%

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

      3. times-frac 50.8%

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

      4. unpow2 50.8%

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

      5. unpow2 50.8%

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

      6. swap-sqr 69.7%

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

      7. unpow2 69.7%

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

      8. associate-*l/ 69.8%

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

      9. associate-/l* 69.8%

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

   9. Simplified 69.8%

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

       1. unpow2 69.8%

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

       2. *-commutative 69.8%

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

       3. *-commutative 69.8%

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

   11. Applied egg-rr 69.8%

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

   12. Taylor expanded in d around 0 72.0%

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

       1. unpow2 72.0%

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

       2. associate-*r/ 69.8%

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

       3. associate-*l* 69.9%

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

   14. Simplified 69.9%

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

   if 1.99999999999999989e119 < (*.f64 M D)

   1. Initial program 68.9%

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

      1. associate-/l* 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in w0 around 0 39.9%

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

   6. Simplified 43.1%

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

   7. Taylor expanded in D around 0 39.9%

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

      1. unpow2 39.9%

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

      2. *-commutative 39.9%

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

      3. unpow2 39.9%

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

      4. *-commutative 39.9%

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

      5. associate-*l/ 43.1%

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

      6. unpow2 43.1%

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

      7. times-frac 56.1%

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

   9. Simplified 56.1%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot M \leq 10^{-148}:\\ \;\;\;\;w0\\ \mathbf{elif}\;D \cdot M \leq 2 \cdot 10^{+119}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(d \cdot \frac{\ell}{h}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(\frac{D}{\ell} \cdot \frac{D}{d \cdot d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\\ \end{array} \]

Alternative 5: 70.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.1 \cdot 10^{-105}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{D \cdot D}{\frac{\ell}{h} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{+111}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(\frac{D}{\ell} \cdot \frac{D}{d \cdot d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 1.1e-105)
   (* w0 (sqrt (+ 1.0 (* -0.25 (/ (* D D) (* (/ l h) (* (/ d M) (/ d M))))))))
   (if (<= d 1.35e+111)
     (*
      w0
      (sqrt (+ 1.0 (* -0.25 (* (* (/ D l) (/ D (* d d))) (* h (* M M)))))))
     w0)))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.1e-105) {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D * D) / ((l / h) * ((d / M) * (d / M)))))));
	} else if (d <= 1.35e+111) {
		tmp = w0 * sqrt((1.0 + (-0.25 * (((D / l) * (D / (d * d))) * (h * (M * M))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 1.1d-105) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((d * d) / ((l / h) * ((d_1 / m) * (d_1 / m)))))))
    else if (d_1 <= 1.35d+111) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (((d / l) * (d / (d_1 * d_1))) * (h * (m * m))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.1e-105) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((D * D) / ((l / h) * ((d / M) * (d / M)))))));
	} else if (d <= 1.35e+111) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (((D / l) * (D / (d * d))) * (h * (M * M))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 1.1e-105:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((D * D) / ((l / h) * ((d / M) * (d / M)))))))
	elif d <= 1.35e+111:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (((D / l) * (D / (d * d))) * (h * (M * M))))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 1.1e-105)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D * D) / Float64(Float64(l / h) * Float64(Float64(d / M) * Float64(d / M))))))));
	elseif (d <= 1.35e+111)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(Float64(D / l) * Float64(D / Float64(d * d))) * Float64(h * Float64(M * M)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 1.1e-105)
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D * D) / ((l / h) * ((d / M) * (d / M)))))));
	elseif (d <= 1.35e+111)
		tmp = w0 * sqrt((1.0 + (-0.25 * (((D / l) * (D / (d * d))) * (h * (M * M))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 1.1e-105], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D * D), $MachinePrecision] / N[(N[(l / h), $MachinePrecision] * N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.35e+111], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * 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]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if d < 1.10000000000000002e-105

   1. Initial program 79.6%

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

      1. associate-/l* 78.4%

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

   3. Simplified 78.4%

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

   4. Taylor expanded in w0 around 0 52.5%

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

   6. Simplified 53.7%

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

   7. Taylor expanded in l around 0 53.7%

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

      1. unpow2 53.7%

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

      2. *-commutative 53.7%

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

      3. times-frac 50.7%

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

      4. unpow2 50.7%

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

      5. times-frac 63.1%

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

   9. Simplified 67.0%

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

   if 1.10000000000000002e-105 < d < 1.3499999999999999e111

   1. Initial program 86.2%

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

      1. associate-/l* 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in w0 around 0 58.6%

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

   6. Simplified 57.2%

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

   7. Taylor expanded in D around 0 58.6%

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

      1. unpow2 58.6%

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

      2. *-commutative 58.6%

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

      3. unpow2 58.6%

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

      4. *-commutative 58.6%

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

      5. associate-*l/ 60.8%

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

      6. unpow2 60.8%

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

      7. times-frac 72.6%

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

   9. Simplified 72.6%

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

   if 1.3499999999999999e111 < d

   1. Initial program 93.6%

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

      1. associate-/l* 93.6%

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

   3. Simplified 93.6%

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

   4. Taylor expanded in M around 0 91.9%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.1 \cdot 10^{-105}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{D \cdot D}{\frac{\ell}{h} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{+111}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(\frac{D}{\ell} \cdot \frac{D}{d \cdot d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 6: 72.6% accurate, 7.4× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \cdot M \leq 10^{-148}:\\ \;\;\;\;w0\\ \mathbf{elif}\;D \cdot M \leq 5 \cdot 10^{+154}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(d \cdot \frac{\ell}{h}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\left(\frac{D}{\ell} \cdot \frac{D}{d \cdot d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

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 M) 1e-148)
   w0
   (if (<= (* D M) 5e+154)
     (* w0 (+ 1.0 (* -0.125 (/ (* (* D M) (* D M)) (* d (* d (/ l h)))))))
     (* w0 (+ 1.0 (* (* (* (/ D l) (/ D (* d d))) (* h (* M M))) -0.125))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((D * M) <= 1e-148) {
		tmp = w0;
	} else if ((D * M) <= 5e+154) {
		tmp = w0 * (1.0 + (-0.125 * (((D * M) * (D * M)) / (d * (d * (l / h))))));
	} else {
		tmp = w0 * (1.0 + ((((D / l) * (D / (d * d))) * (h * (M * M))) * -0.125));
	}
	return tmp;
}

Fortran

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 * m) <= 1d-148) then
        tmp = w0
    else if ((d * m) <= 5d+154) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d * m) * (d * m)) / (d_1 * (d_1 * (l / h))))))
    else
        tmp = w0 * (1.0d0 + ((((d / l) * (d / (d_1 * d_1))) * (h * (m * m))) * (-0.125d0)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((D * M) <= 1e-148) {
		tmp = w0;
	} else if ((D * M) <= 5e+154) {
		tmp = w0 * (1.0 + (-0.125 * (((D * M) * (D * M)) / (d * (d * (l / h))))));
	} else {
		tmp = w0 * (1.0 + ((((D / l) * (D / (d * d))) * (h * (M * M))) * -0.125));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (D * M) <= 1e-148:
		tmp = w0
	elif (D * M) <= 5e+154:
		tmp = w0 * (1.0 + (-0.125 * (((D * M) * (D * M)) / (d * (d * (l / h))))))
	else:
		tmp = w0 * (1.0 + ((((D / l) * (D / (d * d))) * (h * (M * M))) * -0.125))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(D * M) <= 1e-148)
		tmp = w0;
	elseif (Float64(D * M) <= 5e+154)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D * M) * Float64(D * M)) / Float64(d * Float64(d * Float64(l / h)))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(Float64(D / l) * Float64(D / Float64(d * d))) * Float64(h * Float64(M * M))) * -0.125)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((D * M) <= 1e-148)
		tmp = w0;
	elseif ((D * M) <= 5e+154)
		tmp = w0 * (1.0 + (-0.125 * (((D * M) * (D * M)) / (d * (d * (l / h))))));
	else
		tmp = w0 * (1.0 + ((((D / l) * (D / (d * d))) * (h * (M * M))) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(D * M), $MachinePrecision], 1e-148], w0, If[LessEqual[N[(D * M), $MachinePrecision], 5e+154], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(N[(N[(D / l), $MachinePrecision] * N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 9.99999999999999936e-149

   1. Initial program 87.6%

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

      1. associate-/l* 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in M around 0 79.0%

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

   if 9.99999999999999936e-149 < (*.f64 M D) < 5.00000000000000004e154

   1. Initial program 75.6%

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

      1. associate-/l* 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in M around 0 44.7%

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

      1. *-commutative 44.7%

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

      2. associate-/l* 46.6%

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

      3. unpow2 46.6%

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

      4. unpow2 46.6%

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

      5. *-commutative 46.6%

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

      6. unpow2 46.6%

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

   6. Simplified 46.6%

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

   7. Taylor expanded in D around 0 44.7%

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

      1. associate-*r* 48.4%

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

      2. unpow2 48.4%

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

      3. times-frac 48.3%

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

      4. unpow2 48.3%

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

      5. unpow2 48.3%

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

      6. swap-sqr 67.2%

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

      7. unpow2 67.2%

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

      8. associate-*l/ 67.3%

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

      9. associate-/l* 67.4%

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

   9. Simplified 67.4%

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

       1. unpow2 67.4%

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

       2. *-commutative 67.4%

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

       3. *-commutative 67.4%

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

   11. Applied egg-rr 67.4%

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

   12. Taylor expanded in d around 0 69.3%

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

       1. unpow2 69.3%

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

       2. associate-*r/ 67.4%

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

       3. associate-*l* 67.4%

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

   14. Simplified 67.4%

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

   if 5.00000000000000004e154 < (*.f64 M D)

   1. Initial program 70.4%

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

      1. associate-/l* 77.8%

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

   3. Simplified 77.8%

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

      1. associate-*r/ 78.0%

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

      2. associate-/l* 70.6%

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

      3. clear-num 70.6%

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

      4. *-commutative 70.6%

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

      5. associate-*l/ 77.9%

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

      6. *-commutative 77.9%

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

      7. associate-/r* 77.9%

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

      8. div-inv 77.9%

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

      9. metadata-eval 77.9%

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

   5. Applied egg-rr 77.9%

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

      1. associate-/r/ 77.9%

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

   7. Simplified 77.9%

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

   8. Taylor expanded in l around inf 43.3%

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

      1. *-commutative 43.3%

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

      2. associate-*r/ 43.3%

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

      3. unpow2 43.3%

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

      4. *-commutative 43.3%

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

      5. unpow2 43.3%

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

      6. associate-*r/ 43.3%

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

      7. *-commutative 43.3%

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

      8. unpow2 43.3%

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

      9. unpow2 43.3%

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

      10. associate-*l/ 47.1%

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

      11. unpow2 47.1%

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

      12. times-frac 58.7%

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

   10. Simplified 58.7%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot M \leq 10^{-148}:\\ \;\;\;\;w0\\ \mathbf{elif}\;D \cdot M \leq 5 \cdot 10^{+154}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(d \cdot \frac{\ell}{h}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\left(\frac{D}{\ell} \cdot \frac{D}{d \cdot d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 7: 70.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 3.1 \cdot 10^{-111}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\left(\frac{D}{\ell} \cdot \frac{D}{d \cdot d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

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 3.1e-111)
   w0
   (* w0 (+ 1.0 (* (* (* (/ D l) (/ D (* d d))) (* h (* M M))) -0.125)))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.1e-111) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((((D / l) * (D / (d * d))) * (h * (M * M))) * -0.125));
	}
	return tmp;
}

Fortran

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 <= 3.1d-111) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((((d / l) * (d / (d_1 * d_1))) * (h * (m * m))) * (-0.125d0)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.1e-111) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((((D / l) * (D / (d * d))) * (h * (M * M))) * -0.125));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 3.1e-111:
		tmp = w0
	else:
		tmp = w0 * (1.0 + ((((D / l) * (D / (d * d))) * (h * (M * M))) * -0.125))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 3.1e-111)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(Float64(D / l) * Float64(D / Float64(d * d))) * Float64(h * Float64(M * M))) * -0.125)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 3.1e-111)
		tmp = w0;
	else
		tmp = w0 * (1.0 + ((((D / l) * (D / (d * d))) * (h * (M * M))) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 3.1e-111], w0, N[(w0 * N[(1.0 + N[(N[(N[(N[(D / l), $MachinePrecision] * N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.10000000000000014e-111

   1. Initial program 84.9%

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

      1. associate-/l* 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in M around 0 71.9%

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

   if 3.10000000000000014e-111 < M

   1. Initial program 79.7%

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

      1. associate-/l* 79.6%

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

   3. Simplified 79.6%

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

      1. associate-*r/ 83.6%

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

      2. associate-/l* 84.3%

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

      3. clear-num 84.3%

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

      4. *-commutative 84.3%

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

      5. associate-*l/ 83.1%

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

      6. *-commutative 83.1%

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

      7. associate-/r* 83.1%

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

      8. div-inv 83.1%

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

      9. metadata-eval 83.1%

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

   5. Applied egg-rr 83.1%

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

      1. associate-/r/ 83.1%

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

   7. Simplified 83.1%

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

   8. Taylor expanded in l around inf 55.1%

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

      1. *-commutative 55.1%

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

      2. associate-*r/ 55.1%

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

      3. unpow2 55.1%

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

      4. *-commutative 55.1%

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

      5. unpow2 55.1%

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

      6. associate-*r/ 55.1%

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

      7. *-commutative 55.1%

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

      8. unpow2 55.1%

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

      9. unpow2 55.1%

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

      10. associate-*l/ 58.8%

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

      11. unpow2 58.8%

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

      12. times-frac 62.6%

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

   10. Simplified 62.6%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.1 \cdot 10^{-111}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\left(\frac{D}{\ell} \cdot \frac{D}{d \cdot d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 8: 68.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 6.2 \cdot 10^{-62}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot D}{\frac{\ell}{h} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

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 6.2e-62)
   (* w0 (+ 1.0 (* (/ (* D D) (* (/ l h) (* (/ d M) (/ d M)))) -0.125)))
   w0))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 6.2e-62) {
		tmp = w0 * (1.0 + (((D * D) / ((l / h) * ((d / M) * (d / M)))) * -0.125));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

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 <= 6.2d-62) then
        tmp = w0 * (1.0d0 + (((d * d) / ((l / h) * ((d_1 / m) * (d_1 / m)))) * (-0.125d0)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 6.2e-62) {
		tmp = w0 * (1.0 + (((D * D) / ((l / h) * ((d / M) * (d / M)))) * -0.125));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 6.2e-62:
		tmp = w0 * (1.0 + (((D * D) / ((l / h) * ((d / M) * (d / M)))) * -0.125))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 6.2e-62)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D * D) / Float64(Float64(l / h) * Float64(Float64(d / M) * Float64(d / M)))) * -0.125)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 6.2e-62)
		tmp = w0 * (1.0 + (((D * D) / ((l / h) * ((d / M) * (d / M)))) * -0.125));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 6.2e-62], N[(w0 * N[(1.0 + N[(N[(N[(D * D), $MachinePrecision] / N[(N[(l / h), $MachinePrecision] * N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 6.1999999999999999e-62

   1. Initial program 79.6%

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

      1. associate-/l* 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in M around 0 53.5%

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

      1. *-commutative 53.5%

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

      2. associate-/l* 54.6%

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

      3. unpow2 54.6%

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

      4. unpow2 54.6%

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

      5. *-commutative 54.6%

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

      6. unpow2 54.6%

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

   6. Simplified 54.6%

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

   7. Taylor expanded in l around 0 54.6%

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

      1. unpow2 54.6%

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

      2. *-commutative 54.6%

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

      3. times-frac 51.8%

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

      4. unpow2 51.8%

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

      5. times-frac 64.1%

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

   9. Simplified 64.1%

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

   if 6.1999999999999999e-62 < d

   1. Initial program 91.3%

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

      1. associate-/l* 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in M around 0 84.7%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6.2 \cdot 10^{-62}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot D}{\frac{\ell}{h} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 9: 69.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.35 \cdot 10^{+40}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right)\\ \end{array} \end{array} \]

FPCore

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 1.35e+40)
   w0
   (* -0.125 (* (* D (/ D l)) (/ (* w0 (* h (* M M))) (* d d))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.35e+40) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * (D / l)) * ((w0 * (h * (M * M))) / (d * d)));
	}
	return tmp;
}

Fortran

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 <= 1.35d+40) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((d * (d / l)) * ((w0 * (h * (m * m))) / (d_1 * d_1)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.35e+40) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * (D / l)) * ((w0 * (h * (M * M))) / (d * d)));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.35e+40:
		tmp = w0
	else:
		tmp = -0.125 * ((D * (D / l)) * ((w0 * (h * (M * M))) / (d * d)))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.35e+40)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(w0 * Float64(h * Float64(M * M))) / Float64(d * d))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.35e+40)
		tmp = w0;
	else
		tmp = -0.125 * ((D * (D / l)) * ((w0 * (h * (M * M))) / (d * d)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.35e+40], w0, N[(-0.125 * N[(N[(D * N[(D / l), $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 < 1.35000000000000005e40

   1. Initial program 84.4%

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

      1. associate-/l* 83.0%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in M around 0 72.8%

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

   if 1.35000000000000005e40 < M

   1. Initial program 78.0%

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

      1. associate-/l* 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in M around 0 41.7%

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

      1. *-commutative 41.7%

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

      2. associate-/l* 45.9%

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

      3. unpow2 45.9%

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

      4. unpow2 45.9%

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

      5. *-commutative 45.9%

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

      6. unpow2 45.9%

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

   6. Simplified 45.9%

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

   7. Taylor expanded in D around 0 41.7%

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

      1. associate-*r* 46.3%

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

      2. unpow2 46.3%

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

      3. times-frac 44.0%

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

      4. unpow2 44.0%

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

      5. unpow2 44.0%

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

      6. swap-sqr 53.5%

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

      7. unpow2 53.5%

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

      8. associate-*l/ 53.5%

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

      9. associate-/l* 53.5%

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

   9. Simplified 53.5%

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

   10. Taylor expanded in D around inf 25.9%

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

       1. associate-*r/ 25.9%

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

       2. associate-*r* 25.9%

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

       3. *-commutative 25.9%

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

       4. associate-*r* 25.9%

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

       5. unpow2 25.9%

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

       6. associate-*r/ 25.9%

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

       7. *-commutative 25.9%

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

       8. times-frac 25.8%

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

       9. unpow2 25.8%

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

       10. *-commutative 25.8%

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

       11. unpow2 25.8%

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

   12. Simplified 25.8%

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

   13. Taylor expanded in D around 0 25.8%

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

       1. unpow2 25.8%

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

       2. associate-*l/ 28.3%

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

       3. *-commutative 28.3%

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

   15. Simplified 28.3%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.35 \cdot 10^{+40}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot d}\right)\\ \end{array} \]

Alternative 10: 69.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 7.5 \cdot 10^{+17}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{w0}{d} \cdot \frac{h \cdot M}{\frac{d}{M}}\right)\right)\\ \end{array} \end{array} \]

FPCore

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+17)
   w0
   (* -0.125 (* (/ (* D D) l) (* (/ w0 d) (/ (* h M) (/ d M)))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 7.5e+17) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) / l) * ((w0 / d) * ((h * M) / (d / M))));
	}
	return tmp;
}

Fortran

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+17) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d * d) / l) * ((w0 / d_1) * ((h * m) / (d_1 / m))))
    end if
    code = tmp
end function

Java

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+17) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) / l) * ((w0 / d) * ((h * M) / (d / M))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 7.5e+17:
		tmp = w0
	else:
		tmp = -0.125 * (((D * D) / l) * ((w0 / d) * ((h * M) / (d / M))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 7.5e+17)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) / l) * Float64(Float64(w0 / d) * Float64(Float64(h * M) / Float64(d / M)))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 7.5e+17)
		tmp = w0;
	else
		tmp = -0.125 * (((D * D) / l) * ((w0 / d) * ((h * M) / (d / M))));
	end
	tmp_2 = tmp;
end

Wolfram

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+17], w0, N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision] * N[(N[(w0 / d), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.5e17

   1. Initial program 85.4%

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

      1. associate-/l* 84.0%

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

   3. Simplified 84.0%

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

   4. Taylor expanded in M around 0 73.5%

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

   if 7.5e17 < M

   1. Initial program 74.3%

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

      1. associate-/l* 74.2%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in M around 0 41.7%

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

      1. *-commutative 41.7%

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

      2. associate-/l* 47.6%

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

      3. unpow2 47.6%

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

      4. unpow2 47.6%

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

      5. *-commutative 47.6%

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

      6. unpow2 47.6%

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

   6. Simplified 47.6%

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

   7. Taylor expanded in D around 0 41.7%

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

      1. associate-*r* 45.8%

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

      2. unpow2 45.8%

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

      3. times-frac 43.8%

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

      4. unpow2 43.8%

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

      5. unpow2 43.8%

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

      6. swap-sqr 52.3%

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

      7. unpow2 52.3%

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

      8. associate-*l/ 52.3%

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

      9. associate-/l* 52.3%

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

   9. Simplified 52.3%

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

   10. Taylor expanded in D around inf 25.6%

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

       1. associate-*r/ 25.6%

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

       2. associate-*r* 25.5%

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

       3. *-commutative 25.5%

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

       4. associate-*r* 25.6%

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

       5. unpow2 25.6%

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

       6. associate-*r/ 25.6%

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

       7. *-commutative 25.6%

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

       8. times-frac 25.6%

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

       9. unpow2 25.6%

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

       10. *-commutative 25.6%

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

       11. unpow2 25.6%

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

   12. Simplified 25.6%

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

   13. Taylor expanded in w0 around 0 25.6%

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

       1. unpow2 25.6%

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

       2. unpow2 25.6%

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

       3. associate-*l* 25.8%

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

       4. times-frac 27.9%

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

       5. associate-/l* 27.9%

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

   15. Simplified 27.9%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.5 \cdot 10^{+17}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{w0}{d} \cdot \frac{h \cdot M}{\frac{d}{M}}\right)\right)\\ \end{array} \]

Alternative 11: 68.1% accurate, 216.0× speedup

Math

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

FPCore

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

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Fortran

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

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0

Julia

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

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 83.3%

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

   1. associate-/l* 82.1%

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

3. Simplified 82.1%

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

4. Taylor expanded in M around 0 68.2%

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

5. Final simplification 68.2%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023258 
(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))))))