Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 86.5%

Time: 13.3s

Alternatives: 8

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: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2 \cdot 10^{+201}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(h \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}\right) \cdot \frac{-1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(M \cdot \left(\left(M \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{D}{d}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

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 (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 2e+201)
   (* w0 (sqrt (+ 1.0 (* (* h (pow (* (* M D) (/ 0.5 d)) 2.0)) (/ -1.0 l)))))
   (* w0 (sqrt (- 1.0 (* 0.25 (* M (* (* M (/ h l)) (pow (/ D d) 2.0)))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (pow(((M * D) / (2.0 * d)), 2.0) <= 2e+201) {
		tmp = w0 * sqrt((1.0 + ((h * pow(((M * D) * (0.5 / d)), 2.0)) * (-1.0 / l))));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (M * ((M * (h / l)) * pow((D / d), 2.0))))));
	}
	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 * d) / (2.0d0 * d_1)) ** 2.0d0) <= 2d+201) then
        tmp = w0 * sqrt((1.0d0 + ((h * (((m * d) * (0.5d0 / d_1)) ** 2.0d0)) * ((-1.0d0) / l))))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (m * ((m * (h / l)) * ((d / d_1) ** 2.0d0))))))
    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 (Math.pow(((M * D) / (2.0 * d)), 2.0) <= 2e+201) {
		tmp = w0 * Math.sqrt((1.0 + ((h * Math.pow(((M * D) * (0.5 / d)), 2.0)) * (-1.0 / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (M * ((M * (h / l)) * Math.pow((D / d), 2.0))))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if math.pow(((M * D) / (2.0 * d)), 2.0) <= 2e+201:
		tmp = w0 * math.sqrt((1.0 + ((h * math.pow(((M * D) * (0.5 / d)), 2.0)) * (-1.0 / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (M * ((M * (h / l)) * math.pow((D / d), 2.0))))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if ((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) <= 2e+201)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(h * (Float64(Float64(M * D) * Float64(0.5 / d)) ^ 2.0)) * Float64(-1.0 / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(M * Float64(Float64(M * Float64(h / l)) * (Float64(D / d) ^ 2.0)))))));
	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 * D) / (2.0 * d)) ^ 2.0) <= 2e+201)
		tmp = w0 * sqrt((1.0 + ((h * (((M * D) * (0.5 / d)) ^ 2.0)) * (-1.0 / l))));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (M * ((M * (h / l)) * ((D / d) ^ 2.0))))));
	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[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 2e+201], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(h * N[Power[N[(N[(M * D), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(M * N[(N[(M * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(D / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 2.00000000000000008e201

   1. Initial program 86.4%

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

      1. times-frac 86.8%

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

   3. Simplified 86.8%

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

      1. associate-*r/ 98.3%

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

      2. clear-num 98.3%

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

      3. frac-times 98.1%

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

      4. div-inv 98.1%

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

      5. associate-*l* 98.3%

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

      6. associate-/r* 98.3%

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

      7. metadata-eval 98.3%

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

   5. Applied egg-rr 98.3%

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

      1. associate-/r/ 98.3%

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

      2. *-commutative 98.3%

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

      3. associate-*r* 98.1%

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

      4. *-commutative 98.1%

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

      5. *-commutative 98.1%

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

   7. Simplified 98.1%

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

   if 2.00000000000000008e201 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 49.6%

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

      1. times-frac 53.2%

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

   3. Simplified 53.2%

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

      1. associate-*r/ 49.7%

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

      2. frac-times 47.9%

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

      3. div-inv 47.9%

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

      4. associate-*l* 49.7%

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

      5. associate-/r* 49.7%

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

      6. metadata-eval 49.7%

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

   5. Applied egg-rr 49.7%

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

   6. Taylor expanded in M around 0 38.9%

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

      1. *-commutative 38.9%

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

      2. times-frac 39.0%

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

      3. unpow2 39.0%

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

      4. *-commutative 39.0%

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

      5. unpow2 39.0%

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

      6. unpow2 39.0%

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

      7. *-commutative 39.0%

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

      8. associate-/l* 40.6%

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

      9. unpow2 40.6%

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

      10. unpow2 40.6%

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

      11. times-frac 47.0%

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

      12. unpow2 47.0%

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

      13. *-commutative 47.0%

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

      14. associate-/l* 45.4%

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

      15. associate-*r/ 47.1%

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

      16. associate-*l* 52.9%

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

      17. associate-*l* 56.6%

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

   8. Simplified 56.6%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2 \cdot 10^{+201}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(h \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}\right) \cdot \frac{-1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(M \cdot \left(\left(M \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{D}{d}\right)}^{2}\right)\right)}\\ \end{array} \]

Alternative 2: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2 \cdot 10^{+201}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(M \cdot \left(\left(M \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{D}{d}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

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 (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 2e+201)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* 0.5 (/ D (/ d M))) 2.0)) l))))
   (* w0 (sqrt (- 1.0 (* 0.25 (* M (* (* M (/ h l)) (pow (/ D d) 2.0)))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (pow(((M * D) / (2.0 * d)), 2.0) <= 2e+201) {
		tmp = w0 * sqrt((1.0 - ((h * pow((0.5 * (D / (d / M))), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (M * ((M * (h / l)) * pow((D / d), 2.0))))));
	}
	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 * d) / (2.0d0 * d_1)) ** 2.0d0) <= 2d+201) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((0.5d0 * (d / (d_1 / m))) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (m * ((m * (h / l)) * ((d / d_1) ** 2.0d0))))))
    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 (Math.pow(((M * D) / (2.0 * d)), 2.0) <= 2e+201) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((0.5 * (D / (d / M))), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (M * ((M * (h / l)) * Math.pow((D / d), 2.0))))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if math.pow(((M * D) / (2.0 * d)), 2.0) <= 2e+201:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((0.5 * (D / (d / M))), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (M * ((M * (h / l)) * math.pow((D / d), 2.0))))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if ((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) <= 2e+201)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(0.5 * Float64(D / Float64(d / M))) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(M * Float64(Float64(M * Float64(h / l)) * (Float64(D / d) ^ 2.0)))))));
	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 * D) / (2.0 * d)) ^ 2.0) <= 2e+201)
		tmp = w0 * sqrt((1.0 - ((h * ((0.5 * (D / (d / M))) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (M * ((M * (h / l)) * ((D / d) ^ 2.0))))));
	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[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 2e+201], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(0.5 * N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(M * N[(N[(M * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(D / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 2.00000000000000008e201

   1. Initial program 86.4%

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

      1. times-frac 86.8%

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

   3. Simplified 86.8%

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

      1. associate-*r/ 98.3%

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

      2. frac-times 98.1%

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

      3. div-inv 98.1%

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

      4. associate-*l* 98.3%

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

      5. associate-/r* 98.3%

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

      6. metadata-eval 98.3%

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

   5. Applied egg-rr 98.3%

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

   6. Taylor expanded in M around 0 98.1%

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

      1. associate-/l* 97.6%

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

   8. Simplified 97.6%

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

   if 2.00000000000000008e201 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 49.6%

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

      1. times-frac 53.2%

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

   3. Simplified 53.2%

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

      1. associate-*r/ 49.7%

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

      2. frac-times 47.9%

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

      3. div-inv 47.9%

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

      4. associate-*l* 49.7%

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

      5. associate-/r* 49.7%

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

      6. metadata-eval 49.7%

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

   5. Applied egg-rr 49.7%

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

   6. Taylor expanded in M around 0 38.9%

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

      1. *-commutative 38.9%

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

      2. times-frac 39.0%

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

      3. unpow2 39.0%

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

      4. *-commutative 39.0%

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

      5. unpow2 39.0%

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

      6. unpow2 39.0%

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

      7. *-commutative 39.0%

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

      8. associate-/l* 40.6%

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

      9. unpow2 40.6%

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

      10. unpow2 40.6%

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

      11. times-frac 47.0%

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

      12. unpow2 47.0%

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

      13. *-commutative 47.0%

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

      14. associate-/l* 45.4%

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

      15. associate-*r/ 47.1%

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

      16. associate-*l* 52.9%

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

      17. associate-*l* 56.6%

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

   8. Simplified 56.6%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2 \cdot 10^{+201}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(M \cdot \left(\left(M \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{D}{d}\right)}^{2}\right)\right)}\\ \end{array} \]

Alternative 3: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \sqrt{1 - 0.25 \cdot \left(h \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)} \end{array} \]

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 (* 0.25 (* h (/ (pow (* M (/ D 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 - (0.25 * (h * (pow((M * (D / 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 - (0.25d0 * (h * (((m * (d / 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 - (0.25 * (h * (Math.pow((M * (D / d)), 2.0) / l)))));
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (0.25 * (h * (math.pow((M * (D / 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(0.25 * Float64(h * Float64((Float64(M * Float64(D / 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 - (0.25 * (h * (((M * (D / 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[(0.25 * N[(h * N[(N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.8%

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

   1. times-frac 79.8%

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

3. Simplified 79.8%

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

   1. associate-*r/ 88.2%

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

   2. frac-times 87.7%

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

   3. div-inv 87.7%

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

   4. associate-*l* 88.2%

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

   5. associate-/r* 88.2%

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

   6. metadata-eval 88.2%

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

5. Applied egg-rr 88.2%

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

6. Taylor expanded in M around 0 87.7%

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

   1. associate-/l* 87.7%

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

8. Simplified 87.7%

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

9. Taylor expanded in D around 0 56.2%

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

    1. *-commutative 56.2%

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

    2. *-commutative 56.2%

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

    3. times-frac 57.0%

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

    4. unpow2 57.0%

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

    5. *-commutative 57.0%

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

    6. associate-*l/ 58.5%

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

    7. unpow2 58.5%

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

    8. unpow2 58.5%

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

    9. times-frac 72.8%

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

    10. unpow2 72.8%

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

11. Simplified 88.6%

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

12. Final simplification 88.6%

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

Alternative 4: 80.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \left(1 + \left(h \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot -0.125\right) \end{array} \]

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 (+ 1.0 (* (* h (/ (pow (* M (/ D d)) 2.0) l)) -0.125))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + ((h * (pow((M * (D / d)), 2.0) / l)) * -0.125));
}

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 * (1.0d0 + ((h * (((m * (d / d_1)) ** 2.0d0) / l)) * (-0.125d0)))
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + ((h * (Math.pow((M * (D / d)), 2.0) / l)) * -0.125));
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * (1.0 + ((h * (math.pow((M * (D / d)), 2.0) / l)) * -0.125))

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * Float64(1.0 + Float64(Float64(h * Float64((Float64(M * Float64(D / d)) ^ 2.0) / l)) * -0.125)))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * (1.0 + ((h * (((M * (D / d)) ^ 2.0) / l)) * -0.125));
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[(1.0 + N[(N[(h * N[(N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.8%

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

   1. times-frac 79.8%

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

3. Simplified 79.8%

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

4. Taylor expanded in M around 0 55.1%

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

   1. +-commutative 55.1%

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

   2. associate-*r/ 55.1%

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

   3. *-commutative 55.1%

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

   4. associate-*r/ 55.1%

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

   5. *-commutative 55.1%

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

   6. fma-def 55.1%

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

   7. *-commutative 55.1%

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

   8. times-frac 56.7%

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

   9. unpow2 56.7%

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

   10. unpow2 56.7%

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

   11. times-frac 69.1%

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

   12. unpow2 69.1%

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

   13. *-commutative 69.1%

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

   14. associate-/l* 65.5%

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

   15. unpow2 65.5%

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

6. Simplified 65.5%

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

7. Taylor expanded in w0 around 0 55.1%

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

9. Simplified 62.4%

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

10. Taylor expanded in D around 0 55.1%

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

    1. times-frac 56.7%

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

    2. unpow2 56.7%

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

    3. unpow2 56.7%

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

    4. times-frac 69.1%

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

    5. unpow2 69.1%

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

    6. unpow2 69.1%

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

    7. associate-*l/ 70.3%

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

    8. associate-*r* 72.3%

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

    9. *-commutative 72.3%

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

    10. associate-*r/ 73.9%

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

    11. unpow2 73.9%

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

    12. swap-sqr 82.4%

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

    13. unpow2 82.4%

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

    14. *-commutative 82.4%

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

12. Simplified 82.4%

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

13. Final simplification 82.4%

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

Alternative 5: 75.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{+107}:\\ \;\;\;\;w0\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+99}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(M \cdot \left(\frac{D}{d} \cdot \frac{\frac{D}{\ell}}{d}\right)\right) \cdot \left(M \cdot h\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 -5.8e+107)
   w0
   (if (<= d 1.85e+99)
     (* w0 (+ 1.0 (* -0.125 (* (* M (* (/ D d) (/ (/ D l) d))) (* M h)))))
     w0)))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= -5.8e+107) {
		tmp = w0;
	} else if (d <= 1.85e+99) {
		tmp = w0 * (1.0 + (-0.125 * ((M * ((D / d) * ((D / l) / d))) * (M * h))));
	} 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 <= (-5.8d+107)) then
        tmp = w0
    else if (d_1 <= 1.85d+99) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((m * ((d / d_1) * ((d / l) / d_1))) * (m * h))))
    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 <= -5.8e+107) {
		tmp = w0;
	} else if (d <= 1.85e+99) {
		tmp = w0 * (1.0 + (-0.125 * ((M * ((D / d) * ((D / l) / d))) * (M * h))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= -5.8e+107:
		tmp = w0
	elif d <= 1.85e+99:
		tmp = w0 * (1.0 + (-0.125 * ((M * ((D / d) * ((D / l) / d))) * (M * h))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= -5.8e+107)
		tmp = w0;
	elseif (d <= 1.85e+99)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(M * Float64(Float64(D / d) * Float64(Float64(D / l) / d))) * Float64(M * h)))));
	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 <= -5.8e+107)
		tmp = w0;
	elseif (d <= 1.85e+99)
		tmp = w0 * (1.0 + (-0.125 * ((M * ((D / d) * ((D / l) / d))) * (M * h))));
	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, -5.8e+107], w0, If[LessEqual[d, 1.85e+99], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(M * N[(N[(D / d), $MachinePrecision] * N[(N[(D / l), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if d < -5.79999999999999975e107 or 1.85000000000000005e99 < d

   1. Initial program 81.5%

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

      1. times-frac 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in M around 0 87.6%

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

   if -5.79999999999999975e107 < d < 1.85000000000000005e99

   1. Initial program 76.8%

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

      1. times-frac 77.4%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in M around 0 55.3%

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

      1. +-commutative 55.3%

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

      2. associate-*r/ 55.3%

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

      3. *-commutative 55.3%

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

      4. associate-*r/ 55.3%

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

      5. *-commutative 55.3%

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

      6. fma-def 55.3%

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

      7. *-commutative 55.3%

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

      8. times-frac 56.7%

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

      9. unpow2 56.7%

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

      10. unpow2 56.7%

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

      11. times-frac 67.6%

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

      12. unpow2 67.6%

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

      13. *-commutative 67.6%

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

      14. associate-/l* 64.9%

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

      15. unpow2 64.9%

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

   6. Simplified 64.9%

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

   7. Taylor expanded in w0 around 0 55.3%

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

   9. Simplified 61.7%

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

   10. Taylor expanded in D around 0 61.7%

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

       1. unpow2 61.7%

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

       2. associate-*l/ 63.2%

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

       3. *-commutative 63.2%

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

   12. Simplified 63.2%

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

       1. times-frac 73.8%

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

   14. Applied egg-rr 73.8%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{+107}:\\ \;\;\;\;w0\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+99}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(M \cdot \left(\frac{D}{d} \cdot \frac{\frac{D}{\ell}}{d}\right)\right) \cdot \left(M \cdot h\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 6: 69.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -1.9 \cdot 10^{+203}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(M \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{h}{\frac{\ell}{M}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= M -1.9e+203)
   (* w0 (+ 1.0 (* -0.125 (* M (* (/ (* D D) (* d d)) (/ h (/ l M)))))))
   w0))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= -1.9e+203) {
		tmp = w0 * (1.0 + (-0.125 * (M * (((D * D) / (d * d)) * (h / (l / 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 (m <= (-1.9d+203)) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (m * (((d * d) / (d_1 * d_1)) * (h / (l / 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 (M <= -1.9e+203) {
		tmp = w0 * (1.0 + (-0.125 * (M * (((D * D) / (d * d)) * (h / (l / M))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= -1.9e+203:
		tmp = w0 * (1.0 + (-0.125 * (M * (((D * D) / (d * d)) * (h / (l / M))))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= -1.9e+203)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(M * Float64(Float64(Float64(D * D) / Float64(d * d)) * Float64(h / Float64(l / 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 (M <= -1.9e+203)
		tmp = w0 * (1.0 + (-0.125 * (M * (((D * D) / (d * d)) * (h / (l / 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[M, -1.9e+203], N[(w0 * N[(1.0 + N[(-0.125 * N[(M * N[(N[(N[(D * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(h / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if M < -1.90000000000000012e203

   1. Initial program 68.2%

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

      1. times-frac 73.3%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in M around 0 28.7%

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

      1. *-commutative 28.7%

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

      2. *-commutative 28.7%

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

      3. associate-/l* 28.7%

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

      4. unpow2 28.7%

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

      5. unpow2 28.7%

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

      6. *-commutative 28.7%

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

      7. unpow2 28.7%

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

   6. Simplified 28.7%

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

   7. Taylor expanded in D around 0 28.7%

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

      1. times-frac 28.7%

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

      2. unpow2 28.7%

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

      3. associate-/l* 28.4%

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

      4. unpow2 28.4%

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

      5. unpow2 28.4%

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

      6. times-frac 29.1%

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

      7. unpow2 29.1%

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

      8. *-commutative 29.1%

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

      9. associate-/l* 29.3%

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

      10. associate-*r/ 29.1%

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

      11. associate-*l* 40.2%

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

      12. associate-*l* 56.9%

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

   9. Simplified 56.9%

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

   10. Taylor expanded in M around 0 61.8%

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

       1. times-frac 50.7%

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

       2. unpow2 50.7%

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

       3. unpow2 50.7%

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

       4. associate-/l* 56.5%

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

   12. Simplified 56.5%

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

   if -1.90000000000000012e203 < M

   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. times-frac 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in M around 0 75.2%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.9 \cdot 10^{+203}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(M \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{h}{\frac{\ell}{M}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 7: 69.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -2.4 \cdot 10^{+202}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot D}{\frac{\ell}{M} \cdot \frac{d}{\frac{M}{\frac{d}{h}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= M -2.4e+202)
   (* w0 (+ 1.0 (* -0.125 (/ (* D D) (* (/ l M) (/ d (/ M (/ d h))))))))
   w0))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= -2.4e+202) {
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / ((l / M) * (d / (M / (d / h)))))));
	} 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 (m <= (-2.4d+202)) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * d) / ((l / m) * (d_1 / (m / (d_1 / h)))))))
    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 (M <= -2.4e+202) {
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / ((l / M) * (d / (M / (d / h)))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= -2.4e+202:
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / ((l / M) * (d / (M / (d / h)))))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= -2.4e+202)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * D) / Float64(Float64(l / M) * Float64(d / Float64(M / Float64(d / h))))))));
	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 (M <= -2.4e+202)
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / ((l / M) * (d / (M / (d / h)))))));
	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[M, -2.4e+202], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * D), $MachinePrecision] / N[(N[(l / M), $MachinePrecision] * N[(d / N[(M / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if M < -2.4000000000000002e202

   1. Initial program 68.2%

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

      1. times-frac 73.3%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in M around 0 28.7%

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

      1. *-commutative 28.7%

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

      2. *-commutative 28.7%

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

      3. associate-/l* 28.7%

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

      4. unpow2 28.7%

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

      5. unpow2 28.7%

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

      6. *-commutative 28.7%

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

      7. unpow2 28.7%

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

   6. Simplified 28.7%

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

   7. Taylor expanded in l around 0 28.7%

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

      1. unpow2 28.7%

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

      2. associate-*r* 34.0%

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

      3. times-frac 50.7%

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

      4. unpow2 50.7%

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

      5. associate-/l* 51.0%

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

      6. associate-/l* 56.5%

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

   9. Simplified 56.5%

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

   if -2.4000000000000002e202 < M

   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. times-frac 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in M around 0 75.2%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.4 \cdot 10^{+202}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot D}{\frac{\ell}{M} \cdot \frac{d}{\frac{M}{\frac{d}{h}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 8: 67.5% 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 78.8%

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

   1. times-frac 79.8%

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

3. Simplified 79.8%

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

4. Taylor expanded in M around 0 72.8%

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

5. Final simplification 72.8%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023189 
(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))))))