Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 86.6%

Time: 15.9s

Alternatives: 10

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.8% 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: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+235}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(h \cdot \left(M \cdot \frac{M}{d}\right)\right) \cdot \frac{\sqrt[3]{0.015625}}{d}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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) 5e+235)
   (* w0 (sqrt (- 1.0 (/ (* (pow (* M (* 0.5 (/ D d))) 2.0) h) l))))
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* D (/ D l)) (* (* h (* M (/ M d))) (/ (cbrt 0.015625) d))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (pow(((M * D) / (2.0 * d)), 2.0) <= 5e+235) {
		tmp = w0 * sqrt((1.0 - ((pow((M * (0.5 * (D / d))), 2.0) * h) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - ((D * (D / l)) * ((h * (M * (M / d))) * (cbrt(0.015625) / d)))));
	}
	return tmp;
}

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (Math.pow(((M * D) / (2.0 * d)), 2.0) <= 5e+235) {
		tmp = w0 * Math.sqrt((1.0 - ((Math.pow((M * (0.5 * (D / d))), 2.0) * h) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((D * (D / l)) * ((h * (M * (M / d))) * (Math.cbrt(0.015625) / d)))));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if ((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) <= 5e+235)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * h) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D * Float64(D / l)) * Float64(Float64(h * Float64(M * Float64(M / d))) * Float64(cbrt(0.015625) / d))))));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 5e+235], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[0.015625, 1/3], $MachinePrecision] / d), $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) < 5.00000000000000027e235

   1. Initial program 90.0%

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

      1. *-commutative 90.0%

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

      2. times-frac 89.6%

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

   3. Simplified 89.6%

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

      1. unpow2 89.6%

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

      2. associate-*l* 90.2%

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

      3. div-inv 90.2%

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

      4. associate-*l* 89.6%

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

      5. times-frac 89.6%

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

      6. *-un-lft-identity 89.6%

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

      7. *-commutative 89.6%

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

      8. clear-num 89.6%

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

      9. div-inv 89.6%

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

      10. div-inv 89.6%

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

      11. associate-*l* 90.1%

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

      12. times-frac 90.1%

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

      13. *-un-lft-identity 90.1%

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

      14. *-commutative 90.1%

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

      15. clear-num 90.1%

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

      16. div-inv 90.1%

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

      17. associate-*l* 89.5%

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

   5. Applied egg-rr 96.9%

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

   if 5.00000000000000027e235 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   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. *-commutative 46.7%

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

      2. times-frac 49.7%

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

   3. Simplified 49.7%

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

      1. unpow2 49.7%

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

      2. associate-*l* 57.0%

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

      3. div-inv 57.0%

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

      4. associate-*l* 57.0%

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

      5. times-frac 57.1%

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

      6. *-un-lft-identity 57.1%

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

      7. *-commutative 57.1%

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

      8. clear-num 57.0%

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

      9. div-inv 57.0%

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

      10. div-inv 57.0%

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

      11. associate-*l* 59.9%

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

      12. times-frac 60.0%

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

      13. *-un-lft-identity 60.0%

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

      14. *-commutative 60.0%

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

      15. clear-num 59.9%

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

      16. div-inv 59.9%

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

      17. associate-*l* 52.6%

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

   5. Applied egg-rr 51.4%

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

      1. add-cbrt-cube 45.4%

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

   7. Applied egg-rr 45.4%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\sqrt[3]{\left(\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h\right) \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h\right)\right) \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h\right)}}}{\ell}} \]
   8. Step-by-step derivation

      1. associate-*l* 45.4%

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

      2. cube-unmult 45.4%

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

      3. *-commutative 45.4%

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

      4. associate-*r* 45.4%

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

   9. Simplified 45.4%

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

   10. Taylor expanded in h around 0 39.7%

       \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{D}^{2} \cdot \left(\sqrt[3]{0.015625} \cdot \left(h \cdot {M}^{2}\right)\right)}{\ell \cdot {d}^{2}}}} \]
   11. Step-by-step derivation

       1. times-frac 38.3%

          \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{D}^{2}}{\ell} \cdot \frac{\sqrt[3]{0.015625} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}}} \]

       2. unpow2 38.3%

          \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{\sqrt[3]{0.015625} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]

       3. associate-*r/ 42.1%

          \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right)} \cdot \frac{\sqrt[3]{0.015625} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}} \]

       4. *-commutative 42.1%

          \[\leadsto w0 \cdot \sqrt{1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \frac{\color{blue}{\left(h \cdot {M}^{2}\right) \cdot \sqrt[3]{0.015625}}}{{d}^{2}}} \]

       5. unpow2 42.1%

          \[\leadsto w0 \cdot \sqrt{1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \frac{\left(h \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \sqrt[3]{0.015625}}{{d}^{2}}} \]

       6. *-commutative 42.1%

          \[\leadsto w0 \cdot \sqrt{1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \frac{\color{blue}{\left(\left(M \cdot M\right) \cdot h\right)} \cdot \sqrt[3]{0.015625}}{{d}^{2}}} \]

       7. unpow2 42.1%

          \[\leadsto w0 \cdot \sqrt{1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \sqrt[3]{0.015625}}{\color{blue}{d \cdot d}}} \]

       8. times-frac 47.8%

          \[\leadsto w0 \cdot \sqrt{1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot h}{d} \cdot \frac{\sqrt[3]{0.015625}}{d}\right)}} \]

       9. associate-*l/ 49.3%

          \[\leadsto w0 \cdot \sqrt{1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\color{blue}{\left(\frac{M \cdot M}{d} \cdot h\right)} \cdot \frac{\sqrt[3]{0.015625}}{d}\right)} \]

       10. *-commutative 49.3%

           \[\leadsto w0 \cdot \sqrt{1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\color{blue}{\left(h \cdot \frac{M \cdot M}{d}\right)} \cdot \frac{\sqrt[3]{0.015625}}{d}\right)} \]

       11. associate-/l* 55.4%

           \[\leadsto w0 \cdot \sqrt{1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(h \cdot \color{blue}{\frac{M}{\frac{d}{M}}}\right) \cdot \frac{\sqrt[3]{0.015625}}{d}\right)} \]

       12. associate-/r/ 55.4%

           \[\leadsto w0 \cdot \sqrt{1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(h \cdot \color{blue}{\left(\frac{M}{d} \cdot M\right)}\right) \cdot \frac{\sqrt[3]{0.015625}}{d}\right)} \]

   12. Simplified 55.4%

       \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(h \cdot \left(\frac{M}{d} \cdot M\right)\right) \cdot \frac{\sqrt[3]{0.015625}}{d}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+235}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D \cdot \frac{D}{\ell}\right) \cdot \left(\left(h \cdot \left(M \cdot \frac{M}{d}\right)\right) \cdot \frac{\sqrt[3]{0.015625}}{d}\right)}\\ \end{array} \]

Alternative 2: 87.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.8%

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

      1. *-commutative 84.8%

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

      2. times-frac 84.9%

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

   3. Simplified 84.9%

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

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

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 5.6%

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

   3. Simplified 5.6%

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

   4. Taylor expanded in M around 0 33.3%

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

      1. fma-def 33.3%

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

      2. times-frac 38.9%

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

      3. *-commutative 38.9%

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

      4. unpow2 38.9%

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

      5. associate-/l* 38.9%

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

      6. *-commutative 38.9%

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

      7. *-commutative 38.9%

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

      8. unpow2 38.9%

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

      9. times-frac 56.0%

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

      10. *-commutative 56.0%

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

      11. unpow2 56.0%

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

      12. associate-*l* 56.0%

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

   6. Simplified 56.0%

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

      1. associate-*l/ 62.5%

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

      2. frac-times 44.4%

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

   8. Applied egg-rr 44.4%

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

      1. associate-/r/ 44.4%

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

      2. times-frac 62.5%

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

      3. associate-*r* 62.5%

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

      4. unpow2 62.5%

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

      5. associate-/l* 72.6%

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

      6. unpow2 72.6%

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

   10. Simplified 72.6%

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

   11. Taylor expanded in M around 0 62.5%

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

       1. unpow2 62.5%

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

       2. associate-*l/ 78.2%

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

       3. *-commutative 78.2%

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

       4. associate-/l* 83.9%

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

   13. Simplified 83.9%

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

4. Final simplification 84.8%

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

Alternative 3: 85.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. *-commutative 78.9%

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

   2. times-frac 79.3%

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

3. Simplified 79.3%

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

   1. unpow2 79.3%

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

   2. associate-*l* 81.6%

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

   3. div-inv 81.6%

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

   4. associate-*l* 81.2%

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

   5. times-frac 81.2%

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

   6. *-un-lft-identity 81.2%

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

   7. *-commutative 81.2%

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

   8. clear-num 81.2%

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

   9. div-inv 81.2%

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

   10. div-inv 81.2%

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

   11. associate-*l* 82.3%

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

   12. times-frac 82.3%

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

   13. *-un-lft-identity 82.3%

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

   14. *-commutative 82.3%

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

   15. clear-num 82.3%

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

   16. div-inv 82.3%

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

   17. associate-*l* 80.0%

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

5. Applied egg-rr 85.2%

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

6. Final simplification 85.2%

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

Alternative 4: 81.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 2.5 \cdot 10^{-26}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)\\ \mathbf{elif}\;D \leq 1.05 \cdot 10^{+150}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot \left(d \cdot \ell\right)}{M \cdot \left(M \cdot h\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, D \cdot \frac{D \cdot \left(\left(h \cdot \frac{M}{\frac{d}{M}}\right) \cdot \frac{w0}{d}\right)}{\ell}, w0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 2.5e-26)
   (* w0 (+ 1.0 (* -0.125 (* (* D (/ D l)) (* (/ M d) (/ (* M h) d))))))
   (if (<= D 1.05e+150)
     (* w0 (sqrt (- 1.0 (* 0.25 (/ (* D D) (/ (* d (* d l)) (* M (* M h))))))))
     (fma -0.125 (* D (/ (* D (* (* h (/ M (/ d M))) (/ w0 d))) l)) w0))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 2.5e-26) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M / d) * ((M * h) / d)))));
	} else if (D <= 1.05e+150) {
		tmp = w0 * sqrt((1.0 - (0.25 * ((D * D) / ((d * (d * l)) / (M * (M * h)))))));
	} else {
		tmp = fma(-0.125, (D * ((D * ((h * (M / (d / M))) * (w0 / d))) / l)), w0);
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 2.5e-26)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(M / d) * Float64(Float64(M * h) / d))))));
	elseif (D <= 1.05e+150)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(D * D) / Float64(Float64(d * Float64(d * l)) / Float64(M * Float64(M * h))))))));
	else
		tmp = fma(-0.125, Float64(D * Float64(Float64(D * Float64(Float64(h * Float64(M / Float64(d / M))) * Float64(w0 / d))) / l)), w0);
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 2.5e-26], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 1.05e+150], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(D * D), $MachinePrecision] / N[(N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision] / N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.125 * N[(D * N[(N[(D * N[(N[(h * N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if D < 2.5000000000000001e-26

   1. Initial program 81.1%

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

      1. *-commutative 81.1%

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

      2. times-frac 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in M around 0 53.7%

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

      1. associate-*r/ 53.7%

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

      2. *-commutative 53.7%

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

      3. associate-*r/ 53.7%

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

      4. *-commutative 53.7%

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

      5. associate-/l* 52.8%

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

      6. unpow2 52.8%

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

      7. unpow2 52.8%

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

      8. *-commutative 52.8%

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

      9. unpow2 52.8%

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

   6. Simplified 52.8%

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

   7. Taylor expanded in D around 0 53.7%

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

      1. *-commutative 53.7%

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

      2. times-frac 56.7%

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

      3. unpow2 56.7%

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

      4. associate-*r/ 59.5%

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

      5. unpow2 59.5%

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

      6. associate-*r* 62.0%

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

      7. unpow2 62.0%

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

   9. Simplified 62.0%

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

       1. times-frac 74.7%

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

   11. Applied egg-rr 74.7%

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

   if 2.5000000000000001e-26 < D < 1.04999999999999999e150

   1. Initial program 67.5%

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

      1. *-commutative 67.5%

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

      2. times-frac 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in M around 0 58.1%

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

      1. associate-*r/ 58.1%

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

      2. *-commutative 58.1%

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

      3. associate-*r/ 58.1%

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

      4. associate-/l* 67.4%

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

      5. unpow2 67.4%

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

      6. *-commutative 67.4%

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

      7. unpow2 67.4%

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

      8. associate-*l* 76.3%

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

      9. *-commutative 76.3%

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

      10. unpow2 76.3%

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

      11. associate-*l* 76.3%

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

   6. Simplified 76.3%

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

   if 1.04999999999999999e150 < D

   1. Initial program 75.5%

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

      1. *-commutative 75.5%

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

      2. times-frac 75.5%

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

   3. Simplified 75.5%

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

   4. Taylor expanded in M around 0 27.0%

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

      1. fma-def 27.0%

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

      2. times-frac 27.2%

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

      3. *-commutative 27.2%

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

      4. unpow2 27.2%

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

      5. associate-/l* 40.3%

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

      6. *-commutative 40.3%

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

      7. *-commutative 40.3%

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

      8. unpow2 40.3%

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

      9. times-frac 45.0%

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

      10. *-commutative 45.0%

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

      11. unpow2 45.0%

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

      12. associate-*l* 49.4%

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

   6. Simplified 49.4%

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

      1. associate-*l/ 58.1%

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

      2. frac-times 57.7%

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

   8. Applied egg-rr 57.7%

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

      1. associate-/r/ 64.7%

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

      2. times-frac 64.8%

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

      3. associate-*r* 60.6%

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

      4. unpow2 60.6%

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

      5. associate-/l* 56.0%

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

      6. unpow2 56.0%

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

   10. Simplified 56.0%

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

   11. Taylor expanded in M around 0 60.6%

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

       1. unpow2 60.6%

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

       2. associate-*l/ 60.6%

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

       3. *-commutative 60.6%

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

       4. associate-/l* 69.2%

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

   13. Simplified 69.2%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.5 \cdot 10^{-26}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)\\ \mathbf{elif}\;D \leq 1.05 \cdot 10^{+150}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot \left(d \cdot \ell\right)}{M \cdot \left(M \cdot h\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, D \cdot \frac{D \cdot \left(\left(h \cdot \frac{M}{\frac{d}{M}}\right) \cdot \frac{w0}{d}\right)}{\ell}, w0\right)\\ \end{array} \]

Alternative 5: 74.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4.1 \cdot 10^{-75}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, D \cdot \frac{D \cdot \left(\left(h \cdot \frac{M}{\frac{d}{M}}\right) \cdot \frac{w0}{d}\right)}{\ell}, w0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 4.1e-75)
   (* w0 (+ 1.0 (* -0.125 (* (* D (/ D l)) (* (/ M d) (/ (* M h) d))))))
   (fma -0.125 (* D (/ (* D (* (* h (/ M (/ d M))) (/ w0 d))) l)) w0)))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 4.1e-75) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M / d) * ((M * h) / d)))));
	} else {
		tmp = fma(-0.125, (D * ((D * ((h * (M / (d / M))) * (w0 / d))) / l)), w0);
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 4.1e-75)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(M / d) * Float64(Float64(M * h) / d))))));
	else
		tmp = fma(-0.125, Float64(D * Float64(Float64(D * Float64(Float64(h * Float64(M / Float64(d / M))) * Float64(w0 / d))) / l)), w0);
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 4.1e-75], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.125 * N[(D * N[(N[(D * N[(N[(h * N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.10000000000000002e-75

   1. Initial program 76.1%

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

      1. *-commutative 76.1%

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

      2. times-frac 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in M around 0 42.7%

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

      1. associate-*r/ 42.7%

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

      2. *-commutative 42.7%

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

      3. associate-*r/ 42.7%

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

      4. *-commutative 42.7%

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

      5. associate-/l* 42.2%

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

      6. unpow2 42.2%

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

      7. unpow2 42.2%

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

      8. *-commutative 42.2%

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

      9. unpow2 42.2%

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

   6. Simplified 42.2%

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

   7. Taylor expanded in D around 0 42.7%

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

      1. *-commutative 42.7%

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

      2. times-frac 46.1%

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

      3. unpow2 46.1%

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

      4. associate-*r/ 49.1%

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

      5. unpow2 49.1%

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

      6. associate-*r* 52.0%

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

      7. unpow2 52.0%

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

   9. Simplified 52.0%

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

       1. times-frac 66.8%

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

   11. Applied egg-rr 66.8%

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

   if 4.10000000000000002e-75 < d

   1. Initial program 85.0%

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

      1. *-commutative 85.0%

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

      2. times-frac 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in M around 0 67.6%

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

      1. fma-def 67.6%

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

      2. times-frac 68.7%

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

      3. *-commutative 68.7%

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

      4. unpow2 68.7%

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

      5. associate-/l* 73.0%

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

      6. *-commutative 73.0%

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

      7. *-commutative 73.0%

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

      8. unpow2 73.0%

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

      9. times-frac 75.9%

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

      10. *-commutative 75.9%

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

      11. unpow2 75.9%

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

      12. associate-*l* 77.2%

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

   6. Simplified 77.2%

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

      1. associate-*l/ 78.6%

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

      2. frac-times 77.0%

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

   8. Applied egg-rr 77.0%

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

      1. associate-/r/ 78.9%

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

      2. times-frac 80.6%

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

      3. associate-*r* 79.3%

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

      4. unpow2 79.3%

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

      5. associate-/l* 80.5%

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

      6. unpow2 80.5%

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

   10. Simplified 80.5%

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

   11. Taylor expanded in M around 0 79.3%

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

       1. unpow2 79.3%

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

       2. associate-*l/ 80.6%

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

       3. *-commutative 80.6%

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

       4. associate-/l* 86.8%

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

   13. Simplified 86.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.1 \cdot 10^{-75}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, D \cdot \frac{D \cdot \left(\left(h \cdot \frac{M}{\frac{d}{M}}\right) \cdot \frac{w0}{d}\right)}{\ell}, w0\right)\\ \end{array} \]

Alternative 6: 77.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.1 \cdot 10^{-185}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{\frac{D}{d} \cdot \frac{D}{\ell}}{\frac{\frac{d}{M}}{M}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 2.1e-185)
   w0
   (* w0 (+ 1.0 (* -0.125 (* h (/ (* (/ D d) (/ D l)) (/ (/ d M) M))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.1e-185) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (h * (((D / d) * (D / l)) / ((d / M) / M)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 2.1d-185) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (h * (((d / d_1) * (d / l)) / ((d_1 / m) / m)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.1e-185) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (h * (((D / d) * (D / l)) / ((d / M) / M)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.1e-185:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (h * (((D / d) * (D / l)) / ((d / M) / M)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.1e-185)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(h * Float64(Float64(Float64(D / d) * Float64(D / l)) / Float64(Float64(d / M) / M))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.1e-185)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (h * (((D / d) * (D / l)) / ((d / M) / M)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 2.1e-185], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(h * N[(N[(N[(D / d), $MachinePrecision] * N[(D / l), $MachinePrecision]), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.1e-185

   1. Initial program 82.5%

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

      1. *-commutative 82.5%

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

      2. times-frac 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in M around 0 76.5%

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

   if 2.1e-185 < M

   1. Initial program 71.4%

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

      1. *-commutative 71.4%

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

      2. times-frac 73.7%

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

   3. Simplified 73.7%

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

   4. Taylor expanded in M around 0 46.3%

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

      1. associate-*r/ 46.3%

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

      2. *-commutative 46.3%

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

      3. associate-*r/ 46.3%

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

      4. *-commutative 46.3%

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

      5. associate-/l* 48.7%

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

      6. unpow2 48.7%

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

      7. unpow2 48.7%

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

      8. *-commutative 48.7%

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

      9. unpow2 48.7%

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

   6. Simplified 48.7%

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

   7. Taylor expanded in D around 0 46.3%

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

      1. *-commutative 46.3%

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

      2. times-frac 51.0%

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

      3. unpow2 51.0%

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

      4. associate-*r/ 55.3%

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

      5. unpow2 55.3%

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

      6. associate-*r* 57.7%

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

      7. unpow2 57.7%

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

   9. Simplified 57.7%

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

   10. Taylor expanded in D around 0 46.3%

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

       1. times-frac 49.8%

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

       2. unpow2 49.8%

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

       3. times-frac 46.3%

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

       4. unpow2 46.3%

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

       5. associate-*r* 46.9%

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

       6. *-commutative 46.9%

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

       7. times-frac 54.6%

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

       8. unpow2 54.6%

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

       9. *-commutative 54.6%

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

       10. unpow2 54.6%

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

       11. associate-/l* 54.6%

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

       12. unpow2 54.6%

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

   12. Simplified 54.6%

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

   13. Taylor expanded in D around 0 46.3%

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

       1. *-commutative 46.3%

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

       2. unpow2 46.3%

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

       3. *-commutative 46.3%

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

       4. unpow2 46.3%

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

       5. associate-*l* 46.9%

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

       6. times-frac 54.6%

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

       7. associate-/l* 54.6%

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

       8. associate-/l/ 56.0%

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

       9. unpow2 56.0%

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

       10. associate-*l/ 60.5%

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

       11. associate-*r/ 60.5%

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

       12. associate-*l/ 61.2%

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

       13. *-commutative 61.2%

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

       14. associate-/r/ 63.7%

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

       15. *-commutative 63.7%

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

       16. associate-/l* 62.4%

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

       17. *-commutative 62.4%

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

       18. associate-/r* 67.2%

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

   15. Simplified 67.2%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.1 \cdot 10^{-185}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{\frac{D}{d} \cdot \frac{D}{\ell}}{\frac{\frac{d}{M}}{M}}\right)\right)\\ \end{array} \]

Alternative 7: 73.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.05 \cdot 10^{-65}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 1.05e-65)
   (* w0 (+ 1.0 (* -0.125 (* (* D (/ D l)) (* (/ M d) (/ (* M h) d))))))
   (* w0 (+ 1.0 (* -0.125 (/ D (/ (* (/ l (* M M)) (/ (* d d) h)) D)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.05e-65) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M / d) * ((M * h) / d)))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D / (((l / (M * M)) * ((d * d) / h)) / D))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 1.05d-65) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * (d / l)) * ((m / d_1) * ((m * h) / d_1)))))
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (d / (((l / (m * m)) * ((d_1 * d_1) / h)) / d))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.05e-65) {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M / d) * ((M * h) / d)))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D / (((l / (M * M)) * ((d * d) / h)) / D))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 1.05e-65:
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M / d) * ((M * h) / d)))))
	else:
		tmp = w0 * (1.0 + (-0.125 * (D / (((l / (M * M)) * ((d * d) / h)) / D))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 1.05e-65)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(M / d) * Float64(Float64(M * h) / d))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D / Float64(Float64(Float64(l / Float64(M * M)) * Float64(Float64(d * d) / h)) / D)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 1.05e-65)
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / l)) * ((M / d) * ((M * h) / d)))));
	else
		tmp = w0 * (1.0 + (-0.125 * (D / (((l / (M * M)) * ((d * d) / h)) / D))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 1.05e-65], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(D / N[(N[(N[(l / N[(M * M), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.05000000000000001e-65

   1. Initial program 76.4%

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

      1. *-commutative 76.4%

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

      2. times-frac 75.4%

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

   3. Simplified 75.4%

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

   4. Taylor expanded in M around 0 42.8%

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

      1. associate-*r/ 42.8%

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

      2. *-commutative 42.8%

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

      3. associate-*r/ 42.8%

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

      4. *-commutative 42.8%

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

      5. associate-/l* 42.3%

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

      6. unpow2 42.3%

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

      7. unpow2 42.3%

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

      8. *-commutative 42.3%

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

      9. unpow2 42.3%

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

   6. Simplified 42.3%

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

   7. Taylor expanded in D around 0 42.8%

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

      1. *-commutative 42.8%

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

      2. times-frac 46.2%

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

      3. unpow2 46.2%

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

      4. associate-*r/ 49.2%

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

      5. unpow2 49.2%

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

      6. associate-*r* 52.0%

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

      7. unpow2 52.0%

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

   9. Simplified 52.0%

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

       1. times-frac 66.6%

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

   11. Applied egg-rr 66.6%

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

   if 1.05000000000000001e-65 < d

   1. Initial program 84.6%

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

      1. *-commutative 84.6%

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

      2. times-frac 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in M around 0 69.2%

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

      1. associate-*r/ 69.2%

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

      2. *-commutative 69.2%

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

      3. associate-*r/ 69.2%

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

      4. *-commutative 69.2%

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

      5. associate-/l* 71.8%

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

      6. unpow2 71.8%

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

      7. unpow2 71.8%

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

      8. *-commutative 71.8%

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

      9. unpow2 71.8%

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

   6. Simplified 71.8%

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

      1. *-un-lft-identity 71.8%

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

      2. associate-/l* 78.3%

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

      3. times-frac 78.4%

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

   8. Applied egg-rr 78.4%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.05 \cdot 10^{-65}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D}{\frac{\frac{\ell}{M \cdot M} \cdot \frac{d \cdot d}{h}}{D}}\right)\\ \end{array} \]

Alternative 8: 70.7% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.1 \cdot 10^{+147}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \left(\frac{D}{d} \cdot \left(h \cdot \left(M \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{w0}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 2.1e+147)
   w0
   (* -0.125 (* (* D (* (/ D d) (* h (* M (/ M d))))) (/ w0 l)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.1e+147) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * ((D / d) * (h * (M * (M / d))))) * (w0 / l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 2.1d+147) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((d * ((d / d_1) * (h * (m * (m / d_1))))) * (w0 / l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.1e+147) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * ((D / d) * (h * (M * (M / d))))) * (w0 / l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.1e+147:
		tmp = w0
	else:
		tmp = -0.125 * ((D * ((D / d) * (h * (M * (M / d))))) * (w0 / l))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.1e+147)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(D * Float64(Float64(D / d) * Float64(h * Float64(M * Float64(M / d))))) * Float64(w0 / l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.1e+147)
		tmp = w0;
	else
		tmp = -0.125 * ((D * ((D / d) * (h * (M * (M / d))))) * (w0 / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 2.1e+147], w0, N[(-0.125 * N[(N[(D * N[(N[(D / d), $MachinePrecision] * N[(h * N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.10000000000000006e147

   1. Initial program 79.5%

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

      1. *-commutative 79.5%

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

      2. times-frac 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in M around 0 72.5%

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

   if 2.10000000000000006e147 < M

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

      2. times-frac 72.7%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in M around 0 32.8%

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

      1. associate-*r/ 32.8%

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

      2. *-commutative 32.8%

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

      3. associate-*r/ 32.8%

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

      4. *-commutative 32.8%

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

      5. associate-/l* 32.8%

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

      6. unpow2 32.8%

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

      7. unpow2 32.8%

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

      8. *-commutative 32.8%

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

      9. unpow2 32.8%

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

   6. Simplified 32.8%

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

   7. Taylor expanded in D around 0 32.8%

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

      1. *-commutative 32.8%

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

      2. times-frac 33.0%

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

      3. unpow2 33.0%

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

      4. associate-*r/ 37.0%

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

      5. unpow2 37.0%

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

      6. associate-*r* 45.2%

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

      7. unpow2 45.2%

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

   9. Simplified 45.2%

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

   10. Taylor expanded in D around inf 32.8%

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

   12. Simplified 32.9%

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

   13. Taylor expanded in D around 0 32.8%

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

       1. associate-/l* 32.8%

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

       2. *-rgt-identity 32.8%

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

       3. *-commutative 32.8%

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

       4. times-frac 32.4%

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

       5. unpow2 32.4%

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

       6. unpow2 32.4%

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

       7. associate-*l* 32.9%

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

       8. *-commutative 32.9%

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

       9. times-frac 37.6%

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

       10. *-commutative 37.6%

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

       11. times-frac 37.6%

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

   15. Simplified 41.6%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.1 \cdot 10^{+147}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \left(\frac{D}{d} \cdot \left(h \cdot \left(M \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{w0}{\ell}\right)\\ \end{array} \]

Alternative 9: 70.6% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.1 \cdot 10^{+137}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{w0}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.1e+137)
   w0
   (* -0.125 (* (* D (/ D l)) (/ w0 (/ (* d d) (* h (* M M))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.1e+137) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * (D / l)) * (w0 / ((d * d) / (h * (M * M)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 <= 1.1d+137) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((d * (d / l)) * (w0 / ((d_1 * d_1) / (h * (m * m)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.1e+137) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * (D / l)) * (w0 / ((d * d) / (h * (M * M)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.1e+137:
		tmp = w0
	else:
		tmp = -0.125 * ((D * (D / l)) * (w0 / ((d * d) / (h * (M * M)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.1e+137)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(D * Float64(D / l)) * Float64(w0 / Float64(Float64(d * d) / Float64(h * Float64(M * M))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.1e+137)
		tmp = w0;
	else
		tmp = -0.125 * ((D * (D / l)) * (w0 / ((d * d) / (h * (M * M)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.1e+137], w0, N[(-0.125 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(w0 / N[(N[(d * d), $MachinePrecision] / N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.10000000000000008e137

   1. Initial program 79.5%

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

      1. *-commutative 79.5%

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

      2. times-frac 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in M around 0 72.5%

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

   if 1.10000000000000008e137 < M

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

      2. times-frac 72.7%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in M around 0 32.8%

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

      1. associate-*r/ 32.8%

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

      2. *-commutative 32.8%

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

      3. associate-*r/ 32.8%

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

      4. *-commutative 32.8%

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

      5. associate-/l* 32.8%

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

      6. unpow2 32.8%

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

      7. unpow2 32.8%

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

      8. *-commutative 32.8%

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

      9. unpow2 32.8%

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

   6. Simplified 32.8%

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

   7. Taylor expanded in D around 0 32.8%

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

      1. *-commutative 32.8%

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

      2. times-frac 33.0%

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

      3. unpow2 33.0%

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

      4. associate-*r/ 37.0%

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

      5. unpow2 37.0%

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

      6. associate-*r* 45.2%

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

      7. unpow2 45.2%

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

   9. Simplified 45.2%

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

   10. Taylor expanded in D around inf 32.8%

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

   12. Simplified 32.9%

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

   13. Taylor expanded in D around 0 32.8%

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

       1. *-commutative 32.8%

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

       2. times-frac 33.0%

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

       3. unpow2 33.0%

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

       4. associate-*r/ 37.0%

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

       5. associate-/l* 37.0%

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

       6. unpow2 37.0%

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

       7. unpow2 37.0%

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

       8. *-commutative 37.0%

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

   15. Simplified 37.0%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.1 \cdot 10^{+137}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{w0}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)\\ \end{array} \]

Alternative 10: 67.8% accurate, 216.0× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d) :precision binary64 w0)

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. *-commutative 78.9%

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

   2. times-frac 79.3%

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

3. Simplified 79.3%

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

4. Taylor expanded in M around 0 68.8%

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

5. Final simplification 68.8%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023250 
(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))))))