Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.9% → 87.3%

Time: 14.9s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+151}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot e^{0.5 \cdot \mathsf{fma}\left(-2, \log d, \log \left(\frac{-0.25 \cdot \left(h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right)}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ (* M D) (* 2.0 d)) 5e+151)
   (* w0 (sqrt (- 1.0 (/ (* (pow (* M (* 0.5 (/ D d))) 2.0) h) l))))
   (*
    w0
    (exp
     (*
      0.5
      (fma -2.0 (log d) (log (/ (* -0.25 (* h (* (* M D) (* M D)))) l))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 5e+151) {
		tmp = w0 * sqrt((1.0 - ((pow((M * (0.5 * (D / d))), 2.0) * h) / l)));
	} else {
		tmp = w0 * exp((0.5 * fma(-2.0, log(d), log(((-0.25 * (h * ((M * D) * (M * D)))) / l)))));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
d = abs(d)
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(Float64(M * D) / Float64(2.0 * d)) <= 5e+151)
		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 * exp(Float64(0.5 * fma(-2.0, log(d), log(Float64(Float64(-0.25 * Float64(h * Float64(Float64(M * D) * Float64(M * D)))) / l))))));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 5e+151], 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[Exp[N[(0.5 * N[(-2.0 * N[Log[d], $MachinePrecision] + N[Log[N[(N[(-0.25 * N[(h * N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 5.0000000000000002e151

   1. Initial program 87.9%

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

      1. *-commutative 87.9%

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

      2. times-frac 87.1%

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

   3. Simplified 87.1%

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

      1. unpow2 87.1%

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

      2. associate-*l* 87.9%

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

      3. div-inv 88.0%

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

      4. associate-*l* 87.5%

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

      5. times-frac 87.5%

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

      6. *-un-lft-identity 87.5%

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

      7. *-commutative 87.5%

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

      8. clear-num 87.5%

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

      9. div-inv 87.5%

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

      10. div-inv 87.5%

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

      11. associate-*l* 88.4%

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

      12. times-frac 88.4%

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

      13. *-un-lft-identity 88.4%

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

      14. *-commutative 88.4%

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

      15. clear-num 88.4%

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

      16. div-inv 88.4%

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

      17. associate-*l* 87.5%

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

   5. Applied egg-rr 91.4%

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

   if 5.0000000000000002e151 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 54.6%

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

      1. *-commutative 54.6%

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

      2. times-frac 58.8%

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

   3. Simplified 58.8%

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

   5. Applied egg-rr 58.6%

      \[\leadsto w0 \cdot \color{blue}{e^{\mathsf{log1p}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-\frac{h}{\ell}\right)\right) \cdot 0.5}} \]

   6. Taylor expanded in d around 0 22.5%

      \[\leadsto w0 \cdot e^{\color{blue}{\left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right) + -2 \cdot \log d\right)} \cdot 0.5} \]
   7. Step-by-step derivation

      1. +-commutative 22.5%

         \[\leadsto w0 \cdot e^{\color{blue}{\left(-2 \cdot \log d + \log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right)\right)} \cdot 0.5} \]

      2. fma-def 22.5%

         \[\leadsto w0 \cdot e^{\color{blue}{\mathsf{fma}\left(-2, \log d, \log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell}\right)\right)} \cdot 0.5} \]

      3. associate-*r/ 22.5%

         \[\leadsto w0 \cdot e^{\mathsf{fma}\left(-2, \log d, \log \color{blue}{\left(\frac{-0.25 \cdot \left({D}^{2} \cdot \left(h \cdot {M}^{2}\right)\right)}{\ell}\right)}\right) \cdot 0.5} \]

      4. *-commutative 22.5%

         \[\leadsto w0 \cdot e^{\mathsf{fma}\left(-2, \log d, \log \left(\frac{-0.25 \cdot \left({D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}\right)}{\ell}\right)\right) \cdot 0.5} \]

      5. associate-*r* 22.5%

         \[\leadsto w0 \cdot e^{\mathsf{fma}\left(-2, \log d, \log \left(\frac{-0.25 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{\ell}\right)\right) \cdot 0.5} \]

      6. unpow2 22.5%

         \[\leadsto w0 \cdot e^{\mathsf{fma}\left(-2, \log d, \log \left(\frac{-0.25 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h\right)}{\ell}\right)\right) \cdot 0.5} \]

      7. unpow2 22.5%

         \[\leadsto w0 \cdot e^{\mathsf{fma}\left(-2, \log d, \log \left(\frac{-0.25 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{\ell}\right)\right) \cdot 0.5} \]

      8. unswap-sqr 26.9%

         \[\leadsto w0 \cdot e^{\mathsf{fma}\left(-2, \log d, \log \left(\frac{-0.25 \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right)}{\ell}\right)\right) \cdot 0.5} \]

   8. Simplified 26.9%

      \[\leadsto w0 \cdot e^{\color{blue}{\mathsf{fma}\left(-2, \log d, \log \left(\frac{-0.25 \cdot \left(\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h\right)}{\ell}\right)\right)} \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+151}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot e^{0.5 \cdot \mathsf{fma}\left(-2, \log d, \log \left(\frac{-0.25 \cdot \left(h \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)\right)}{\ell}\right)\right)}\\ \end{array} \]

Alternative 2: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{-48}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\frac{D \cdot w0}{\frac{d}{M} \cdot \frac{\frac{d}{h}}{M}}}{\frac{\ell}{D}}, w0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) 5e-48)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
   (fma -0.125 (/ (/ (* D w0) (* (/ d M) (/ (/ d h) M))) (/ l D)) w0)))

C

M = abs(M);
D = abs(D);
d = abs(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) * (h / l)) <= 5e-48) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = fma(-0.125, (((D * w0) / ((d / M) * ((d / h) / M))) / (l / D)), w0);
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
d = abs(d)
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= 5e-48)
		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(Float64(Float64(D * w0) / Float64(Float64(d / M) * Float64(Float64(d / h) / M))) / Float64(l / D)), w0);
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 5e-48], 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[(N[(N[(D * w0), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] * N[(N[(d / h), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 4.9999999999999999e-48

   1. Initial program 89.8%

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

      1. *-commutative 89.8%

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

      2. times-frac 89.4%

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

   3. Simplified 89.4%

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

   if 4.9999999999999999e-48 < (*.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 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in M around 0 43.5%

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

      1. fma-def 43.5%

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

      2. times-frac 43.7%

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

      3. *-commutative 43.7%

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

      4. unpow2 43.7%

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

      5. associate-/l* 43.7%

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

      6. *-commutative 43.7%

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

      7. *-commutative 43.7%

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

      8. unpow2 43.7%

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

      9. times-frac 51.1%

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

      10. *-commutative 51.1%

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

      11. unpow2 51.1%

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

      12. associate-*l* 51.3%

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

   6. Simplified 51.3%

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

      1. associate-*l/ 51.3%

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

      2. frac-times 43.9%

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

   8. Applied egg-rr 43.9%

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

      1. *-commutative 43.9%

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

      2. associate-*r* 43.7%

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

      3. unpow2 43.7%

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

      4. *-commutative 43.7%

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

      5. unpow2 43.7%

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

      6. associate-/l* 50.9%

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

      7. associate-*r/ 50.9%

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

      8. unpow2 50.9%

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

      9. *-commutative 50.9%

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

      10. unpow2 50.9%

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

      11. associate-*r* 51.0%

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

   10. Simplified 51.0%

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

   11. Taylor expanded in d around 0 50.9%

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

       1. unpow2 50.9%

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

       2. unpow2 50.9%

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

       3. associate-*r* 51.0%

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

       4. times-frac 51.4%

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

       5. associate-/l/ 72.8%

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

   13. Simplified 72.8%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{-48}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\frac{D \cdot w0}{\frac{d}{M} \cdot \frac{\frac{d}{h}}{M}}}{\frac{\ell}{D}}, w0\right)\\ \end{array} \]

Alternative 3: 85.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive 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);
d = abs(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: d should be positive 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);
d = Math.abs(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)
d = abs(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)
d = abs(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)
d = abs(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: d should be positive 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 84.9%

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

   1. *-commutative 84.9%

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

   2. times-frac 84.5%

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

3. Simplified 84.5%

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

   1. unpow2 84.5%

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

   2. associate-*l* 85.7%

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

   3. div-inv 85.7%

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

   4. associate-*l* 85.3%

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

   5. times-frac 85.4%

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

   6. *-un-lft-identity 85.4%

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

   7. *-commutative 85.4%

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

   8. clear-num 85.4%

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

   9. div-inv 85.4%

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

   10. div-inv 85.3%

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

   11. associate-*l* 86.1%

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

   12. times-frac 86.1%

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

   13. *-un-lft-identity 86.1%

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

   14. *-commutative 86.1%

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

   15. clear-num 86.1%

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

   16. div-inv 86.1%

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

   17. associate-*l* 84.9%

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

5. Applied egg-rr 88.5%

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

6. Final simplification 88.5%

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

Alternative 4: 79.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-160}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \cdot D \leq 2 \cdot 10^{+130}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{{\left(M \cdot D\right)}^{2}}{\frac{d \cdot \left(d \cdot \ell\right)}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\left(M \cdot h\right) \cdot \left(M \cdot \left(\frac{D}{d} \cdot \frac{w0}{d}\right)\right)}{\frac{\ell}{D}}, w0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* M D) 1e-160)
   w0
   (if (<= (* M D) 2e+130)
     (* w0 (+ 1.0 (* -0.125 (/ (pow (* M D) 2.0) (/ (* d (* d l)) h)))))
     (fma -0.125 (/ (* (* M h) (* M (* (/ D d) (/ w0 d)))) (/ l D)) w0))))

C

M = abs(M);
D = abs(D);
d = abs(d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= 1e-160) {
		tmp = w0;
	} else if ((M * D) <= 2e+130) {
		tmp = w0 * (1.0 + (-0.125 * (pow((M * D), 2.0) / ((d * (d * l)) / h))));
	} else {
		tmp = fma(-0.125, (((M * h) * (M * ((D / d) * (w0 / d)))) / (l / D)), w0);
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
d = abs(d)
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(M * D) <= 1e-160)
		tmp = w0;
	elseif (Float64(M * D) <= 2e+130)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64((Float64(M * D) ^ 2.0) / Float64(Float64(d * Float64(d * l)) / h)))));
	else
		tmp = fma(-0.125, Float64(Float64(Float64(M * h) * Float64(M * Float64(Float64(D / d) * Float64(w0 / d)))) / Float64(l / D)), w0);
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(M * D), $MachinePrecision], 1e-160], w0, If[LessEqual[N[(M * D), $MachinePrecision], 2e+130], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.125 * N[(N[(N[(M * h), $MachinePrecision] * N[(M * N[(N[(D / d), $MachinePrecision] * N[(w0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 9.9999999999999999e-161

   1. Initial program 86.6%

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

      1. *-commutative 86.6%

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

      2. times-frac 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in M around 0 73.6%

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

   if 9.9999999999999999e-161 < (*.f64 M D) < 2.0000000000000001e130

   1. Initial program 91.3%

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

      1. *-commutative 91.3%

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

      2. times-frac 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in M around 0 54.0%

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

      1. associate-*r/ 54.0%

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

      2. *-commutative 54.0%

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

      3. associate-*r/ 54.0%

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

      4. *-commutative 54.0%

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

      5. associate-/l* 55.8%

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

      6. unpow2 55.8%

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

      7. unpow2 55.8%

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

      8. *-commutative 55.8%

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

      9. unpow2 55.8%

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

   6. Simplified 55.8%

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

      1. associate-*r* 59.3%

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

      2. associate-/r/ 59.3%

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

   8. Applied egg-rr 59.3%

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

   9. Taylor expanded in D around 0 54.0%

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

       1. *-commutative 54.0%

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

       2. associate-*r* 59.4%

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

       3. *-commutative 59.4%

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

       4. unpow2 59.4%

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

       5. unpow2 59.4%

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

       6. swap-sqr 81.1%

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

       7. unpow2 81.1%

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

       8. *-commutative 81.1%

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

       9. associate-/l* 84.7%

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

       10. *-commutative 84.7%

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

       11. unpow2 84.7%

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

       12. associate-*l* 86.9%

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

   11. Simplified 86.9%

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

   if 2.0000000000000001e130 < (*.f64 M D)

   1. Initial program 66.3%

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

      1. *-commutative 66.3%

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

      2. times-frac 66.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in M around 0 43.1%

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

      1. fma-def 43.1%

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

      2. times-frac 43.0%

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

      3. *-commutative 43.0%

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

      4. unpow2 43.0%

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

      5. associate-/l* 51.8%

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

      6. *-commutative 51.8%

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

      7. *-commutative 51.8%

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

      8. unpow2 51.8%

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

      9. times-frac 51.8%

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

      10. *-commutative 51.8%

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

      11. unpow2 51.8%

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

      12. associate-*l* 54.8%

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

   6. Simplified 54.8%

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

      1. associate-*l/ 58.2%

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

      2. frac-times 55.2%

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

   8. Applied egg-rr 55.2%

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

   9. Taylor expanded in D around 0 52.2%

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

       1. associate-*r* 46.4%

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

       2. *-commutative 46.4%

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

       3. unpow2 46.4%

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

       4. associate-*r* 46.4%

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

       5. associate-*l/ 49.4%

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

       6. unpow2 49.4%

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

       7. associate-*r* 55.6%

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

       8. times-frac 64.2%

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

       9. *-commutative 64.2%

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

   11. Simplified 64.2%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-160}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \cdot D \leq 2 \cdot 10^{+130}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{{\left(M \cdot D\right)}^{2}}{\frac{d \cdot \left(d \cdot \ell\right)}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\left(M \cdot h\right) \cdot \left(M \cdot \left(\frac{D}{d} \cdot \frac{w0}{d}\right)\right)}{\frac{\ell}{D}}, w0\right)\\ \end{array} \]

Alternative 5: 78.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-160}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\frac{D \cdot w0}{\frac{d}{M} \cdot \frac{d}{M \cdot h}}}{\frac{\ell}{D}}, w0\right)\\ \mathbf{elif}\;M \cdot D \leq 2 \cdot 10^{+130}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{{\left(M \cdot D\right)}^{2}}{\frac{d \cdot \left(d \cdot \ell\right)}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\left(M \cdot h\right) \cdot \left(M \cdot \left(\frac{D}{d} \cdot \frac{w0}{d}\right)\right)}{\frac{\ell}{D}}, w0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* M D) 1e-160)
   (fma -0.125 (/ (/ (* D w0) (* (/ d M) (/ d (* M h)))) (/ l D)) w0)
   (if (<= (* M D) 2e+130)
     (* w0 (+ 1.0 (* -0.125 (/ (pow (* M D) 2.0) (/ (* d (* d l)) h)))))
     (fma -0.125 (/ (* (* M h) (* M (* (/ D d) (/ w0 d)))) (/ l D)) w0))))

C

M = abs(M);
D = abs(D);
d = abs(d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= 1e-160) {
		tmp = fma(-0.125, (((D * w0) / ((d / M) * (d / (M * h)))) / (l / D)), w0);
	} else if ((M * D) <= 2e+130) {
		tmp = w0 * (1.0 + (-0.125 * (pow((M * D), 2.0) / ((d * (d * l)) / h))));
	} else {
		tmp = fma(-0.125, (((M * h) * (M * ((D / d) * (w0 / d)))) / (l / D)), w0);
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
d = abs(d)
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(M * D) <= 1e-160)
		tmp = fma(-0.125, Float64(Float64(Float64(D * w0) / Float64(Float64(d / M) * Float64(d / Float64(M * h)))) / Float64(l / D)), w0);
	elseif (Float64(M * D) <= 2e+130)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64((Float64(M * D) ^ 2.0) / Float64(Float64(d * Float64(d * l)) / h)))));
	else
		tmp = fma(-0.125, Float64(Float64(Float64(M * h) * Float64(M * Float64(Float64(D / d) * Float64(w0 / d)))) / Float64(l / D)), w0);
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(M * D), $MachinePrecision], 1e-160], N[(-0.125 * N[(N[(N[(D * w0), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] * N[(d / N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], If[LessEqual[N[(M * D), $MachinePrecision], 2e+130], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.125 * N[(N[(N[(M * h), $MachinePrecision] * N[(M * N[(N[(D / d), $MachinePrecision] * N[(w0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 9.9999999999999999e-161

   1. Initial program 86.6%

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

      1. *-commutative 86.6%

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

      2. times-frac 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in M around 0 54.6%

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

      1. fma-def 54.6%

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

      2. times-frac 51.5%

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

      3. *-commutative 51.5%

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

      4. unpow2 51.5%

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

      5. associate-/l* 56.5%

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

      6. *-commutative 56.5%

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

      7. *-commutative 56.5%

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

      8. unpow2 56.5%

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

      9. times-frac 63.0%

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

      10. *-commutative 63.0%

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

      11. unpow2 63.0%

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

      12. associate-*l* 66.5%

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

   6. Simplified 66.5%

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

      1. associate-*l/ 69.7%

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

      2. frac-times 61.5%

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

   8. Applied egg-rr 61.5%

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

      1. *-commutative 61.5%

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

      2. associate-*r* 59.6%

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

      3. unpow2 59.6%

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

      4. *-commutative 59.6%

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

      5. unpow2 59.6%

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

      6. associate-/l* 60.2%

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

      7. associate-*r/ 57.8%

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

      8. unpow2 57.8%

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

      9. *-commutative 57.8%

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

      10. unpow2 57.8%

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

      11. associate-*r* 62.0%

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

   10. Simplified 62.0%

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

       1. times-frac 75.7%

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

   12. Applied egg-rr 75.7%

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

   if 9.9999999999999999e-161 < (*.f64 M D) < 2.0000000000000001e130

   1. Initial program 91.3%

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

      1. *-commutative 91.3%

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

      2. times-frac 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in M around 0 54.0%

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

      1. associate-*r/ 54.0%

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

      2. *-commutative 54.0%

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

      3. associate-*r/ 54.0%

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

      4. *-commutative 54.0%

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

      5. associate-/l* 55.8%

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

      6. unpow2 55.8%

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

      7. unpow2 55.8%

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

      8. *-commutative 55.8%

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

      9. unpow2 55.8%

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

   6. Simplified 55.8%

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

      1. associate-*r* 59.3%

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

      2. associate-/r/ 59.3%

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

   8. Applied egg-rr 59.3%

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

   9. Taylor expanded in D around 0 54.0%

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

       1. *-commutative 54.0%

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

       2. associate-*r* 59.4%

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

       3. *-commutative 59.4%

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

       4. unpow2 59.4%

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

       5. unpow2 59.4%

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

       6. swap-sqr 81.1%

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

       7. unpow2 81.1%

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

       8. *-commutative 81.1%

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

       9. associate-/l* 84.7%

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

       10. *-commutative 84.7%

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

       11. unpow2 84.7%

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

       12. associate-*l* 86.9%

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

   11. Simplified 86.9%

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

   if 2.0000000000000001e130 < (*.f64 M D)

   1. Initial program 66.3%

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

      1. *-commutative 66.3%

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

      2. times-frac 66.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in M around 0 43.1%

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

      1. fma-def 43.1%

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

      2. times-frac 43.0%

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

      3. *-commutative 43.0%

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

      4. unpow2 43.0%

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

      5. associate-/l* 51.8%

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

      6. *-commutative 51.8%

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

      7. *-commutative 51.8%

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

      8. unpow2 51.8%

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

      9. times-frac 51.8%

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

      10. *-commutative 51.8%

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

      11. unpow2 51.8%

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

      12. associate-*l* 54.8%

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

   6. Simplified 54.8%

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

      1. associate-*l/ 58.2%

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

      2. frac-times 55.2%

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

   8. Applied egg-rr 55.2%

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

   9. Taylor expanded in D around 0 52.2%

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

       1. associate-*r* 46.4%

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

       2. *-commutative 46.4%

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

       3. unpow2 46.4%

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

       4. associate-*r* 46.4%

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

       5. associate-*l/ 49.4%

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

       6. unpow2 49.4%

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

       7. associate-*r* 55.6%

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

       8. times-frac 64.2%

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

       9. *-commutative 64.2%

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

   11. Simplified 64.2%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-160}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\frac{D \cdot w0}{\frac{d}{M} \cdot \frac{d}{M \cdot h}}}{\frac{\ell}{D}}, w0\right)\\ \mathbf{elif}\;M \cdot D \leq 2 \cdot 10^{+130}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{{\left(M \cdot D\right)}^{2}}{\frac{d \cdot \left(d \cdot \ell\right)}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\left(M \cdot h\right) \cdot \left(M \cdot \left(\frac{D}{d} \cdot \frac{w0}{d}\right)\right)}{\frac{\ell}{D}}, w0\right)\\ \end{array} \]

Alternative 6: 77.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ \\ \begin{array}{l} t_0 := d \cdot \left(d \cdot \ell\right)\\ \mathbf{if}\;M \cdot D \leq 10^{-160}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \cdot D \leq 5 \cdot 10^{+102}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{{\left(M \cdot D\right)}^{2}}{\frac{t_0}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{t_0}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* d (* d l))))
   (if (<= (* M D) 1e-160)
     w0
     (if (<= (* M D) 5e+102)
       (* w0 (+ 1.0 (* -0.125 (/ (pow (* M D) 2.0) (/ t_0 h)))))
       (* w0 (+ 1.0 (* -0.125 (* (* D (/ D t_0)) (* M (* M h))))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = d * (d * l);
	double tmp;
	if ((M * D) <= 1e-160) {
		tmp = w0;
	} else if ((M * D) <= 5e+102) {
		tmp = w0 * (1.0 + (-0.125 * (pow((M * D), 2.0) / (t_0 / h))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / t_0)) * (M * (M * h)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_1 * (d_1 * l)
    if ((m * d) <= 1d-160) then
        tmp = w0
    else if ((m * d) <= 5d+102) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((m * d) ** 2.0d0) / (t_0 / h))))
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * (d / t_0)) * (m * (m * h)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = d * (d * l);
	double tmp;
	if ((M * D) <= 1e-160) {
		tmp = w0;
	} else if ((M * D) <= 5e+102) {
		tmp = w0 * (1.0 + (-0.125 * (Math.pow((M * D), 2.0) / (t_0 / h))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / t_0)) * (M * (M * h)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
def code(w0, M, D, h, l, d):
	t_0 = d * (d * l)
	tmp = 0
	if (M * D) <= 1e-160:
		tmp = w0
	elif (M * D) <= 5e+102:
		tmp = w0 * (1.0 + (-0.125 * (math.pow((M * D), 2.0) / (t_0 / h))))
	else:
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / t_0)) * (M * (M * h)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
function code(w0, M, D, h, l, d)
	t_0 = Float64(d * Float64(d * l))
	tmp = 0.0
	if (Float64(M * D) <= 1e-160)
		tmp = w0;
	elseif (Float64(M * D) <= 5e+102)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64((Float64(M * D) ^ 2.0) / Float64(t_0 / h)))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D / t_0)) * Float64(M * Float64(M * h))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = d * (d * l);
	tmp = 0.0;
	if ((M * D) <= 1e-160)
		tmp = w0;
	elseif ((M * D) <= 5e+102)
		tmp = w0 * (1.0 + (-0.125 * (((M * D) ^ 2.0) / (t_0 / h))));
	else
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / t_0)) * (M * (M * h)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M * D), $MachinePrecision], 1e-160], w0, If[LessEqual[N[(M * D), $MachinePrecision], 5e+102], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 9.9999999999999999e-161

   1. Initial program 86.6%

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

      1. *-commutative 86.6%

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

      2. times-frac 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in M around 0 73.6%

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

   if 9.9999999999999999e-161 < (*.f64 M D) < 5e102

   1. Initial program 93.7%

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

      1. *-commutative 93.7%

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

      2. times-frac 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in M around 0 58.6%

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

      1. associate-*r/ 58.6%

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

      2. *-commutative 58.6%

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

      3. associate-*r/ 58.6%

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

      4. *-commutative 58.6%

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

      5. associate-/l* 58.6%

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

      6. unpow2 58.6%

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

      7. unpow2 58.6%

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

      8. *-commutative 58.6%

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

      9. unpow2 58.6%

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

   6. Simplified 58.6%

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

      1. associate-*r* 62.7%

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

      2. associate-/r/ 62.7%

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

   8. Applied egg-rr 62.7%

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

   9. Taylor expanded in D around 0 58.6%

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

       1. *-commutative 58.6%

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

       2. associate-*r* 64.8%

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

       3. *-commutative 64.8%

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

       4. unpow2 64.8%

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

       5. unpow2 64.8%

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

       6. swap-sqr 87.7%

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

       7. unpow2 87.7%

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

       8. *-commutative 87.7%

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

       9. associate-/l* 89.8%

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

       10. *-commutative 89.8%

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

       11. unpow2 89.8%

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

       12. associate-*l* 90.4%

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

   11. Simplified 90.4%

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

   if 5e102 < (*.f64 M D)

   1. Initial program 68.3%

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

      1. *-commutative 68.3%

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

      2. times-frac 68.3%

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

   3. Simplified 68.3%

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

   4. Taylor expanded in M around 0 39.9%

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

      1. associate-*r/ 39.9%

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

      2. *-commutative 39.9%

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

      3. associate-*r/ 39.9%

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

      4. *-commutative 39.9%

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

      5. associate-/l* 42.4%

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

      6. unpow2 42.4%

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

      7. unpow2 42.4%

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

      8. *-commutative 42.4%

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

      9. unpow2 42.4%

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

   6. Simplified 42.4%

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

      1. associate-*r* 44.8%

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

      2. associate-/r/ 44.7%

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

   8. Applied egg-rr 44.7%

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

   9. Taylor expanded in D around 0 44.7%

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

       1. *-commutative 44.7%

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

       2. *-rgt-identity 44.7%

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

       3. unpow2 44.7%

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

       4. associate-*r/ 44.7%

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

       5. unpow2 44.7%

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

       6. associate-*l* 54.6%

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

       7. unpow2 54.6%

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

       8. *-commutative 54.6%

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

       9. associate-*r/ 54.6%

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

       10. *-commutative 54.6%

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

       11. *-lft-identity 54.6%

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

       12. *-commutative 54.6%

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

       13. unpow2 54.6%

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

       14. associate-*l* 56.8%

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

   11. Simplified 56.8%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-160}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \cdot D \leq 5 \cdot 10^{+102}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{{\left(M \cdot D\right)}^{2}}{\frac{d \cdot \left(d \cdot \ell\right)}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 70.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.9 \cdot 10^{-154}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 2.9e-154)
   w0
   (* w0 (+ 1.0 (* -0.125 (* (* D (/ D (* d (* d l)))) (* M (* M h))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.9e-154) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / (d * (d * l)))) * (M * (M * h)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
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 <= 2.9d-154) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * (d / (d_1 * (d_1 * l)))) * (m * (m * h)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.9e-154) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / (d * (d * l)))) * (M * (M * h)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.9e-154:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / (d * (d * l)))) * (M * (M * h)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.9e-154)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * Float64(D / Float64(d * Float64(d * l)))) * Float64(M * Float64(M * h))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.9e-154)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((D * (D / (d * (d * l)))) * (M * (M * h)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 2.9e-154], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * N[(D / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.9e-154

   1. Initial program 87.9%

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

      1. *-commutative 87.9%

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

      2. times-frac 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in M around 0 72.4%

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

   if 2.9e-154 < M

   1. Initial program 79.8%

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

      1. *-commutative 79.8%

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

      2. times-frac 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in M around 0 47.0%

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

      1. associate-*r/ 47.0%

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

      2. *-commutative 47.0%

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

      3. associate-*r/ 47.0%

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

      4. *-commutative 47.0%

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

      5. associate-/l* 48.0%

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

      6. unpow2 48.0%

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

      7. unpow2 48.0%

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

      8. *-commutative 48.0%

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

      9. unpow2 48.0%

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

   6. Simplified 48.0%

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

      1. associate-*r* 56.4%

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

      2. associate-/r/ 56.5%

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

   8. Applied egg-rr 56.5%

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

   9. Taylor expanded in D around 0 56.5%

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

       1. *-commutative 56.5%

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

       2. *-rgt-identity 56.5%

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

       3. unpow2 56.5%

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

       4. associate-*r/ 56.5%

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

       5. unpow2 56.5%

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

       6. associate-*l* 62.2%

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

       7. unpow2 62.2%

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

       8. *-commutative 62.2%

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

       9. associate-*r/ 62.2%

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

       10. *-commutative 62.2%

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

       11. *-lft-identity 62.2%

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

       12. *-commutative 62.2%

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

       13. unpow2 62.2%

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

       14. associate-*l* 63.5%

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

   11. Simplified 63.5%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.9 \cdot 10^{-154}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right)\\ \end{array} \]

Alternative 8: 62.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4.2 \cdot 10^{+72}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \frac{\frac{D}{d}}{\frac{d}{w0} \cdot \frac{\ell}{h \cdot \left(M \cdot M\right)}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 4.2e+72)
   w0
   (* -0.125 (* D (/ (/ D d) (* (/ d w0) (/ l (* h (* M M)))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.2e+72) {
		tmp = w0;
	} else {
		tmp = -0.125 * (D * ((D / d) / ((d / w0) * (l / (h * (M * M))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
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 <= 4.2d+72) then
        tmp = w0
    else
        tmp = (-0.125d0) * (d * ((d / d_1) / ((d_1 / w0) * (l / (h * (m * m))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.2e+72) {
		tmp = w0;
	} else {
		tmp = -0.125 * (D * ((D / d) / ((d / w0) * (l / (h * (M * M))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 4.2e+72:
		tmp = w0
	else:
		tmp = -0.125 * (D * ((D / d) / ((d / w0) * (l / (h * (M * M))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 4.2e+72)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(D * Float64(Float64(D / d) / Float64(Float64(d / w0) * Float64(l / Float64(h * Float64(M * M)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 4.2e+72)
		tmp = w0;
	else
		tmp = -0.125 * (D * ((D / d) / ((d / w0) * (l / (h * (M * M))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 4.2e+72], w0, N[(-0.125 * N[(D * N[(N[(D / d), $MachinePrecision] / N[(N[(d / w0), $MachinePrecision] * N[(l / N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.2000000000000003e72

   1. Initial program 88.3%

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

      1. *-commutative 88.3%

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

      2. times-frac 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in M around 0 73.5%

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

   if 4.2000000000000003e72 < M

   1. Initial program 70.2%

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

      1. *-commutative 70.2%

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

      2. times-frac 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in M around 0 24.1%

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

      1. associate-*r/ 24.1%

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

      2. *-commutative 24.1%

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

      3. associate-*r/ 24.1%

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

      4. *-commutative 24.1%

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

      5. associate-/l* 24.1%

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

      6. unpow2 24.1%

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

      7. unpow2 24.1%

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

      8. *-commutative 24.1%

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

      9. unpow2 24.1%

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

   6. Simplified 24.1%

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

      1. associate-*r* 40.8%

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

      2. associate-/r/ 40.8%

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

   8. Applied egg-rr 40.8%

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

   9. Taylor expanded in D around inf 18.2%

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

   11. Simplified 20.7%

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

   12. Taylor expanded in l around 0 21.9%

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

       1. *-commutative 21.9%

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

       2. *-commutative 21.9%

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

       3. times-frac 24.0%

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

       4. *-commutative 24.0%

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

       5. unpow2 24.0%

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

   14. Simplified 24.0%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.2 \cdot 10^{+72}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \frac{\frac{D}{d}}{\frac{d}{w0} \cdot \frac{\ell}{h \cdot \left(M \cdot M\right)}}\right)\\ \end{array} \]

Alternative 9: 62.9% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.1 \cdot 10^{+74}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \frac{\frac{D}{d}}{\frac{\ell}{\frac{w0}{d} \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 2.1e+74)
   w0
   (* -0.125 (* D (/ (/ D d) (/ l (* (/ w0 d) (* h (* M M)))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.1e+74) {
		tmp = w0;
	} else {
		tmp = -0.125 * (D * ((D / d) / (l / ((w0 / d) * (h * (M * M))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
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+74) then
        tmp = w0
    else
        tmp = (-0.125d0) * (d * ((d / d_1) / (l / ((w0 / d_1) * (h * (m * m))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.1e+74) {
		tmp = w0;
	} else {
		tmp = -0.125 * (D * ((D / d) / (l / ((w0 / d) * (h * (M * M))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.1e+74:
		tmp = w0
	else:
		tmp = -0.125 * (D * ((D / d) / (l / ((w0 / d) * (h * (M * M))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.1e+74)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(D * Float64(Float64(D / d) / Float64(l / Float64(Float64(w0 / d) * Float64(h * Float64(M * M)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.1e+74)
		tmp = w0;
	else
		tmp = -0.125 * (D * ((D / d) / (l / ((w0 / d) * (h * (M * M))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 2.1e+74], w0, N[(-0.125 * N[(D * N[(N[(D / d), $MachinePrecision] / N[(l / N[(N[(w0 / d), $MachinePrecision] * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.0999999999999999e74

   1. Initial program 88.3%

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

      1. *-commutative 88.3%

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

      2. times-frac 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in M around 0 73.5%

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

   if 2.0999999999999999e74 < M

   1. Initial program 70.2%

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

      1. *-commutative 70.2%

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

      2. times-frac 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in M around 0 24.1%

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

      1. associate-*r/ 24.1%

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

      2. *-commutative 24.1%

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

      3. associate-*r/ 24.1%

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

      4. *-commutative 24.1%

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

      5. associate-/l* 24.1%

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

      6. unpow2 24.1%

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

      7. unpow2 24.1%

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

      8. *-commutative 24.1%

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

      9. unpow2 24.1%

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

   6. Simplified 24.1%

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

      1. associate-*r* 40.8%

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

      2. associate-/r/ 40.8%

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

   8. Applied egg-rr 40.8%

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

   9. Taylor expanded in D around inf 18.2%

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

   11. Simplified 20.7%

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

   12. Taylor expanded in w0 around 0 24.3%

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

       1. associate-/l* 24.3%

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

       2. *-commutative 24.3%

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

       3. unpow2 24.3%

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

       4. associate-*r* 24.6%

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

       5. associate-/r/ 24.6%

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

       6. associate-*r* 24.3%

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

       7. unpow2 24.3%

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

       8. *-commutative 24.3%

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

       9. unpow2 24.3%

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

   14. Simplified 24.3%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.1 \cdot 10^{+74}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \frac{\frac{D}{d}}{\frac{\ell}{\frac{w0}{d} \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\right)\\ \end{array} \]

Alternative 10: 63.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 9 \cdot 10^{+72}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \frac{\frac{D}{d}}{\frac{\ell}{M \cdot \frac{w0}{\frac{d}{M \cdot h}}}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 9e+72)
   w0
   (* -0.125 (* D (/ (/ D d) (/ l (* M (/ w0 (/ d (* M h))))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 9e+72) {
		tmp = w0;
	} else {
		tmp = -0.125 * (D * ((D / d) / (l / (M * (w0 / (d / (M * h)))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
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 <= 9d+72) then
        tmp = w0
    else
        tmp = (-0.125d0) * (d * ((d / d_1) / (l / (m * (w0 / (d_1 / (m * h)))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 9e+72) {
		tmp = w0;
	} else {
		tmp = -0.125 * (D * ((D / d) / (l / (M * (w0 / (d / (M * h)))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 9e+72:
		tmp = w0
	else:
		tmp = -0.125 * (D * ((D / d) / (l / (M * (w0 / (d / (M * h)))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 9e+72)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(D * Float64(Float64(D / d) / Float64(l / Float64(M * Float64(w0 / Float64(d / Float64(M * h))))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 9e+72)
		tmp = w0;
	else
		tmp = -0.125 * (D * ((D / d) / (l / (M * (w0 / (d / (M * h)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 9e+72], w0, N[(-0.125 * N[(D * N[(N[(D / d), $MachinePrecision] / N[(l / N[(M * N[(w0 / N[(d / N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 8.9999999999999997e72

   1. Initial program 88.3%

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

      1. *-commutative 88.3%

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

      2. times-frac 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in M around 0 73.5%

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

   if 8.9999999999999997e72 < M

   1. Initial program 70.2%

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

      1. *-commutative 70.2%

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

      2. times-frac 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in M around 0 24.1%

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

      1. associate-*r/ 24.1%

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

      2. *-commutative 24.1%

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

      3. associate-*r/ 24.1%

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

      4. *-commutative 24.1%

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

      5. associate-/l* 24.1%

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

      6. unpow2 24.1%

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

      7. unpow2 24.1%

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

      8. *-commutative 24.1%

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

      9. unpow2 24.1%

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

   6. Simplified 24.1%

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

      1. associate-*r* 40.8%

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

      2. associate-/r/ 40.8%

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

   8. Applied egg-rr 40.8%

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

   9. Taylor expanded in D around inf 18.2%

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

   11. Simplified 20.7%

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

       1. associate-/r/ 20.7%

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

       2. associate-/l/ 24.8%

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

   13. Applied egg-rr 24.8%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9 \cdot 10^{+72}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \frac{\frac{D}{d}}{\frac{\ell}{M \cdot \frac{w0}{\frac{d}{M \cdot h}}}}\right)\\ \end{array} \]

Alternative 11: 67.3% accurate, 216.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ \\ w0 \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
(FPCore (w0 M D h l d) :precision binary64 w0)

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
function tmp = code(w0, M, D, h, l, d)
	tmp = w0;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 84.9%

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

   1. *-commutative 84.9%

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

   2. times-frac 84.5%

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

3. Simplified 84.5%

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

4. Taylor expanded in M around 0 67.9%

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

5. Final simplification 67.9%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023194 
(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))))))