Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.5% → 87.4%

Time: 14.3s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.5% 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: 87.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 10^{+283}:\\ \;\;\;\;w0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(h \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \frac{-1}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
   (if (<= t_0 1e+283)
     (* w0 (sqrt t_0))
     (*
      w0
      (sqrt (+ 1.0 (* (* h (pow (* 0.5 (/ M (/ d D))) 2.0)) (/ -1.0 l))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = 1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 1e+283) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 + ((h * pow((0.5 * (M / (d / D))), 2.0)) * (-1.0 / l))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))
    if (t_0 <= 1d+283) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 + ((h * ((0.5d0 * (m / (d_1 / d))) ** 2.0d0)) * ((-1.0d0) / l))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = 1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 1e+283) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((h * Math.pow((0.5 * (M / (d / D))), 2.0)) * (-1.0 / l))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = 1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 1e+283:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 + ((h * math.pow((0.5 * (M / (d / D))), 2.0)) * (-1.0 / l))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 1e+283)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(h * (Float64(0.5 * Float64(M / Float64(d / D))) ^ 2.0)) * Float64(-1.0 / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = 1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 1e+283)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 + ((h * ((0.5 * (M / (d / D))) ^ 2.0)) * (-1.0 / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 1e+283], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(h * N[Power[N[(0.5 * N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 9.99999999999999955e282

   1. Initial program 99.9%

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

   if 9.99999999999999955e282 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 38.1%

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

      1. times-frac 40.7%

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

   3. Simplified 40.7%

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

      1. associate-*r/ 64.6%

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

      2. clear-num 64.6%

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

      3. frac-times 60.3%

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

      4. div-inv 60.3%

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

      5. associate-*l* 64.6%

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

      6. associate-/r* 64.6%

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

      7. metadata-eval 64.6%

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

   5. Applied egg-rr 64.6%

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

      1. associate-/r/ 64.6%

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

      2. *-commutative 64.6%

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

      3. *-commutative 64.6%

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

      4. associate-*r* 63.3%

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

      5. associate-*r/ 63.3%

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

      6. associate-/r/ 64.6%

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

      7. *-commutative 64.6%

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

      8. associate-*r/ 64.6%

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

   7. Simplified 64.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{+283}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(h \cdot {\left(0.5 \cdot \frac{M}{\frac{d}{D}}\right)}^{2}\right) \cdot \frac{-1}{\ell}}\\ \end{array} \]

Alternative 2: 86.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
   (if (<= t_0 INFINITY)
     (* w0 (sqrt t_0))
     (* w0 (+ 1.0 (* (/ (* D (* (/ D d) (* M (* h (/ M d))))) l) -0.125))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = 1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * (1.0 + (((D * ((D / d) * (M * (h * (M / d))))) / l) * -0.125));
	}
	return tmp;
}

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = 1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * (1.0 + (((D * ((D / d) * (M * (h * (M / d))))) / l) * -0.125));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = 1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= math.inf:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * (1.0 + (((D * ((D / d) * (M * (h * (M / d))))) / l) * -0.125))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D * Float64(Float64(D / d) * Float64(M * Float64(h * Float64(M / d))))) / l) * -0.125)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = 1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * (1.0 + (((D * ((D / d) * (M * (h * (M / d))))) / l) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, Infinity], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(N[(D * N[(N[(D / d), $MachinePrecision] * N[(M * N[(h * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 89.3%

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

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

   1. Initial program 0.0%

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

      1. times-frac 4.5%

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

   3. Simplified 4.5%

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

   4. Taylor expanded in M around 0 41.1%

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

      1. associate-*r/ 41.1%

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

      2. *-commutative 41.1%

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

      3. associate-*r/ 41.1%

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

      4. *-commutative 41.1%

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

      5. times-frac 44.1%

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

      6. unpow2 44.1%

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

      7. *-commutative 44.1%

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

      8. unpow2 44.1%

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

      9. unpow2 44.1%

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

   6. Simplified 44.1%

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

      1. div-inv 40.0%

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

      2. *-commutative 40.0%

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

   8. Applied egg-rr 40.0%

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

   9. Taylor expanded in D around 0 41.1%

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

       1. unpow2 41.1%

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

       2. *-commutative 41.1%

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

       3. times-frac 44.1%

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

       4. associate-*l/ 44.3%

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

       5. unpow2 44.3%

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

       6. *-commutative 44.3%

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

       7. associate-*r* 48.5%

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

       8. unpow2 48.5%

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

       9. associate-*l* 59.1%

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

       10. times-frac 67.8%

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

       11. associate-*r* 73.2%

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

       12. associate-*r/ 72.8%

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

       13. associate-/l* 73.2%

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

       14. associate-/r/ 77.6%

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

       15. associate-/l* 72.9%

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

   11. Simplified 72.9%

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

       1. associate-*l/ 81.6%

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

       2. associate-*l* 72.1%

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

       3. associate-/r/ 76.8%

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

   13. Applied egg-rr 76.8%

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

4. Final simplification 88.2%

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

Alternative 3: 87.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 10^{+283}:\\ \;\;\;\;w0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
   (if (<= t_0 1e+283)
     (* w0 (sqrt t_0))
     (* w0 (sqrt (- 1.0 (/ (* h (* 0.25 (pow (* D (/ M d)) 2.0))) l)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = 1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 1e+283) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - ((h * (0.25 * pow((D * (M / d)), 2.0))) / l)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))
    if (t_0 <= 1d+283) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - ((h * (0.25d0 * ((d * (m / d_1)) ** 2.0d0))) / l)))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = 1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 1e+283) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * (0.25 * Math.pow((D * (M / d)), 2.0))) / l)));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = 1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 1e+283:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - ((h * (0.25 * math.pow((D * (M / d)), 2.0))) / l)))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 1e+283)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(0.25 * (Float64(D * Float64(M / d)) ^ 2.0))) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = 1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 1e+283)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - ((h * (0.25 * ((D * (M / d)) ^ 2.0))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 1e+283], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(0.25 * N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 9.99999999999999955e282

   1. Initial program 99.9%

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

   if 9.99999999999999955e282 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 38.1%

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

      1. times-frac 40.7%

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

   3. Simplified 40.7%

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

      1. associate-*r/ 64.6%

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

      2. clear-num 64.6%

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

      3. frac-times 60.3%

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

      4. div-inv 60.3%

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

      5. associate-*l* 64.6%

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

      6. associate-/r* 64.6%

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

      7. metadata-eval 64.6%

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

   5. Applied egg-rr 64.6%

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

      1. associate-/r/ 64.6%

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

      2. *-commutative 64.6%

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

      3. *-commutative 64.6%

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

      4. associate-*r* 63.3%

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

      5. associate-*r/ 63.3%

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

      6. associate-/r/ 64.6%

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

      7. *-commutative 64.6%

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

      8. associate-*r/ 64.6%

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

   7. Simplified 64.6%

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

      1. associate-*l/ 64.6%

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

      2. *-un-lft-identity 64.6%

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

      3. unpow-prod-down 64.6%

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

      4. metadata-eval 64.6%

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

      5. associate-/r/ 63.3%

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

   9. Applied egg-rr 63.3%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{+283}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}{\ell}}\\ \end{array} \]

Alternative 4: 87.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 10^{+283}:\\ \;\;\;\;w0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
   (if (<= t_0 1e+283)
     (* w0 (sqrt t_0))
     (* w0 (sqrt (- 1.0 (/ (* h (pow (* M (* D (/ 0.5 d))) 2.0)) l)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = 1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 1e+283) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow((M * (D * (0.5 / d))), 2.0)) / l)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))
    if (t_0 <= 1d+283) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - ((h * ((m * (d * (0.5d0 / d_1))) ** 2.0d0)) / l)))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = 1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 1e+283) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M * (D * (0.5 / d))), 2.0)) / l)));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = 1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 1e+283:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((M * (D * (0.5 / d))), 2.0)) / l)))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 1e+283)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0)) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = 1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 1e+283)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - ((h * ((M * (D * (0.5 / d))) ^ 2.0)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 1e+283], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 9.99999999999999955e282

   1. Initial program 99.9%

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

   if 9.99999999999999955e282 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 38.1%

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

      1. times-frac 40.7%

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

   3. Simplified 40.7%

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

      1. unpow2 40.7%

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

      2. unpow2 40.7%

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

      3. frac-times 38.1%

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

      4. associate-*r/ 60.3%

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

      5. frac-times 64.6%

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

      6. unpow2 64.6%

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

      7. unpow2 64.6%

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

      8. frac-times 60.3%

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

      9. div-inv 60.3%

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

      10. associate-*l* 64.6%

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

      11. associate-/r* 64.6%

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

      12. metadata-eval 64.6%

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

   5. Applied egg-rr 64.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{+283}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 5: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-270}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) (- INFINITY))
   (* w0 (sqrt (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (/ (* M (* M h)) l))))))
   (if (<= (/ h l) -1e-270)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ D d) (/ M 2.0)) 2.0)))))
     w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	} else if ((h / l) <= -1e-270) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D / d) * (M / 2.0)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	} else if ((h / l) <= -1e-270) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D / d) * (M / 2.0)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -math.inf:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))))
	elif (h / l) <= -1e-270:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D / d) * (M / 2.0)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * Float64(M * h)) / l))))));
	elseif (Float64(h / l) <= -1e-270)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -Inf)
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	elseif ((h / l) <= -1e-270)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D / d) * (M / 2.0)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-270], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 40.4%

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

      1. times-frac 40.5%

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

   3. Simplified 40.5%

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

   4. Taylor expanded in M around 0 35.0%

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

      1. *-commutative 35.0%

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

      2. times-frac 44.1%

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

      3. *-commutative 44.1%

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

      4. unpow2 44.1%

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

      5. unpow2 44.1%

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

      6. times-frac 70.3%

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

      7. unpow2 70.3%

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

      8. *-commutative 70.3%

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

      9. unpow2 70.3%

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

      10. associate-*l* 74.5%

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

   6. Simplified 74.5%

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

      1. unpow2 74.5%

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

   8. Applied egg-rr 74.5%

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

   if -inf.0 < (/.f64 h l) < -1e-270

   1. Initial program 85.1%

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

      1. times-frac 85.2%

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

   3. Simplified 85.2%

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

   if -1e-270 < (/.f64 h l)

   1. Initial program 85.6%

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

      1. times-frac 85.6%

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

   3. Simplified 85.6%

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

   4. Taylor expanded in M around 0 94.9%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-270}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 6: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-308}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(h \cdot \frac{M}{d}\right)\right)\right)}{\ell} \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) (- INFINITY))
   (* w0 (sqrt (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (/ (* M (* M h)) l))))))
   (if (<= (/ h l) -1e-308)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
     (* w0 (+ 1.0 (* (/ (* D (* (/ D d) (* M (* h (/ M d))))) l) -0.125))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	} else if ((h / l) <= -1e-308) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * (1.0 + (((D * ((D / d) * (M * (h * (M / d))))) / l) * -0.125));
	}
	return tmp;
}

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	} else if ((h / l) <= -1e-308) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * (1.0 + (((D * ((D / d) * (M * (h * (M / d))))) / l) * -0.125));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -math.inf:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))))
	elif (h / l) <= -1e-308:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	else:
		tmp = w0 * (1.0 + (((D * ((D / d) * (M * (h * (M / d))))) / l) * -0.125))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * Float64(M * h)) / l))))));
	elseif (Float64(h / l) <= -1e-308)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0)))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D * Float64(Float64(D / d) * Float64(M * Float64(h * Float64(M / d))))) / l) * -0.125)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -Inf)
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	elseif ((h / l) <= -1e-308)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	else
		tmp = w0 * (1.0 + (((D * ((D / d) * (M * (h * (M / d))))) / l) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-308], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(N[(D * N[(N[(D / d), $MachinePrecision] * N[(M * N[(h * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 40.4%

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

      1. times-frac 40.5%

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

   3. Simplified 40.5%

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

   4. Taylor expanded in M around 0 35.0%

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

      1. *-commutative 35.0%

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

      2. times-frac 44.1%

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

      3. *-commutative 44.1%

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

      4. unpow2 44.1%

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

      5. unpow2 44.1%

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

      6. times-frac 70.3%

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

      7. unpow2 70.3%

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

      8. *-commutative 70.3%

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

      9. unpow2 70.3%

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

      10. associate-*l* 74.5%

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

   6. Simplified 74.5%

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

      1. unpow2 74.5%

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

   8. Applied egg-rr 74.5%

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

   if -inf.0 < (/.f64 h l) < -9.9999999999999991e-309

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. times-frac 83.2%

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

   3. Simplified 83.2%

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

   if -9.9999999999999991e-309 < (/.f64 h l)

   1. Initial program 85.2%

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

      1. times-frac 85.2%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in M around 0 64.1%

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

      1. associate-*r/ 64.1%

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

      2. *-commutative 64.1%

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

      3. associate-*r/ 64.1%

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

      4. *-commutative 64.1%

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

      5. times-frac 60.3%

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

      6. unpow2 60.3%

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

      7. *-commutative 60.3%

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

      8. unpow2 60.3%

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

      9. unpow2 60.3%

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

   6. Simplified 60.3%

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

      1. div-inv 59.4%

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

      2. *-commutative 59.4%

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

   8. Applied egg-rr 59.4%

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

   9. Taylor expanded in D around 0 64.1%

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

       1. unpow2 64.1%

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

       2. *-commutative 64.1%

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

       3. times-frac 60.3%

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

       4. associate-*l/ 63.1%

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

       5. unpow2 63.1%

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

       6. *-commutative 63.1%

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

       7. associate-*r* 69.5%

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

       8. unpow2 69.5%

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

       9. associate-*l* 75.5%

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

       10. times-frac 85.7%

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

       11. associate-*r* 88.7%

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

       12. associate-*r/ 88.6%

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

       13. associate-/l* 88.7%

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

       14. associate-/r/ 88.6%

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

       15. associate-/l* 88.7%

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

   11. Simplified 88.7%

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

       1. associate-*l/ 93.3%

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

       2. associate-*l* 87.5%

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

       3. associate-/r/ 89.4%

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

   13. Applied egg-rr 89.4%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-308}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(h \cdot \frac{M}{d}\right)\right)\right)}{\ell} \cdot -0.125\right)\\ \end{array} \]

Alternative 7: 78.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{-84}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 5e-84)
   (* w0 (+ 1.0 (* -0.125 (* h (/ (pow (* D (/ M d)) 2.0) l)))))
   (* w0 (sqrt (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (/ (* M (* M h)) l))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 5e-84) {
		tmp = w0 * (1.0 + (-0.125 * (h * (pow((D * (M / d)), 2.0) / l))));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= 5d-84) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (h * (((d * (m / d_1)) ** 2.0d0) / l))))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d / d_1) * (d / d_1)) * ((m * (m * h)) / l)))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 5e-84) {
		tmp = w0 * (1.0 + (-0.125 * (h * (Math.pow((D * (M / d)), 2.0) / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 5e-84:
		tmp = w0 * (1.0 + (-0.125 * (h * (math.pow((D * (M / d)), 2.0) / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 5e-84)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(h * Float64((Float64(D * Float64(M / d)) ^ 2.0) / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M * Float64(M * h)) / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 5e-84)
		tmp = w0 * (1.0 + (-0.125 * (h * (((D * (M / d)) ^ 2.0) / l))));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * ((M * (M * h)) / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 5e-84], N[(w0 * N[(1.0 + N[(-0.125 * N[(h * N[(N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.0000000000000002e-84

   1. Initial program 80.7%

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

      1. times-frac 80.7%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in M around 0 57.7%

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

      1. associate-*r/ 57.7%

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

      2. *-commutative 57.7%

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

      3. associate-*r/ 57.7%

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

      4. *-commutative 57.7%

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

      5. times-frac 57.8%

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

      6. unpow2 57.8%

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

      7. *-commutative 57.8%

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

      8. unpow2 57.8%

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

      9. unpow2 57.8%

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

   6. Simplified 57.8%

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

   7. Taylor expanded in D around 0 57.7%

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

      1. times-frac 61.3%

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

      2. unpow2 61.3%

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

      3. unpow2 61.3%

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

      4. times-frac 69.8%

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

      5. unpow2 69.8%

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

      6. unpow2 69.8%

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

      7. associate-/l* 65.6%

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

      8. associate-*r/ 67.3%

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

      9. unpow2 67.3%

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

      10. swap-sqr 76.3%

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

      11. *-commutative 76.3%

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

      12. associate-*r/ 75.8%

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

      13. associate-*l/ 75.2%

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

      14. *-commutative 75.2%

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

      15. associate-*r/ 75.2%

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

      16. associate-*l/ 75.7%

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

      17. unpow2 75.7%

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

      18. *-commutative 75.7%

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

   9. Simplified 75.7%

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

       1. associate-/r/ 81.6%

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

   11. Applied egg-rr 81.6%

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

   if 5.0000000000000002e-84 < M

   1. Initial program 82.7%

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

      1. times-frac 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in M around 0 48.9%

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

      1. *-commutative 48.9%

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

      2. times-frac 47.6%

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

      3. *-commutative 47.6%

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

      4. unpow2 47.6%

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

      5. unpow2 47.6%

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

      6. times-frac 62.7%

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

      7. unpow2 62.7%

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

      8. *-commutative 62.7%

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

      9. unpow2 62.7%

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

      10. associate-*l* 74.7%

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

   6. Simplified 74.7%

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

      1. unpow2 74.7%

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

   8. Applied egg-rr 74.7%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{-84}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{\ell}\right)}\\ \end{array} \]

Alternative 8: 81.1% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (+ 1.0 (* -0.125 (* h (/ (pow (* D (/ M d)) 2.0) l))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * (h * (pow((D * (M / d)), 2.0) / 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 * (1.0d0 + ((-0.125d0) * (h * (((d * (m / d_1)) ** 2.0d0) / l))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * (1.0 + (-0.125 * (h * (((D * (M / d)) ^ 2.0) / l))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[(1.0 + N[(-0.125 * N[(h * N[(N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.3%

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

   1. times-frac 81.4%

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

3. Simplified 81.4%

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

4. Taylor expanded in M around 0 54.3%

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

   1. associate-*r/ 54.3%

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

   2. *-commutative 54.3%

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

   3. associate-*r/ 54.3%

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

   4. *-commutative 54.3%

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

   5. times-frac 53.5%

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

   6. unpow2 53.5%

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

   7. *-commutative 53.5%

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

   8. unpow2 53.5%

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

   9. unpow2 53.5%

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

6. Simplified 53.5%

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

7. Taylor expanded in D around 0 54.3%

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

   1. times-frac 56.0%

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

   2. unpow2 56.0%

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

   3. unpow2 56.0%

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

   4. times-frac 65.0%

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

   5. unpow2 65.0%

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

   6. unpow2 65.0%

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

   7. associate-/l* 62.0%

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

   8. associate-*r/ 64.4%

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

   9. unpow2 64.4%

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

   10. swap-sqr 74.8%

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

   11. *-commutative 74.8%

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

   12. associate-*r/ 74.4%

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

   13. associate-*l/ 73.9%

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

   14. *-commutative 73.9%

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

   15. associate-*r/ 73.9%

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

   16. associate-*l/ 74.3%

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

   17. unpow2 74.3%

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

   18. *-commutative 74.3%

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

9. Simplified 74.3%

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

    1. associate-/r/ 79.3%

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

11. Applied egg-rr 79.3%

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

12. Final simplification 79.3%

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

Alternative 9: 73.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 9 \cdot 10^{-128}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{M}{\frac{d}{h}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 9e-128)
   w0
   (* w0 (+ 1.0 (* -0.125 (* (/ D l) (* (* M (/ D d)) (/ M (/ d h)))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 9e-128) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D / l) * ((M * (D / d)) * (M / (d / h))))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= 9d-128) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d / l) * ((m * (d / d_1)) * (m / (d_1 / h))))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 9e-128) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D / l) * ((M * (D / d)) * (M / (d / h))))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 9e-128:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((D / l) * ((M * (D / d)) * (M / (d / h))))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 9e-128)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D / l) * Float64(Float64(M * Float64(D / d)) * Float64(M / Float64(d / h)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 9e-128)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((D / l) * ((M * (D / d)) * (M / (d / h))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 9e-128], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D / l), $MachinePrecision] * N[(N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(M / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 8.9999999999999998e-128

   1. Initial program 80.6%

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

      1. times-frac 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in M around 0 73.2%

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

   if 8.9999999999999998e-128 < M

   1. Initial program 82.5%

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

      1. times-frac 82.7%

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

   3. Simplified 82.7%

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

   4. Taylor expanded in M around 0 49.9%

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

      1. associate-*r/ 49.9%

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

      2. *-commutative 49.9%

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

      3. associate-*r/ 49.9%

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

      4. *-commutative 49.9%

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

      5. times-frac 47.7%

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

      6. unpow2 47.7%

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

      7. *-commutative 47.7%

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

      8. unpow2 47.7%

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

      9. unpow2 47.7%

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

   6. Simplified 47.7%

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

      1. div-inv 46.7%

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

      2. *-commutative 46.7%

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

   8. Applied egg-rr 46.7%

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

   9. Taylor expanded in D around 0 49.9%

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

       1. unpow2 49.9%

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

       2. *-commutative 49.9%

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

       3. times-frac 47.7%

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

       4. associate-*l/ 49.1%

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

       5. unpow2 49.1%

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

       6. *-commutative 49.1%

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

       7. associate-*r* 54.6%

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

       8. unpow2 54.6%

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

       9. associate-*l* 60.5%

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

       10. times-frac 67.3%

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

       11. associate-*r* 71.7%

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

       12. associate-*r/ 72.7%

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

       13. associate-/l* 71.7%

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

       14. associate-/r/ 72.7%

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

       15. associate-/l* 71.6%

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

   11. Simplified 71.6%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9 \cdot 10^{-128}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{\ell} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{M}{\frac{d}{h}}\right)\right)\right)\\ \end{array} \]

Alternative 10: 72.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 2.5 \cdot 10^{-201}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(h \cdot \frac{M}{d}\right)\right)\right)}{\ell} \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 2.5e-201)
   w0
   (* w0 (+ 1.0 (* (/ (* D (* (/ D d) (* M (* h (/ M d))))) l) -0.125)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.5e-201) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (((D * ((D / d) * (M * (h * (M / d))))) / l) * -0.125));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= 2.5d-201) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + (((d * ((d / d_1) * (m * (h * (m / d_1))))) / l) * (-0.125d0)))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.5e-201) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (((D * ((D / d) * (M * (h * (M / d))))) / l) * -0.125));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.5e-201:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (((D * ((D / d) * (M * (h * (M / d))))) / l) * -0.125))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.5e-201)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D * Float64(Float64(D / d) * Float64(M * Float64(h * Float64(M / d))))) / l) * -0.125)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.5e-201)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (((D * ((D / d) * (M * (h * (M / d))))) / l) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 2.5e-201], w0, N[(w0 * N[(1.0 + N[(N[(N[(D * N[(N[(D / d), $MachinePrecision] * N[(M * N[(h * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.5e-201

   1. Initial program 80.4%

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

      1. times-frac 80.4%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in M around 0 73.3%

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

   if 2.5e-201 < M

   1. Initial program 82.6%

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

      1. times-frac 82.7%

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

   3. Simplified 82.7%

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

   4. Taylor expanded in M around 0 50.4%

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

      1. associate-*r/ 50.4%

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

      2. *-commutative 50.4%

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

      3. associate-*r/ 50.4%

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

      4. *-commutative 50.4%

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

      5. times-frac 50.5%

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

      6. unpow2 50.5%

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

      7. *-commutative 50.5%

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

      8. unpow2 50.5%

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

      9. unpow2 50.5%

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

   6. Simplified 50.5%

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

      1. div-inv 49.6%

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

      2. *-commutative 49.6%

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

   8. Applied egg-rr 49.6%

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

   9. Taylor expanded in D around 0 50.4%

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

       1. unpow2 50.4%

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

       2. *-commutative 50.4%

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

       3. times-frac 50.5%

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

       4. associate-*l/ 51.6%

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

       5. unpow2 51.6%

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

       6. *-commutative 51.6%

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

       7. associate-*r* 56.3%

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

       8. unpow2 56.3%

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

       9. associate-*l* 61.2%

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

       10. times-frac 68.1%

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

       11. associate-*r* 71.8%

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

       12. associate-*r/ 72.7%

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

       13. associate-/l* 71.8%

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

       14. associate-/r/ 72.7%

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

       15. associate-/l* 71.7%

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

   11. Simplified 71.7%

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

       1. associate-*l/ 74.7%

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

       2. associate-*l* 71.0%

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

       3. associate-/r/ 72.9%

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

   13. Applied egg-rr 72.9%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.5 \cdot 10^{-201}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot \left(\frac{D}{d} \cdot \left(M \cdot \left(h \cdot \frac{M}{d}\right)\right)\right)}{\ell} \cdot -0.125\right)\\ \end{array} \]

Alternative 11: 65.7% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 3 \cdot 10^{+74}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{h}{d} \cdot \left(M \cdot \left(M \cdot \frac{w0}{d}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 3e+74)
   w0
   (* -0.125 (* (* D (/ D l)) (* (/ h d) (* M (* M (/ w0 d))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3e+74) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * (D / l)) * ((h / d) * (M * (M * (w0 / d)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= 3d+74) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((d * (d / l)) * ((h / d_1) * (m * (m * (w0 / d_1)))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3e+74) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * (D / l)) * ((h / d) * (M * (M * (w0 / d)))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 3e+74:
		tmp = w0
	else:
		tmp = -0.125 * ((D * (D / l)) * ((h / d) * (M * (M * (w0 / d)))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 3e+74)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(h / d) * Float64(M * Float64(M * Float64(w0 / d))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 3e+74)
		tmp = w0;
	else
		tmp = -0.125 * ((D * (D / l)) * ((h / d) * (M * (M * (w0 / d)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 3e+74], w0, N[(-0.125 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(h / d), $MachinePrecision] * N[(M * N[(M * N[(w0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3e74

   1. Initial program 81.6%

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

      1. times-frac 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in M around 0 71.4%

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

   if 3e74 < M

   1. Initial program 79.6%

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

      1. times-frac 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in M around 0 43.7%

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

      1. +-commutative 43.7%

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

      2. associate-*r/ 43.7%

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

      3. *-commutative 43.7%

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

      4. associate-*r/ 43.7%

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

      5. *-commutative 43.7%

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

      6. fma-def 43.7%

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

      7. *-commutative 43.7%

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

      8. times-frac 38.7%

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

      9. unpow2 38.7%

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

      10. unpow2 38.7%

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

      11. times-frac 44.3%

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

      12. unpow2 44.3%

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

      13. *-commutative 44.3%

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

      14. unpow2 44.3%

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

      15. associate-*l* 62.1%

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

   6. Simplified 62.1%

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

   7. Taylor expanded in D around inf 26.6%

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

      1. associate-/l* 26.6%

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

      2. *-commutative 26.6%

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

      3. associate-/l* 26.6%

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

      4. times-frac 26.7%

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

      5. unpow2 26.7%

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

      6. unpow2 26.7%

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

      7. *-commutative 26.7%

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

      8. associate-*r* 26.8%

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

      9. unpow2 26.8%

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

   9. Simplified 26.8%

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

   10. Taylor expanded in D around 0 26.6%

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

       1. unpow2 26.6%

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

       2. *-commutative 26.6%

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

       3. times-frac 26.7%

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

       4. associate-*l/ 26.7%

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

       5. *-commutative 26.7%

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

       6. *-commutative 26.7%

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

       7. unpow2 26.7%

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

       8. associate-*r* 27.4%

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

       9. unpow2 27.4%

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

       10. times-frac 32.5%

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

       11. associate-*r* 31.9%

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

       12. associate-*r/ 31.7%

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

       13. *-commutative 31.7%

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

       14. associate-*l* 31.7%

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

   12. Simplified 31.7%

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

   13. Taylor expanded in M around 0 31.7%

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

       1. unpow2 31.7%

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

       2. *-commutative 31.7%

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

       3. associate-*r/ 31.7%

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

       4. associate-*l* 32.4%

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

   15. Simplified 32.4%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3 \cdot 10^{+74}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \left(\frac{h}{d} \cdot \left(M \cdot \left(M \cdot \frac{w0}{d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 12: 65.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 3 \cdot 10^{+74}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M \cdot \left(M \cdot h\right)}{d} \cdot \frac{w0}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 3e+74)
   w0
   (* -0.125 (* (/ (* D D) l) (* (/ (* M (* M h)) d) (/ w0 d))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3e+74) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) / l) * (((M * (M * h)) / d) * (w0 / d)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= 3d+74) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d * d) / l) * (((m * (m * h)) / d_1) * (w0 / d_1)))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3e+74) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) / l) * (((M * (M * h)) / d) * (w0 / d)));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 3e+74:
		tmp = w0
	else:
		tmp = -0.125 * (((D * D) / l) * (((M * (M * h)) / d) * (w0 / d)))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 3e+74)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) / l) * Float64(Float64(Float64(M * Float64(M * h)) / d) * Float64(w0 / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 3e+74)
		tmp = w0;
	else
		tmp = -0.125 * (((D * D) / l) * (((M * (M * h)) / d) * (w0 / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 3e+74], w0, N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(w0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3e74

   1. Initial program 81.6%

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

      1. times-frac 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in M around 0 71.4%

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

   if 3e74 < M

   1. Initial program 79.6%

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

      1. times-frac 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in M around 0 43.7%

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

      1. +-commutative 43.7%

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

      2. associate-*r/ 43.7%

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

      3. *-commutative 43.7%

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

      4. associate-*r/ 43.7%

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

      5. *-commutative 43.7%

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

      6. fma-def 43.7%

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

      7. *-commutative 43.7%

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

      8. times-frac 38.7%

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

      9. unpow2 38.7%

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

      10. unpow2 38.7%

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

      11. times-frac 44.3%

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

      12. unpow2 44.3%

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

      13. *-commutative 44.3%

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

      14. unpow2 44.3%

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

      15. associate-*l* 62.1%

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

   6. Simplified 62.1%

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

   7. Taylor expanded in D around inf 26.6%

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

      1. associate-/l* 26.6%

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

      2. *-commutative 26.6%

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

      3. associate-/l* 26.6%

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

      4. times-frac 26.7%

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

      5. unpow2 26.7%

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

      6. unpow2 26.7%

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

      7. *-commutative 26.7%

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

      8. associate-*r* 26.8%

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

      9. unpow2 26.8%

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

   9. Simplified 26.8%

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

       1. *-commutative 26.8%

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

       2. *-commutative 26.8%

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

       3. associate-*l* 26.7%

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

       4. times-frac 31.9%

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

       5. associate-*l* 32.5%

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

   11. Applied egg-rr 32.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3 \cdot 10^{+74}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(\frac{M \cdot \left(M \cdot h\right)}{d} \cdot \frac{w0}{d}\right)\right)\\ \end{array} \]

Alternative 13: 68.7% accurate, 216.0× speedup

Math

\[\begin{array}{l} \\ w0 \end{array} \]

FPCore

(FPCore (w0 M D h l d) :precision binary64 w0)

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 81.3%

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

   1. times-frac 81.4%

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

3. Simplified 81.4%

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

4. Taylor expanded in M around 0 69.1%

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

5. Final simplification 69.1%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023243 
(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))))))