Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.3%

Time: 11.0s

Alternatives: 11

Speedup: 2.0×

• 44.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    t_1 = cos(th) / sqrt(2.0d0)
    code = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	return Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    t_1 = cos(th) / sqrt(2.0d0)
    code = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	return Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \cos th \cdot \frac{{2}^{-0.5}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (cos th) (/ (pow 2.0 -0.5) (/ 1.0 (+ (* a1 a1) (* a2 a2))))))

C

double code(double a1, double a2, double th) {
	return cos(th) * (pow(2.0, -0.5) / (1.0 / ((a1 * a1) + (a2 * a2))));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = cos(th) * ((2.0d0 ** (-0.5d0)) / (1.0d0 / ((a1 * a1) + (a2 * a2))))
end function

Java

public static double code(double a1, double a2, double th) {
	return Math.cos(th) * (Math.pow(2.0, -0.5) / (1.0 / ((a1 * a1) + (a2 * a2))));
}

Python

def code(a1, a2, th):
	return math.cos(th) * (math.pow(2.0, -0.5) / (1.0 / ((a1 * a1) + (a2 * a2))))

Julia

function code(a1, a2, th)
	return Float64(cos(th) * Float64((2.0 ^ -0.5) / Float64(1.0 / Float64(Float64(a1 * a1) + Float64(a2 * a2)))))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = cos(th) * ((2.0 ^ -0.5) / (1.0 / ((a1 * a1) + (a2 * a2))));
end

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[Power[2.0, -0.5], $MachinePrecision] / N[(1.0 / N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

   3. associate-/l* N/A

      \[\leadsto \cos th \cdot \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\color{blue}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right) \]

   3. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

   5. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\frac{1}{\sqrt{2}}}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{\sqrt{2}}\right), \color{blue}{\left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)}\right)\right) \]

   7. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{{2}^{\frac{1}{2}}}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   8. pow-flip N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   11. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{\color{blue}{\frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}}\right)\right)\right) \]

   12. flip3-+ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{a1 \cdot a1 + \color{blue}{a2 \cdot a2}}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{/.f64}\left(1, \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \cos th \cdot \color{blue}{\frac{{2}^{-0.5}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}} \]
7. Add Preprocessing

Alternative 2: 99.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {2}^{-0.5} \cdot \left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (pow 2.0 -0.5) (* (cos th) (+ (* a1 a1) (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	return pow(2.0, -0.5) * (cos(th) * ((a1 * a1) + (a2 * a2)));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (2.0d0 ** (-0.5d0)) * (cos(th) * ((a1 * a1) + (a2 * a2)))
end function

Java

public static double code(double a1, double a2, double th) {
	return Math.pow(2.0, -0.5) * (Math.cos(th) * ((a1 * a1) + (a2 * a2)));
}

Python

def code(a1, a2, th):
	return math.pow(2.0, -0.5) * (math.cos(th) * ((a1 * a1) + (a2 * a2)))

Julia

function code(a1, a2, th)
	return Float64((2.0 ^ -0.5) * Float64(cos(th) * Float64(Float64(a1 * a1) + Float64(a2 * a2))))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (2.0 ^ -0.5) * (cos(th) * ((a1 * a1) + (a2 * a2)));
end

Wolfram

code[a1_, a2_, th_] := N[(N[Power[2.0, -0.5], $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

   3. associate-/l* N/A

      \[\leadsto \cos th \cdot \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   3. clear-num N/A

      \[\leadsto \frac{1}{\frac{\sqrt{2}}{\cos th}} \cdot \left(\color{blue}{a1 \cdot a1} + a2 \cdot a2\right) \]

   4. associate-/r/ N/A

      \[\leadsto \left(\frac{1}{\sqrt{2}} \cdot \cos th\right) \cdot \left(\color{blue}{a1 \cdot a1} + a2 \cdot a2\right) \]

   5. associate-*l* N/A

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{\left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\sqrt{2}}\right), \color{blue}{\left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)}\right) \]

   7. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{2}^{\frac{1}{2}}}\right), \left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

   8. pow-flip N/A

      \[\leadsto \mathsf{*.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\color{blue}{\cos th} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{\cos th} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{*.f64}\left(\cos th, \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right) \]

   12. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\color{blue}{a1 \cdot a1} + a2 \cdot a2\right)\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{+.f64}\left(\left(a1 \cdot a1\right), \color{blue}{\left(a2 \cdot a2\right)}\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(\color{blue}{a2} \cdot a2\right)\right)\right)\right) \]

   15. *-lowering-*.f64 99.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right)\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{{2}^{-0.5} \cdot \left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)} \]
7. Add Preprocessing

Alternative 3: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos th \cdot \left(\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (cos th) (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5))))

C

double code(double a1, double a2, double th) {
	return cos(th) * (((a1 * a1) + (a2 * a2)) * sqrt(0.5));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = cos(th) * (((a1 * a1) + (a2 * a2)) * sqrt(0.5d0))
end function

Java

public static double code(double a1, double a2, double th) {
	return Math.cos(th) * (((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5));
}

Python

def code(a1, a2, th):
	return math.cos(th) * (((a1 * a1) + (a2 * a2)) * math.sqrt(0.5))

Julia

function code(a1, a2, th)
	return Float64(cos(th) * Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = cos(th) * (((a1 * a1) + (a2 * a2)) * sqrt(0.5));
end

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

   3. associate-/l* N/A

      \[\leadsto \cos th \cdot \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\color{blue}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right) \]

   3. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

   5. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\frac{1}{\sqrt{2}}}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{\sqrt{2}}\right), \color{blue}{\left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)}\right)\right) \]

   7. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{{2}^{\frac{1}{2}}}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   8. pow-flip N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   11. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{\color{blue}{\frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}}\right)\right)\right) \]

   12. flip3-+ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{a1 \cdot a1 + \color{blue}{a2 \cdot a2}}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{/.f64}\left(1, \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \cos th \cdot \color{blue}{\frac{{2}^{-0.5}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}} \]

7. Taylor expanded in a1 around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \color{blue}{\left({a1}^{2} \cdot \sqrt{\frac{1}{2}} + {a2}^{2} \cdot \sqrt{\frac{1}{2}}\right)}\right) \]
8. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\sqrt{\frac{1}{2}} \cdot \color{blue}{\left({a1}^{2} + {a2}^{2}\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left({a1}^{2} + {a2}^{2}\right)}\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\color{blue}{{a1}^{2}} + {a2}^{2}\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{+.f64}\left(\left({a1}^{2}\right), \color{blue}{\left({a2}^{2}\right)}\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left({\color{blue}{a2}}^{2}\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left({\color{blue}{a2}}^{2}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot \color{blue}{a2}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 99.6%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right)\right)\right) \]

9. Simplified 99.6%

   \[\leadsto \cos th \cdot \color{blue}{\left(\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)} \]

10. Final simplification 99.6%

    \[\leadsto \cos th \cdot \left(\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\right) \]
11. Add Preprocessing

Alternative 4: 57.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5} \cdot \left(a2 \cdot \left(\cos th \cdot a2\right)\right) \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* (sqrt 0.5) (* a2 (* (cos th) a2))))

C

double code(double a1, double a2, double th) {
	return sqrt(0.5) * (a2 * (cos(th) * a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = sqrt(0.5d0) * (a2 * (cos(th) * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	return Math.sqrt(0.5) * (a2 * (Math.cos(th) * a2));
}

Python

def code(a1, a2, th):
	return math.sqrt(0.5) * (a2 * (math.cos(th) * a2))

Julia

function code(a1, a2, th)
	return Float64(sqrt(0.5) * Float64(a2 * Float64(cos(th) * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = sqrt(0.5) * (a2 * (cos(th) * a2));
end

Wolfram

code[a1_, a2_, th_] := N[(N[Sqrt[0.5], $MachinePrecision] * N[(a2 * N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

   3. associate-/l* N/A

      \[\leadsto \cos th \cdot \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\color{blue}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right) \]

   3. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

   5. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\frac{1}{\sqrt{2}}}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{\sqrt{2}}\right), \color{blue}{\left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)}\right)\right) \]

   7. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{{2}^{\frac{1}{2}}}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   8. pow-flip N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   11. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{\color{blue}{\frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}}\right)\right)\right) \]

   12. flip3-+ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{a1 \cdot a1 + \color{blue}{a2 \cdot a2}}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{/.f64}\left(1, \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \cos th \cdot \color{blue}{\frac{{2}^{-0.5}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}} \]

7. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{{a2}^{2} \cdot \left(\cos th \cdot \sqrt{\frac{1}{2}}\right)} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left({a2}^{2} \cdot \cos th\right) \cdot \color{blue}{\sqrt{\frac{1}{2}}} \]

   2. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{2}} \cdot \color{blue}{\left({a2}^{2} \cdot \cos th\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left({a2}^{2} \cdot \cos th\right)}\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\color{blue}{{a2}^{2}} \cdot \cos th\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\cos th \cdot \color{blue}{{a2}^{2}}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\cos th \cdot \left(a2 \cdot \color{blue}{a2}\right)\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\left(\cos th \cdot a2\right) \cdot \color{blue}{a2}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\left(\cos th \cdot a2\right), \color{blue}{a2}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos th, a2\right), a2\right)\right) \]

   10. cos-lowering-cos.f64 58.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), a2\right), a2\right)\right) \]

9. Simplified 58.8%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left(\left(\cos th \cdot a2\right) \cdot a2\right)} \]

10. Final simplification 58.8%

    \[\leadsto \sqrt{0.5} \cdot \left(a2 \cdot \left(\cos th \cdot a2\right)\right) \]
11. Add Preprocessing

Alternative 5: 57.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right) \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* (cos th) (* (* a2 a2) (sqrt 0.5))))

C

double code(double a1, double a2, double th) {
	return cos(th) * ((a2 * a2) * sqrt(0.5));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = cos(th) * ((a2 * a2) * sqrt(0.5d0))
end function

Java

public static double code(double a1, double a2, double th) {
	return Math.cos(th) * ((a2 * a2) * Math.sqrt(0.5));
}

Python

def code(a1, a2, th):
	return math.cos(th) * ((a2 * a2) * math.sqrt(0.5))

Julia

function code(a1, a2, th)
	return Float64(cos(th) * Float64(Float64(a2 * a2) * sqrt(0.5)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = cos(th) * ((a2 * a2) * sqrt(0.5));
end

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

   3. associate-/l* N/A

      \[\leadsto \cos th \cdot \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\color{blue}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right) \]

   3. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

   5. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\frac{1}{\sqrt{2}}}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{\sqrt{2}}\right), \color{blue}{\left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)}\right)\right) \]

   7. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{{2}^{\frac{1}{2}}}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   8. pow-flip N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   11. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{\color{blue}{\frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}}\right)\right)\right) \]

   12. flip3-+ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{a1 \cdot a1 + \color{blue}{a2 \cdot a2}}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{/.f64}\left(1, \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \cos th \cdot \color{blue}{\frac{{2}^{-0.5}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}} \]

7. Taylor expanded in a1 around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \color{blue}{\left({a2}^{2} \cdot \sqrt{\frac{1}{2}}\right)}\right) \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\sqrt{\frac{1}{2}} \cdot \color{blue}{{a2}^{2}}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left({a2}^{2}\right)}\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left({\color{blue}{a2}}^{2}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(a2 \cdot \color{blue}{a2}\right)\right)\right) \]

   5. *-lowering-*.f64 58.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right)\right) \]

9. Simplified 58.8%

   \[\leadsto \cos th \cdot \color{blue}{\left(\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\right)} \]

10. Final simplification 58.8%

    \[\leadsto \cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right) \]
11. Add Preprocessing

Alternative 6: 64.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 2.05 \cdot 10^{+40}:\\ \;\;\;\;\frac{{2}^{-0.5}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}\\ \mathbf{elif}\;th \leq 3.3 \cdot 10^{+148}:\\ \;\;\;\;\left(-0.5 \cdot \left(th \cdot th\right)\right) \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 2.05e+40)
   (/ (pow 2.0 -0.5) (/ 1.0 (+ (* a1 a1) (* a2 a2))))
   (if (<= th 3.3e+148)
     (* (* -0.5 (* th th)) (/ (* a2 a2) (sqrt 2.0)))
     (* a2 (* a2 (sqrt 0.5))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 2.05e+40) {
		tmp = pow(2.0, -0.5) / (1.0 / ((a1 * a1) + (a2 * a2)));
	} else if (th <= 3.3e+148) {
		tmp = (-0.5 * (th * th)) * ((a2 * a2) / sqrt(2.0));
	} else {
		tmp = a2 * (a2 * sqrt(0.5));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 2.05d+40) then
        tmp = (2.0d0 ** (-0.5d0)) / (1.0d0 / ((a1 * a1) + (a2 * a2)))
    else if (th <= 3.3d+148) then
        tmp = ((-0.5d0) * (th * th)) * ((a2 * a2) / sqrt(2.0d0))
    else
        tmp = a2 * (a2 * sqrt(0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 2.05e+40) {
		tmp = Math.pow(2.0, -0.5) / (1.0 / ((a1 * a1) + (a2 * a2)));
	} else if (th <= 3.3e+148) {
		tmp = (-0.5 * (th * th)) * ((a2 * a2) / Math.sqrt(2.0));
	} else {
		tmp = a2 * (a2 * Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 2.05e+40:
		tmp = math.pow(2.0, -0.5) / (1.0 / ((a1 * a1) + (a2 * a2)))
	elif th <= 3.3e+148:
		tmp = (-0.5 * (th * th)) * ((a2 * a2) / math.sqrt(2.0))
	else:
		tmp = a2 * (a2 * math.sqrt(0.5))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 2.05e+40)
		tmp = Float64((2.0 ^ -0.5) / Float64(1.0 / Float64(Float64(a1 * a1) + Float64(a2 * a2))));
	elseif (th <= 3.3e+148)
		tmp = Float64(Float64(-0.5 * Float64(th * th)) * Float64(Float64(a2 * a2) / sqrt(2.0)));
	else
		tmp = Float64(a2 * Float64(a2 * sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 2.05e+40)
		tmp = (2.0 ^ -0.5) / (1.0 / ((a1 * a1) + (a2 * a2)));
	elseif (th <= 3.3e+148)
		tmp = (-0.5 * (th * th)) * ((a2 * a2) / sqrt(2.0));
	else
		tmp = a2 * (a2 * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 2.05e+40], N[(N[Power[2.0, -0.5], $MachinePrecision] / N[(1.0 / N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 3.3e+148], N[(N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 2.0500000000000001e40

   1. Initial program 99.5%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

      3. associate-/l* N/A

         \[\leadsto \cos th \cdot \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\frac{{a1}^{2} + {a2}^{2}}{\sqrt{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({a1}^{2} + {a2}^{2}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({a1}^{2}\right), \left({a2}^{2}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left({a2}^{2}\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left({a2}^{2}\right)\right), \left(\sqrt{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 72.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   7. Simplified 72.0%

      \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}} \]

      2. div-inv N/A

         \[\leadsto \frac{1}{\sqrt{2} \cdot \color{blue}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{\sqrt{2}}}{\color{blue}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}} \]

      4. pow1/2 N/A

         \[\leadsto \frac{\frac{1}{{2}^{\frac{1}{2}}}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}} \]

      5. pow-flip N/A

         \[\leadsto \frac{{2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\frac{\color{blue}{1}}{a1 \cdot a1 + a2 \cdot a2}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{{2}^{\frac{-1}{2}}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({2}^{\frac{-1}{2}}\right), \color{blue}{\left(\frac{1}{a1 \cdot a1 + a2 \cdot a2}\right)}\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{\color{blue}{1}}{a1 \cdot a1 + a2 \cdot a2}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{/.f64}\left(1, \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(a1 \cdot a1\right), \color{blue}{\left(a2 \cdot a2\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(\color{blue}{a2} \cdot a2\right)\right)\right)\right) \]

      12. *-lowering-*.f64 72.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right)\right)\right) \]

   9. Applied egg-rr 72.1%

      \[\leadsto \color{blue}{\frac{{2}^{-0.5}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}} \]

   if 2.0500000000000001e40 < th < 3.3000000000000001e148

   1. Initial program 99.8%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

      3. associate-/l* N/A

         \[\leadsto \cos th \cdot \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \color{blue}{\left(\frac{{a2}^{2}}{\sqrt{2}}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left({a2}^{2}\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a2 \cdot a2\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 45.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   7. Simplified 45.9%

      \[\leadsto \cos th \cdot \color{blue}{\frac{a2 \cdot a2}{\sqrt{2}}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {th}^{2}\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {th}^{2}\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(a2, a2\right)}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({th}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, \color{blue}{a2}\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(th \cdot th\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      4. *-lowering-*.f64 31.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(th, th\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   10. Simplified 31.5%

       \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(th \cdot th\right)\right)} \cdot \frac{a2 \cdot a2}{\sqrt{2}} \]

   11. Taylor expanded in th around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot {th}^{2}\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left({th}^{2}\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(a2, a2\right)}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(th \cdot th\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, \color{blue}{a2}\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       3. *-lowering-*.f64 31.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(th, th\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, \color{blue}{a2}\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   13. Simplified 31.5%

       \[\leadsto \color{blue}{\left(-0.5 \cdot \left(th \cdot th\right)\right)} \cdot \frac{a2 \cdot a2}{\sqrt{2}} \]

   if 3.3000000000000001e148 < th

   1. Initial program 99.5%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

      3. associate-/l* N/A

         \[\leadsto \cos th \cdot \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\color{blue}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right) \]

      3. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\frac{1}{\sqrt{2}}}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{\sqrt{2}}\right), \color{blue}{\left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{{2}^{\frac{1}{2}}}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

      8. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{\color{blue}{\frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}}\right)\right)\right) \]

      12. flip3-+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{a1 \cdot a1 + \color{blue}{a2 \cdot a2}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{/.f64}\left(1, \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \cos th \cdot \color{blue}{\frac{{2}^{-0.5}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}} \]

   7. Taylor expanded in a1 around 0

      \[\leadsto \color{blue}{{a2}^{2} \cdot \left(\cos th \cdot \sqrt{\frac{1}{2}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left({a2}^{2} \cdot \cos th\right) \cdot \color{blue}{\sqrt{\frac{1}{2}}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2}} \cdot \color{blue}{\left({a2}^{2} \cdot \cos th\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left({a2}^{2} \cdot \cos th\right)}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\color{blue}{{a2}^{2}} \cdot \cos th\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\cos th \cdot \color{blue}{{a2}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\cos th \cdot \left(a2 \cdot \color{blue}{a2}\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\left(\cos th \cdot a2\right) \cdot \color{blue}{a2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\left(\cos th \cdot a2\right), \color{blue}{a2}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos th, a2\right), a2\right)\right) \]

      10. cos-lowering-cos.f64 64.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), a2\right), a2\right)\right) \]

   9. Simplified 64.7%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left(\left(\cos th \cdot a2\right) \cdot a2\right)} \]

   10. Taylor expanded in th around 0

       \[\leadsto \color{blue}{{a2}^{2} \cdot \sqrt{\frac{1}{2}}} \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \left(a2 \cdot a2\right) \cdot \sqrt{\color{blue}{\frac{1}{2}}} \]

       2. associate-*l* N/A

          \[\leadsto a2 \cdot \color{blue}{\left(a2 \cdot \sqrt{\frac{1}{2}}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a2, \color{blue}{\left(a2 \cdot \sqrt{\frac{1}{2}}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \color{blue}{\left(\sqrt{\frac{1}{2}}\right)}\right)\right) \]

       5. sqrt-lowering-sqrt.f64 38.6%

          \[\leadsto \mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right) \]

   12. Simplified 38.6%

       \[\leadsto \color{blue}{a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 64.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 2.05 \cdot 10^{+40}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;th \leq 3.3 \cdot 10^{+148}:\\ \;\;\;\;\left(-0.5 \cdot \left(th \cdot th\right)\right) \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 2.05e+40)
   (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5))
   (if (<= th 3.3e+148)
     (* (* -0.5 (* th th)) (/ (* a2 a2) (sqrt 2.0)))
     (* a2 (* a2 (sqrt 0.5))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 2.05e+40) {
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	} else if (th <= 3.3e+148) {
		tmp = (-0.5 * (th * th)) * ((a2 * a2) / sqrt(2.0));
	} else {
		tmp = a2 * (a2 * sqrt(0.5));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 2.05d+40) then
        tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
    else if (th <= 3.3d+148) then
        tmp = ((-0.5d0) * (th * th)) * ((a2 * a2) / sqrt(2.0d0))
    else
        tmp = a2 * (a2 * sqrt(0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 2.05e+40) {
		tmp = ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
	} else if (th <= 3.3e+148) {
		tmp = (-0.5 * (th * th)) * ((a2 * a2) / Math.sqrt(2.0));
	} else {
		tmp = a2 * (a2 * Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 2.05e+40:
		tmp = ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)
	elif th <= 3.3e+148:
		tmp = (-0.5 * (th * th)) * ((a2 * a2) / math.sqrt(2.0))
	else:
		tmp = a2 * (a2 * math.sqrt(0.5))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 2.05e+40)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5));
	elseif (th <= 3.3e+148)
		tmp = Float64(Float64(-0.5 * Float64(th * th)) * Float64(Float64(a2 * a2) / sqrt(2.0)));
	else
		tmp = Float64(a2 * Float64(a2 * sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 2.05e+40)
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	elseif (th <= 3.3e+148)
		tmp = (-0.5 * (th * th)) * ((a2 * a2) / sqrt(2.0));
	else
		tmp = a2 * (a2 * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 2.05e+40], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 3.3e+148], N[(N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 2.0500000000000001e40

   1. Initial program 99.5%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

      3. associate-/l* N/A

         \[\leadsto \cos th \cdot \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\color{blue}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right) \]

      3. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\frac{1}{\sqrt{2}}}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{\sqrt{2}}\right), \color{blue}{\left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{{2}^{\frac{1}{2}}}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

      8. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{\color{blue}{\frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}}\right)\right)\right) \]

      12. flip3-+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{a1 \cdot a1 + \color{blue}{a2 \cdot a2}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{/.f64}\left(1, \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \cos th \cdot \color{blue}{\frac{{2}^{-0.5}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}} \]

   7. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)} \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left({a1}^{2} + {a2}^{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\color{blue}{{a1}^{2}} + {a2}^{2}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{+.f64}\left(\left({a1}^{2}\right), \color{blue}{\left({a2}^{2}\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left({\color{blue}{a2}}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left({\color{blue}{a2}}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot \color{blue}{a2}\right)\right)\right) \]

      7. *-lowering-*.f64 72.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right)\right) \]

   9. Simplified 72.1%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   if 2.0500000000000001e40 < th < 3.3000000000000001e148

   1. Initial program 99.8%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

      3. associate-/l* N/A

         \[\leadsto \cos th \cdot \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \color{blue}{\left(\frac{{a2}^{2}}{\sqrt{2}}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left({a2}^{2}\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a2 \cdot a2\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 45.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   7. Simplified 45.9%

      \[\leadsto \cos th \cdot \color{blue}{\frac{a2 \cdot a2}{\sqrt{2}}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {th}^{2}\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {th}^{2}\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(a2, a2\right)}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({th}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, \color{blue}{a2}\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(th \cdot th\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      4. *-lowering-*.f64 31.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(th, th\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   10. Simplified 31.5%

       \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(th \cdot th\right)\right)} \cdot \frac{a2 \cdot a2}{\sqrt{2}} \]

   11. Taylor expanded in th around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot {th}^{2}\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left({th}^{2}\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(a2, a2\right)}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(th \cdot th\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, \color{blue}{a2}\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       3. *-lowering-*.f64 31.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(th, th\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, \color{blue}{a2}\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   13. Simplified 31.5%

       \[\leadsto \color{blue}{\left(-0.5 \cdot \left(th \cdot th\right)\right)} \cdot \frac{a2 \cdot a2}{\sqrt{2}} \]

   if 3.3000000000000001e148 < th

   1. Initial program 99.5%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

      3. associate-/l* N/A

         \[\leadsto \cos th \cdot \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\color{blue}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right) \]

      3. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\frac{1}{\sqrt{2}}}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{\sqrt{2}}\right), \color{blue}{\left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)}\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{{2}^{\frac{1}{2}}}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

      8. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{\color{blue}{\frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}}\right)\right)\right) \]

      12. flip3-+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{a1 \cdot a1 + \color{blue}{a2 \cdot a2}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{/.f64}\left(1, \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \cos th \cdot \color{blue}{\frac{{2}^{-0.5}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}} \]

   7. Taylor expanded in a1 around 0

      \[\leadsto \color{blue}{{a2}^{2} \cdot \left(\cos th \cdot \sqrt{\frac{1}{2}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left({a2}^{2} \cdot \cos th\right) \cdot \color{blue}{\sqrt{\frac{1}{2}}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2}} \cdot \color{blue}{\left({a2}^{2} \cdot \cos th\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left({a2}^{2} \cdot \cos th\right)}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\color{blue}{{a2}^{2}} \cdot \cos th\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\cos th \cdot \color{blue}{{a2}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\cos th \cdot \left(a2 \cdot \color{blue}{a2}\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\left(\cos th \cdot a2\right) \cdot \color{blue}{a2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\left(\cos th \cdot a2\right), \color{blue}{a2}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos th, a2\right), a2\right)\right) \]

      10. cos-lowering-cos.f64 64.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), a2\right), a2\right)\right) \]

   9. Simplified 64.7%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left(\left(\cos th \cdot a2\right) \cdot a2\right)} \]

   10. Taylor expanded in th around 0

       \[\leadsto \color{blue}{{a2}^{2} \cdot \sqrt{\frac{1}{2}}} \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \left(a2 \cdot a2\right) \cdot \sqrt{\color{blue}{\frac{1}{2}}} \]

       2. associate-*l* N/A

          \[\leadsto a2 \cdot \color{blue}{\left(a2 \cdot \sqrt{\frac{1}{2}}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a2, \color{blue}{\left(a2 \cdot \sqrt{\frac{1}{2}}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \color{blue}{\left(\sqrt{\frac{1}{2}}\right)}\right)\right) \]

       5. sqrt-lowering-sqrt.f64 38.6%

          \[\leadsto \mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right) \]

   12. Simplified 38.6%

       \[\leadsto \color{blue}{a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 2.05 \cdot 10^{+40}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;th \leq 3.3 \cdot 10^{+148}:\\ \;\;\;\;\left(-0.5 \cdot \left(th \cdot th\right)\right) \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 5 \cdot 10^{+224}:\\ \;\;\;\;\frac{{2}^{-0.5}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(th \cdot th\right)\right) \cdot \left(a2 \cdot \frac{a2}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 5e+224)
   (/ (pow 2.0 -0.5) (/ 1.0 (+ (* a1 a1) (* a2 a2))))
   (* (+ 1.0 (* -0.5 (* th th))) (* a2 (/ a2 (sqrt 2.0))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 5e+224) {
		tmp = pow(2.0, -0.5) / (1.0 / ((a1 * a1) + (a2 * a2)));
	} else {
		tmp = (1.0 + (-0.5 * (th * th))) * (a2 * (a2 / sqrt(2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (a2 <= 5d+224) then
        tmp = (2.0d0 ** (-0.5d0)) / (1.0d0 / ((a1 * a1) + (a2 * a2)))
    else
        tmp = (1.0d0 + ((-0.5d0) * (th * th))) * (a2 * (a2 / sqrt(2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 5e+224) {
		tmp = Math.pow(2.0, -0.5) / (1.0 / ((a1 * a1) + (a2 * a2)));
	} else {
		tmp = (1.0 + (-0.5 * (th * th))) * (a2 * (a2 / Math.sqrt(2.0)));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 5e+224:
		tmp = math.pow(2.0, -0.5) / (1.0 / ((a1 * a1) + (a2 * a2)))
	else:
		tmp = (1.0 + (-0.5 * (th * th))) * (a2 * (a2 / math.sqrt(2.0)))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 5e+224)
		tmp = Float64((2.0 ^ -0.5) / Float64(1.0 / Float64(Float64(a1 * a1) + Float64(a2 * a2))));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(th * th))) * Float64(a2 * Float64(a2 / sqrt(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 5e+224)
		tmp = (2.0 ^ -0.5) / (1.0 / ((a1 * a1) + (a2 * a2)));
	else
		tmp = (1.0 + (-0.5 * (th * th))) * (a2 * (a2 / sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 5e+224], N[(N[Power[2.0, -0.5], $MachinePrecision] / N[(1.0 / N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 4.99999999999999964e224

   1. Initial program 99.5%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

      3. associate-/l* N/A

         \[\leadsto \cos th \cdot \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\frac{{a1}^{2} + {a2}^{2}}{\sqrt{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({a1}^{2} + {a2}^{2}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({a1}^{2}\right), \left({a2}^{2}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left({a2}^{2}\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left({a2}^{2}\right)\right), \left(\sqrt{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 62.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   7. Simplified 62.5%

      \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}} \]

      2. div-inv N/A

         \[\leadsto \frac{1}{\sqrt{2} \cdot \color{blue}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{\sqrt{2}}}{\color{blue}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}} \]

      4. pow1/2 N/A

         \[\leadsto \frac{\frac{1}{{2}^{\frac{1}{2}}}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}} \]

      5. pow-flip N/A

         \[\leadsto \frac{{2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\frac{\color{blue}{1}}{a1 \cdot a1 + a2 \cdot a2}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{{2}^{\frac{-1}{2}}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({2}^{\frac{-1}{2}}\right), \color{blue}{\left(\frac{1}{a1 \cdot a1 + a2 \cdot a2}\right)}\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{\color{blue}{1}}{a1 \cdot a1 + a2 \cdot a2}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{/.f64}\left(1, \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(a1 \cdot a1\right), \color{blue}{\left(a2 \cdot a2\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(\color{blue}{a2} \cdot a2\right)\right)\right)\right) \]

      12. *-lowering-*.f64 62.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right)\right)\right) \]

   9. Applied egg-rr 62.6%

      \[\leadsto \color{blue}{\frac{{2}^{-0.5}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}} \]

   if 4.99999999999999964e224 < a2

   1. Initial program 100.0%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

      3. associate-/l* N/A

         \[\leadsto \cos th \cdot \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \color{blue}{\left(\frac{{a2}^{2}}{\sqrt{2}}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left({a2}^{2}\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a2 \cdot a2\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \cos th \cdot \color{blue}{\frac{a2 \cdot a2}{\sqrt{2}}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {th}^{2}\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {th}^{2}\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(a2, a2\right)}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({th}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, \color{blue}{a2}\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(th \cdot th\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      4. *-lowering-*.f64 85.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(th, th\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   10. Simplified 85.0%

       \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(th \cdot th\right)\right)} \cdot \frac{a2 \cdot a2}{\sqrt{2}} \]
   11. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(th, th\right)\right)\right), \left(a2 \cdot \color{blue}{\frac{a2}{\sqrt{2}}}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(th, th\right)\right)\right), \left(\frac{a2}{\sqrt{2}} \cdot \color{blue}{a2}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(th, th\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{a2}{\sqrt{2}}\right), \color{blue}{a2}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(th, th\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(a2, \left(\sqrt{2}\right)\right), a2\right)\right) \]

       5. sqrt-lowering-sqrt.f64 85.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(th, th\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(a2, \mathsf{sqrt.f64}\left(2\right)\right), a2\right)\right) \]

   12. Applied egg-rr 85.0%

       \[\leadsto \left(1 + -0.5 \cdot \left(th \cdot th\right)\right) \cdot \color{blue}{\left(\frac{a2}{\sqrt{2}} \cdot a2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 5 \cdot 10^{+224}:\\ \;\;\;\;\frac{{2}^{-0.5}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(th \cdot th\right)\right) \cdot \left(a2 \cdot \frac{a2}{\sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5)))

C

double code(double a1, double a2, double th) {
	return ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
end function

Java

public static double code(double a1, double a2, double th) {
	return ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
}

Python

def code(a1, a2, th):
	return ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)

Julia

function code(a1, a2, th)
	return Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

   3. associate-/l* N/A

      \[\leadsto \cos th \cdot \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\color{blue}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right) \]

   3. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

   5. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\frac{1}{\sqrt{2}}}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{\sqrt{2}}\right), \color{blue}{\left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)}\right)\right) \]

   7. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{{2}^{\frac{1}{2}}}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   8. pow-flip N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   11. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{\color{blue}{\frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}}\right)\right)\right) \]

   12. flip3-+ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{a1 \cdot a1 + \color{blue}{a2 \cdot a2}}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{/.f64}\left(1, \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \cos th \cdot \color{blue}{\frac{{2}^{-0.5}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}} \]

7. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)} \]
8. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left({a1}^{2} + {a2}^{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\color{blue}{{a1}^{2}} + {a2}^{2}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{+.f64}\left(\left({a1}^{2}\right), \color{blue}{\left({a2}^{2}\right)}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left({\color{blue}{a2}}^{2}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left({\color{blue}{a2}}^{2}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot \color{blue}{a2}\right)\right)\right) \]

   7. *-lowering-*.f64 63.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right)\right) \]

9. Simplified 63.9%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

10. Final simplification 63.9%

    \[\leadsto \left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5} \]
11. Add Preprocessing

Alternative 10: 39.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \left(a2 \cdot a2\right) \cdot \sqrt{0.5} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* (* a2 a2) (sqrt 0.5)))

C

double code(double a1, double a2, double th) {
	return (a2 * a2) * sqrt(0.5);
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (a2 * a2) * sqrt(0.5d0)
end function

Java

public static double code(double a1, double a2, double th) {
	return (a2 * a2) * Math.sqrt(0.5);
}

Python

def code(a1, a2, th):
	return (a2 * a2) * math.sqrt(0.5)

Julia

function code(a1, a2, th)
	return Float64(Float64(a2 * a2) * sqrt(0.5))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (a2 * a2) * sqrt(0.5);
end

Wolfram

code[a1_, a2_, th_] := N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

   3. associate-/l* N/A

      \[\leadsto \cos th \cdot \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\color{blue}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right) \]

   3. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

   5. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\frac{1}{\sqrt{2}}}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{\sqrt{2}}\right), \color{blue}{\left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)}\right)\right) \]

   7. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{{2}^{\frac{1}{2}}}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   8. pow-flip N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   11. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{\color{blue}{\frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}}\right)\right)\right) \]

   12. flip3-+ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{a1 \cdot a1 + \color{blue}{a2 \cdot a2}}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{/.f64}\left(1, \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \cos th \cdot \color{blue}{\frac{{2}^{-0.5}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}} \]

7. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{{a2}^{2} \cdot \left(\cos th \cdot \sqrt{\frac{1}{2}}\right)} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left({a2}^{2} \cdot \cos th\right) \cdot \color{blue}{\sqrt{\frac{1}{2}}} \]

   2. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{2}} \cdot \color{blue}{\left({a2}^{2} \cdot \cos th\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left({a2}^{2} \cdot \cos th\right)}\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\color{blue}{{a2}^{2}} \cdot \cos th\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\cos th \cdot \color{blue}{{a2}^{2}}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\cos th \cdot \left(a2 \cdot \color{blue}{a2}\right)\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\left(\cos th \cdot a2\right) \cdot \color{blue}{a2}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\left(\cos th \cdot a2\right), \color{blue}{a2}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos th, a2\right), a2\right)\right) \]

   10. cos-lowering-cos.f64 58.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), a2\right), a2\right)\right) \]

9. Simplified 58.8%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left(\left(\cos th \cdot a2\right) \cdot a2\right)} \]

10. Taylor expanded in th around 0

    \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \color{blue}{\left({a2}^{2}\right)}\right) \]
11. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(a2 \cdot \color{blue}{a2}\right)\right) \]

    2. *-lowering-*.f64 41.1%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right) \]

12. Simplified 41.1%

    \[\leadsto \sqrt{0.5} \cdot \color{blue}{\left(a2 \cdot a2\right)} \]

13. Final simplification 41.1%

    \[\leadsto \left(a2 \cdot a2\right) \cdot \sqrt{0.5} \]
14. Add Preprocessing

Alternative 11: 39.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ a2 \cdot \left(a2 \cdot \sqrt{0.5}\right) \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* a2 (* a2 (sqrt 0.5))))

C

double code(double a1, double a2, double th) {
	return a2 * (a2 * sqrt(0.5));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 * (a2 * sqrt(0.5d0))
end function

Java

public static double code(double a1, double a2, double th) {
	return a2 * (a2 * Math.sqrt(0.5));
}

Python

def code(a1, a2, th):
	return a2 * (a2 * math.sqrt(0.5))

Julia

function code(a1, a2, th)
	return Float64(a2 * Float64(a2 * sqrt(0.5)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = a2 * (a2 * sqrt(0.5));
end

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

   3. associate-/l* N/A

      \[\leadsto \cos th \cdot \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\color{blue}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right) \]

   3. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

   5. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\frac{1}{\sqrt{2}}}{\color{blue}{\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{\sqrt{2}}\right), \color{blue}{\left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)}\right)\right) \]

   7. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(\frac{1}{{2}^{\frac{1}{2}}}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   8. pow-flip N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\frac{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}\right)\right)\right) \]

   11. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{\color{blue}{\frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}}\right)\right)\right) \]

   12. flip3-+ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \left(\frac{1}{a1 \cdot a1 + \color{blue}{a2 \cdot a2}}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{/.f64}\left(1, \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \cos th \cdot \color{blue}{\frac{{2}^{-0.5}}{\frac{1}{a1 \cdot a1 + a2 \cdot a2}}} \]

7. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{{a2}^{2} \cdot \left(\cos th \cdot \sqrt{\frac{1}{2}}\right)} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left({a2}^{2} \cdot \cos th\right) \cdot \color{blue}{\sqrt{\frac{1}{2}}} \]

   2. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{2}} \cdot \color{blue}{\left({a2}^{2} \cdot \cos th\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left({a2}^{2} \cdot \cos th\right)}\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\color{blue}{{a2}^{2}} \cdot \cos th\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\cos th \cdot \color{blue}{{a2}^{2}}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\cos th \cdot \left(a2 \cdot \color{blue}{a2}\right)\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\left(\cos th \cdot a2\right) \cdot \color{blue}{a2}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\left(\cos th \cdot a2\right), \color{blue}{a2}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos th, a2\right), a2\right)\right) \]

   10. cos-lowering-cos.f64 58.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), a2\right), a2\right)\right) \]

9. Simplified 58.8%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left(\left(\cos th \cdot a2\right) \cdot a2\right)} \]

10. Taylor expanded in th around 0

    \[\leadsto \color{blue}{{a2}^{2} \cdot \sqrt{\frac{1}{2}}} \]
11. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto \left(a2 \cdot a2\right) \cdot \sqrt{\color{blue}{\frac{1}{2}}} \]

    2. associate-*l* N/A

       \[\leadsto a2 \cdot \color{blue}{\left(a2 \cdot \sqrt{\frac{1}{2}}\right)} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(a2, \color{blue}{\left(a2 \cdot \sqrt{\frac{1}{2}}\right)}\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \color{blue}{\left(\sqrt{\frac{1}{2}}\right)}\right)\right) \]

    5. sqrt-lowering-sqrt.f64 41.1%

       \[\leadsto \mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right) \]

12. Simplified 41.1%

    \[\leadsto \color{blue}{a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))