Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 10.3s

Alternatives: 12

Speedup: 2.0×

• 43.4% 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: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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)) * ((a1 * a1) + (a2 * a2))) * cos(th)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing
3. 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. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

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

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

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

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

   9. pow1/2 N/A

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

   10. pow-flip N/A

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

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

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

   12. metadata-eval N/A

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

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

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

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

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

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

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

   16. cos-lowering-cos.f64 99.7%

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

4. Applied egg-rr 99.7%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{a1 \cdot a1 + a2 \cdot a2}{\frac{\sqrt{2}}{\cos th}} \end{array} \]

FPCore

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

C

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

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(2.0d0) / cos(th))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing
3. 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. *-commutative N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. cos-lowering-cos.f64 99.6%

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

4. Applied egg-rr 99.6%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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)) * cos(th)) / sqrt(2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

   9. sqrt-lowering-sqrt.f64 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

   \[\leadsto \frac{\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \cos th}{\sqrt{2}} \]
6. Add Preprocessing

Alternative 4: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}} \end{array} \]

FPCore

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

C

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

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(2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing
3. 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. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. cos-lowering-cos.f64 99.6%

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 5: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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)) * (cos(th) / sqrt(2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

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

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

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

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

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

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

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

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

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

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

   8. *-lowering-*.f64 99.5%

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

   \[\leadsto \left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \]
6. Add Preprocessing

Alternative 6: 57.3% accurate, 2.0× speedup

Math

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

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(a2 * Float64(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[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Add Preprocessing
3. 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. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

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

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

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

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

   9. pow1/2 N/A

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

   10. pow-flip N/A

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

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

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

   12. metadata-eval N/A

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

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

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

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

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

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

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

   16. cos-lowering-cos.f64 99.7%

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. sqrt-lowering-sqrt.f64 58.8%

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

7. Simplified 58.8%

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

8. Final simplification 58.8%

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

Alternative 7: 57.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return (a2 * a2) * (cos(th) * 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) * (cos(th) * sqrt(0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing
3. 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. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

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

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

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

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

   9. pow1/2 N/A

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

   10. pow-flip N/A

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

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

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

   12. metadata-eval N/A

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

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

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

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

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

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

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

   16. cos-lowering-cos.f64 99.7%

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in a1 around 0

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

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

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

   2. unpow2 N/A

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

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

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

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

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

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

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

   6. sqrt-lowering-sqrt.f64 58.8%

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

7. Simplified 58.8%

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

Alternative 8: 66.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 8 \cdot 10^{+195}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(th \cdot th\right)\right) \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 8e+195)
   (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5))
   (* (+ 1.0 (* -0.5 (* th th))) (/ (* a2 a2) (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 8e+195) {
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	} 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 <= 8d+195) then
        tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
    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 <= 8e+195) {
		tmp = ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
	} 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 <= 8e+195:
		tmp = ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)
	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 <= 8e+195)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(th * th))) * Float64(Float64(a2 * a2) / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 8e+195)
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	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, 8e+195], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 7.99999999999999982e195

   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. Add Preprocessing
   3. 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. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. pow1/2 N/A

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

      10. pow-flip N/A

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

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

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

      12. metadata-eval N/A

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

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

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

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

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

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

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

      16. cos-lowering-cos.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)} \]
   6. 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. +-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 66.4%

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

   7. Simplified 66.4%

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

   if 7.99999999999999982e195 < 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. Add Preprocessing

   3. Taylor expanded in a1 around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. sqrt-lowering-sqrt.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in th around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. distribute-lft1-in N/A

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

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

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

      15. sqrt-lowering-sqrt.f64 87.0%

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

   8. Simplified 87.0%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 8 \cdot 10^{+195}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(th \cdot th\right)\right) \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 7.2 \cdot 10^{+16}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{th \cdot \left(th \cdot \left(-0.5 \cdot \left(a2 \cdot a2\right)\right)\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 7.2e+16)
   (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5))
   (/ (* th (* th (* -0.5 (* a2 a2)))) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 7.2e+16) {
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	} else {
		tmp = (th * (th * (-0.5 * (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 (th <= 7.2d+16) then
        tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
    else
        tmp = (th * (th * ((-0.5d0) * (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 (th <= 7.2e+16) {
		tmp = ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
	} else {
		tmp = (th * (th * (-0.5 * (a2 * a2)))) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 7.2e+16:
		tmp = ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)
	else:
		tmp = (th * (th * (-0.5 * (a2 * a2)))) / math.sqrt(2.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 7.2e+16)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5));
	else
		tmp = Float64(Float64(th * Float64(th * Float64(-0.5 * Float64(a2 * a2)))) / sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 7.2e+16)
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	else
		tmp = (th * (th * (-0.5 * (a2 * a2)))) / sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 7.2e+16], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[(th * N[(th * N[(-0.5 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 7.2e16

   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. Add Preprocessing
   3. 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. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. pow1/2 N/A

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

      10. pow-flip N/A

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

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

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

      12. metadata-eval N/A

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

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

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

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

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

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

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

      16. cos-lowering-cos.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)} \]
   6. 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. +-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 77.5%

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

   7. Simplified 77.5%

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

   if 7.2e16 < 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. Add Preprocessing

   3. Taylor expanded in a1 around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. sqrt-lowering-sqrt.f64 60.9%

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

   5. Simplified 60.9%

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

   6. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 30.1%

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

   8. Simplified 30.1%

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

   9. Taylor expanded in th around inf

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

       1. associate-*r/ N/A

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

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

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

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

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

       7. unpow2 N/A

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

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

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

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

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

       10. unpow2 N/A

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

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

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

       12. sqrt-lowering-sqrt.f64 29.9%

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

   11. Simplified 29.9%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

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

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

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

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

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

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

       6. *-lowering-*.f64 33.6%

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

   13. Applied egg-rr 33.6%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 7.2 \cdot 10^{+16}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{th \cdot \left(th \cdot \left(-0.5 \cdot \left(a2 \cdot a2\right)\right)\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.1% 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 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing
3. 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. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

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

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

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

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

   9. pow1/2 N/A

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

   10. pow-flip N/A

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

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

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

   12. metadata-eval N/A

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

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

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

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

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

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

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

   16. cos-lowering-cos.f64 99.7%

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)} \]
6. 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. +-commutative N/A

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

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

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

   5. unpow2 N/A

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 65.1%

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

7. Simplified 65.1%

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

8. Final simplification 65.1%

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

Alternative 11: 39.7% 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 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing
3. 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. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

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

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

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

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

   9. pow1/2 N/A

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

   10. pow-flip N/A

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

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

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

   12. metadata-eval N/A

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

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

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

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

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

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

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

   16. cos-lowering-cos.f64 99.7%

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. sqrt-lowering-sqrt.f64 58.8%

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

7. Simplified 58.8%

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

8. Taylor expanded in th around 0

   \[\leadsto \color{blue}{{a2}^{2} \cdot \sqrt{\frac{1}{2}}} \]
9. 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 39.6%

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

10. Simplified 39.6%

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

    1. associate-*r* N/A

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

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

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

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

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

    4. sqrt-lowering-sqrt.f64 39.6%

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

12. Applied egg-rr 39.6%

    \[\leadsto \color{blue}{\left(a2 \cdot a2\right) \cdot \sqrt{0.5}} \]
13. Add Preprocessing

Alternative 12: 39.7% 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 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing
3. 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. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

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

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

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

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

   9. pow1/2 N/A

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

   10. pow-flip N/A

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

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

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

   12. metadata-eval N/A

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

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

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

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

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

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

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

   16. cos-lowering-cos.f64 99.7%

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. sqrt-lowering-sqrt.f64 58.8%

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

7. Simplified 58.8%

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

8. Taylor expanded in th around 0

   \[\leadsto \color{blue}{{a2}^{2} \cdot \sqrt{\frac{1}{2}}} \]
9. 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 39.6%

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

10. Simplified 39.6%

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

Reproduce

herbie shell --seed 2024139 
(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))))