Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 12.4s

Alternatives: 14

Speedup: 2.0×

• 43.6% 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: 99.6% 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, 1.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \frac{\cos th}{\sqrt{2}} \cdot \left(a1\_m \cdot a1\_m\right) + \left(\cos th \cdot {2}^{-0.5}\right) \cdot \left(a2 \cdot a2\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (+
  (* (/ (cos th) (sqrt 2.0)) (* a1_m a1_m))
  (* (* (cos th) (pow 2.0 -0.5)) (* a2 a2))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return ((cos(th) / sqrt(2.0)) * (a1_m * a1_m)) + ((cos(th) * pow(2.0, -0.5)) * (a2 * a2));
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = ((cos(th) / sqrt(2.0d0)) * (a1_m * a1_m)) + ((cos(th) * (2.0d0 ** (-0.5d0))) * (a2 * a2))
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return ((Math.cos(th) / Math.sqrt(2.0)) * (a1_m * a1_m)) + ((Math.cos(th) * Math.pow(2.0, -0.5)) * (a2 * a2));
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return ((math.cos(th) / math.sqrt(2.0)) * (a1_m * a1_m)) + ((math.cos(th) * math.pow(2.0, -0.5)) * (a2 * a2))

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(Float64(cos(th) / sqrt(2.0)) * Float64(a1_m * a1_m)) + Float64(Float64(cos(th) * (2.0 ^ -0.5)) * Float64(a2 * a2)))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = ((cos(th) / sqrt(2.0)) * (a1_m * a1_m)) + ((cos(th) * (2.0 ^ -0.5)) * (a2 * a2));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[th], $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision] * N[(a2 * a2), $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. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

   4. pow1/2 N/A

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

   5. pow-flip N/A

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

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

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

   7. metadata-eval N/A

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

   8. cos-lowering-cos.f64 99.6%

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

   \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(\cos th \cdot {2}^{-0.5}\right) \cdot \left(a2 \cdot a2\right) \]
6. Add Preprocessing

Alternative 2: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \frac{\cos th \cdot \left(a1\_m \cdot a1\_m + a2 \cdot a2\right)}{\sqrt{2}} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (/ (* (cos th) (+ (* a1_m a1_m) (* a2 a2))) (sqrt 2.0)))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return (cos(th) * ((a1_m * a1_m) + (a2 * a2))) / sqrt(2.0);
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (cos(th) * ((a1_m * a1_m) + (a2 * a2))) / sqrt(2.0d0)
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return (Math.cos(th) * ((a1_m * a1_m) + (a2 * a2))) / Math.sqrt(2.0);
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return (math.cos(th) * ((a1_m * a1_m) + (a2 * a2))) / math.sqrt(2.0)

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(cos(th) * Float64(Float64(a1_m * a1_m) + Float64(a2 * a2))) / sqrt(2.0))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = (cos(th) * ((a1_m * a1_m) + (a2 * a2))) / sqrt(2.0);
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] * N[(N[(a1$95$m * a1$95$m), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $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. 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. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   8. sqrt-lowering-sqrt.f64 99.6%

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

6. Applied egg-rr 99.6%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \cos th \cdot \frac{a1\_m \cdot a1\_m + a2 \cdot a2}{\sqrt{2}} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (* (cos th) (/ (+ (* a1_m a1_m) (* a2 a2)) (sqrt 2.0))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return cos(th) * (((a1_m * a1_m) + (a2 * a2)) / sqrt(2.0));
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = cos(th) * (((a1_m * a1_m) + (a2 * a2)) / sqrt(2.0d0))
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return Math.cos(th) * (((a1_m * a1_m) + (a2 * a2)) / Math.sqrt(2.0));
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return math.cos(th) * (((a1_m * a1_m) + (a2 * a2)) / math.sqrt(2.0))

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(cos(th) * Float64(Float64(Float64(a1_m * a1_m) + Float64(a2 * a2)) / sqrt(2.0)))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = cos(th) * (((a1_m * a1_m) + (a2 * a2)) / sqrt(2.0));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(N[(a1$95$m * a1$95$m), $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. 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. Add Preprocessing

Alternative 4: 78.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ a2 \cdot \left(\left(\cos th \cdot a2\right) \cdot \sqrt{0.5}\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th) :precision binary64 (* a2 (* (* (cos th) a2) (sqrt 0.5))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return a2 * ((cos(th) * a2) * sqrt(0.5));
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 * ((cos(th) * a2) * sqrt(0.5d0))
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return a2 * ((Math.cos(th) * a2) * Math.sqrt(0.5));
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return a2 * ((math.cos(th) * a2) * math.sqrt(0.5))

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(a2 * Float64(Float64(cos(th) * a2) * sqrt(0.5)))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = a2 * ((cos(th) * a2) * sqrt(0.5));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(a2 * N[(N[(N[Cos[th], $MachinePrecision] * a2), $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. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

   4. pow1/2 N/A

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

   5. pow-flip N/A

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

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

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

   7. metadata-eval N/A

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

   8. cos-lowering-cos.f64 99.6%

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

4. Applied egg-rr 99.6%

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

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. *-commutative N/A

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

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

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

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

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

   7. cos-lowering-cos.f64 55.0%

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

7. Simplified 55.0%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

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

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

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

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

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

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

   9. sqrt-lowering-sqrt.f64 55.0%

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

9. Applied egg-rr 55.0%

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

10. Final simplification 55.0%

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

Alternative 5: 78.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \left(a2 \cdot a2\right) \cdot \left(\cos th \cdot \sqrt{0.5}\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th) :precision binary64 (* (* a2 a2) (* (cos th) (sqrt 0.5))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return (a2 * a2) * (cos(th) * sqrt(0.5));
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (a2 * a2) * (cos(th) * sqrt(0.5d0))
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return (a2 * a2) * (Math.cos(th) * Math.sqrt(0.5));
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return (a2 * a2) * (math.cos(th) * math.sqrt(0.5))

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(a2 * a2) * Float64(cos(th) * sqrt(0.5)))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = (a2 * a2) * (cos(th) * sqrt(0.5));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, 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. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

   4. pow1/2 N/A

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

   5. pow-flip N/A

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

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

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

   7. metadata-eval N/A

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

   8. cos-lowering-cos.f64 99.6%

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

4. Applied egg-rr 99.6%

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

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. *-commutative N/A

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

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

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

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

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

   7. cos-lowering-cos.f64 55.0%

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

7. Simplified 55.0%

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

8. Final simplification 55.0%

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

Alternative 6: 78.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th) :precision binary64 (* (cos th) (* (* a2 a2) (sqrt 0.5))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return cos(th) * ((a2 * a2) * sqrt(0.5));
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = cos(th) * ((a2 * a2) * sqrt(0.5d0))
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return Math.cos(th) * ((a2 * a2) * Math.sqrt(0.5));
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return math.cos(th) * ((a2 * a2) * math.sqrt(0.5))

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(cos(th) * Float64(Float64(a2 * a2) * sqrt(0.5)))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = cos(th) * ((a2 * a2) * sqrt(0.5));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 * a2), $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. 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. *-lowering-*.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-flip N/A

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

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

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

   7. metadata-eval N/A

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

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

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

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

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

   10. *-lowering-*.f64 99.6%

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

6. Applied egg-rr 99.6%

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

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 55.0%

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

9. Simplified 55.0%

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

10. Final simplification 55.0%

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

Alternative 7: 64.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := a1\_m \cdot a1\_m + a2 \cdot a2\\ \mathbf{if}\;th \leq 910:\\ \;\;\;\;\frac{t\_1 \cdot \left(1 + th \cdot \left(th \cdot \left(-0.5 + \left(th \cdot th\right) \cdot \left(0.041666666666666664 + th \cdot \left(th \cdot -0.001388888888888889\right)\right)\right)\right)\right)}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 4.8 \cdot 10^{+141}:\\ \;\;\;\;\frac{-1}{\frac{0 - \sqrt{2}}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1_m a1_m) (* a2 a2))))
   (if (<= th 910.0)
     (/
      (*
       t_1
       (+
        1.0
        (*
         th
         (*
          th
          (+
           -0.5
           (*
            (* th th)
            (+ 0.041666666666666664 (* th (* th -0.001388888888888889)))))))))
      (sqrt 2.0))
     (if (<= th 4.8e+141)
       (/ -1.0 (/ (- 0.0 (sqrt 2.0)) t_1))
       (* a2 (* a2 (sqrt 0.5)))))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	double t_1 = (a1_m * a1_m) + (a2 * a2);
	double tmp;
	if (th <= 910.0) {
		tmp = (t_1 * (1.0 + (th * (th * (-0.5 + ((th * th) * (0.041666666666666664 + (th * (th * -0.001388888888888889))))))))) / sqrt(2.0);
	} else if (th <= 4.8e+141) {
		tmp = -1.0 / ((0.0 - sqrt(2.0)) / t_1);
	} else {
		tmp = a2 * (a2 * sqrt(0.5));
	}
	return tmp;
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a1_m * a1_m) + (a2 * a2)
    if (th <= 910.0d0) then
        tmp = (t_1 * (1.0d0 + (th * (th * ((-0.5d0) + ((th * th) * (0.041666666666666664d0 + (th * (th * (-0.001388888888888889d0)))))))))) / sqrt(2.0d0)
    else if (th <= 4.8d+141) then
        tmp = (-1.0d0) / ((0.0d0 - sqrt(2.0d0)) / t_1)
    else
        tmp = a2 * (a2 * sqrt(0.5d0))
    end if
    code = tmp
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	double t_1 = (a1_m * a1_m) + (a2 * a2);
	double tmp;
	if (th <= 910.0) {
		tmp = (t_1 * (1.0 + (th * (th * (-0.5 + ((th * th) * (0.041666666666666664 + (th * (th * -0.001388888888888889))))))))) / Math.sqrt(2.0);
	} else if (th <= 4.8e+141) {
		tmp = -1.0 / ((0.0 - Math.sqrt(2.0)) / t_1);
	} else {
		tmp = a2 * (a2 * Math.sqrt(0.5));
	}
	return tmp;
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	t_1 = (a1_m * a1_m) + (a2 * a2)
	tmp = 0
	if th <= 910.0:
		tmp = (t_1 * (1.0 + (th * (th * (-0.5 + ((th * th) * (0.041666666666666664 + (th * (th * -0.001388888888888889))))))))) / math.sqrt(2.0)
	elif th <= 4.8e+141:
		tmp = -1.0 / ((0.0 - math.sqrt(2.0)) / t_1)
	else:
		tmp = a2 * (a2 * math.sqrt(0.5))
	return tmp

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	t_1 = Float64(Float64(a1_m * a1_m) + Float64(a2 * a2))
	tmp = 0.0
	if (th <= 910.0)
		tmp = Float64(Float64(t_1 * Float64(1.0 + Float64(th * Float64(th * Float64(-0.5 + Float64(Float64(th * th) * Float64(0.041666666666666664 + Float64(th * Float64(th * -0.001388888888888889))))))))) / sqrt(2.0));
	elseif (th <= 4.8e+141)
		tmp = Float64(-1.0 / Float64(Float64(0.0 - sqrt(2.0)) / t_1));
	else
		tmp = Float64(a2 * Float64(a2 * sqrt(0.5)));
	end
	return tmp
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp_2 = code(a1_m, a2, th)
	t_1 = (a1_m * a1_m) + (a2 * a2);
	tmp = 0.0;
	if (th <= 910.0)
		tmp = (t_1 * (1.0 + (th * (th * (-0.5 + ((th * th) * (0.041666666666666664 + (th * (th * -0.001388888888888889))))))))) / sqrt(2.0);
	elseif (th <= 4.8e+141)
		tmp = -1.0 / ((0.0 - sqrt(2.0)) / t_1);
	else
		tmp = a2 * (a2 * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := Block[{t$95$1 = N[(N[(a1$95$m * a1$95$m), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[th, 910.0], N[(N[(t$95$1 * N[(1.0 + N[(th * N[(th * N[(-0.5 + N[(N[(th * th), $MachinePrecision] * N[(0.041666666666666664 + N[(th * N[(th * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 4.8e+141], N[(-1.0 / N[(N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if th < 910

   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

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {th}^{2} \cdot \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({th}^{2} \cdot \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right)}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(th \cdot th\right) \cdot \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(\color{blue}{a2}, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(th \cdot \left(th \cdot \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \color{blue}{\mathsf{*.f64}\left(a2, a2\right)}\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(th \cdot \left(\left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right) \cdot th\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \left(\left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right) \cdot th\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \color{blue}{\mathsf{*.f64}\left(a2, a2\right)}\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \left(th \cdot \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) + \frac{-1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{-1}{2} + {th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({th}^{2}\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(th \cdot th\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {th}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({th}^{2} \cdot \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({th}^{2}\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(th \cdot th\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      19. *-lowering-*.f64 69.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   7. Simplified 69.5%

      \[\leadsto \color{blue}{\left(1 + th \cdot \left(th \cdot \left(-0.5 + \left(th \cdot th\right) \cdot \left(0.041666666666666664 + \left(th \cdot th\right) \cdot -0.001388888888888889\right)\right)\right)\right)} \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\left(1 + th \cdot \left(th \cdot \left(\frac{-1}{2} + \left(th \cdot th\right) \cdot \left(\frac{1}{24} + \left(th \cdot th\right) \cdot \frac{-1}{720}\right)\right)\right)\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + th \cdot \left(th \cdot \left(\frac{-1}{2} + \left(th \cdot th\right) \cdot \left(\frac{1}{24} + \left(th \cdot th\right) \cdot \frac{-1}{720}\right)\right)\right)\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   9. Applied egg-rr 69.5%

      \[\leadsto \color{blue}{\frac{\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(1 + th \cdot \left(th \cdot \left(-0.5 + \left(th \cdot th\right) \cdot \left(0.041666666666666664 + th \cdot \left(th \cdot -0.001388888888888889\right)\right)\right)\right)\right)}{\sqrt{2}}} \]

   if 910 < th < 4.79999999999999995e141

   1. Initial program 99.4%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   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 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   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 17.4%

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

   7. Simplified 17.4%

      \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   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. frac-2neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \color{blue}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\sqrt{2}\right), \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\color{blue}{a1 \cdot a1} + a2 \cdot a2\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\left(a1 \cdot a1\right), \color{blue}{\left(a2 \cdot a2\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(\color{blue}{a2} \cdot a2\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 34.9%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 34.9%

      \[\leadsto \color{blue}{\frac{-1}{0 - \frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}} \]

   if 4.79999999999999995e141 < 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. Step-by-step derivation

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

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

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

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

      7. metadata-eval N/A

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

      8. cos-lowering-cos.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   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. *-commutative N/A

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

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

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

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

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

      7. cos-lowering-cos.f64 51.6%

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

   7. Simplified 51.6%

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

   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 18.7%

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

   10. Simplified 18.7%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 910:\\ \;\;\;\;\frac{\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(1 + th \cdot \left(th \cdot \left(-0.5 + \left(th \cdot th\right) \cdot \left(0.041666666666666664 + th \cdot \left(th \cdot -0.001388888888888889\right)\right)\right)\right)\right)}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 4.8 \cdot 10^{+141}:\\ \;\;\;\;\frac{-1}{\frac{0 - \sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := a1\_m \cdot a1\_m + a2 \cdot a2\\ \mathbf{if}\;th \leq 910:\\ \;\;\;\;\frac{t\_1}{\sqrt{2}} \cdot \left(1 + th \cdot \left(th \cdot \left(-0.5 + \left(th \cdot th\right) \cdot \left(0.041666666666666664 + \left(th \cdot th\right) \cdot -0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;th \leq 4.8 \cdot 10^{+141}:\\ \;\;\;\;\frac{-1}{\frac{0 - \sqrt{2}}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1_m a1_m) (* a2 a2))))
   (if (<= th 910.0)
     (*
      (/ t_1 (sqrt 2.0))
      (+
       1.0
       (*
        th
        (*
         th
         (+
          -0.5
          (*
           (* th th)
           (+ 0.041666666666666664 (* (* th th) -0.001388888888888889))))))))
     (if (<= th 4.8e+141)
       (/ -1.0 (/ (- 0.0 (sqrt 2.0)) t_1))
       (* a2 (* a2 (sqrt 0.5)))))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	double t_1 = (a1_m * a1_m) + (a2 * a2);
	double tmp;
	if (th <= 910.0) {
		tmp = (t_1 / sqrt(2.0)) * (1.0 + (th * (th * (-0.5 + ((th * th) * (0.041666666666666664 + ((th * th) * -0.001388888888888889)))))));
	} else if (th <= 4.8e+141) {
		tmp = -1.0 / ((0.0 - sqrt(2.0)) / t_1);
	} else {
		tmp = a2 * (a2 * sqrt(0.5));
	}
	return tmp;
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a1_m * a1_m) + (a2 * a2)
    if (th <= 910.0d0) then
        tmp = (t_1 / sqrt(2.0d0)) * (1.0d0 + (th * (th * ((-0.5d0) + ((th * th) * (0.041666666666666664d0 + ((th * th) * (-0.001388888888888889d0))))))))
    else if (th <= 4.8d+141) then
        tmp = (-1.0d0) / ((0.0d0 - sqrt(2.0d0)) / t_1)
    else
        tmp = a2 * (a2 * sqrt(0.5d0))
    end if
    code = tmp
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	double t_1 = (a1_m * a1_m) + (a2 * a2);
	double tmp;
	if (th <= 910.0) {
		tmp = (t_1 / Math.sqrt(2.0)) * (1.0 + (th * (th * (-0.5 + ((th * th) * (0.041666666666666664 + ((th * th) * -0.001388888888888889)))))));
	} else if (th <= 4.8e+141) {
		tmp = -1.0 / ((0.0 - Math.sqrt(2.0)) / t_1);
	} else {
		tmp = a2 * (a2 * Math.sqrt(0.5));
	}
	return tmp;
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	t_1 = (a1_m * a1_m) + (a2 * a2)
	tmp = 0
	if th <= 910.0:
		tmp = (t_1 / math.sqrt(2.0)) * (1.0 + (th * (th * (-0.5 + ((th * th) * (0.041666666666666664 + ((th * th) * -0.001388888888888889)))))))
	elif th <= 4.8e+141:
		tmp = -1.0 / ((0.0 - math.sqrt(2.0)) / t_1)
	else:
		tmp = a2 * (a2 * math.sqrt(0.5))
	return tmp

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	t_1 = Float64(Float64(a1_m * a1_m) + Float64(a2 * a2))
	tmp = 0.0
	if (th <= 910.0)
		tmp = Float64(Float64(t_1 / sqrt(2.0)) * Float64(1.0 + Float64(th * Float64(th * Float64(-0.5 + Float64(Float64(th * th) * Float64(0.041666666666666664 + Float64(Float64(th * th) * -0.001388888888888889))))))));
	elseif (th <= 4.8e+141)
		tmp = Float64(-1.0 / Float64(Float64(0.0 - sqrt(2.0)) / t_1));
	else
		tmp = Float64(a2 * Float64(a2 * sqrt(0.5)));
	end
	return tmp
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp_2 = code(a1_m, a2, th)
	t_1 = (a1_m * a1_m) + (a2 * a2);
	tmp = 0.0;
	if (th <= 910.0)
		tmp = (t_1 / sqrt(2.0)) * (1.0 + (th * (th * (-0.5 + ((th * th) * (0.041666666666666664 + ((th * th) * -0.001388888888888889)))))));
	elseif (th <= 4.8e+141)
		tmp = -1.0 / ((0.0 - sqrt(2.0)) / t_1);
	else
		tmp = a2 * (a2 * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := Block[{t$95$1 = N[(N[(a1$95$m * a1$95$m), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[th, 910.0], N[(N[(t$95$1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(th * N[(th * N[(-0.5 + N[(N[(th * th), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(th * th), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 4.8e+141], N[(-1.0 / N[(N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if th < 910

   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

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {th}^{2} \cdot \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({th}^{2} \cdot \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right)}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(th \cdot th\right) \cdot \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(\color{blue}{a2}, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(th \cdot \left(th \cdot \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \color{blue}{\mathsf{*.f64}\left(a2, a2\right)}\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(th \cdot \left(\left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right) \cdot th\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \left(\left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right) \cdot th\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \color{blue}{\mathsf{*.f64}\left(a2, a2\right)}\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \left(th \cdot \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) + \frac{-1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{-1}{2} + {th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({th}^{2}\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(th \cdot th\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {th}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({th}^{2} \cdot \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({th}^{2}\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(th \cdot th\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      19. *-lowering-*.f64 69.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   7. Simplified 69.5%

      \[\leadsto \color{blue}{\left(1 + th \cdot \left(th \cdot \left(-0.5 + \left(th \cdot th\right) \cdot \left(0.041666666666666664 + \left(th \cdot th\right) \cdot -0.001388888888888889\right)\right)\right)\right)} \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}} \]

   if 910 < th < 4.79999999999999995e141

   1. Initial program 99.4%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   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 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   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 17.4%

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

   7. Simplified 17.4%

      \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   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. frac-2neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \color{blue}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\sqrt{2}\right), \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\color{blue}{a1 \cdot a1} + a2 \cdot a2\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\left(a1 \cdot a1\right), \color{blue}{\left(a2 \cdot a2\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(\color{blue}{a2} \cdot a2\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 34.9%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 34.9%

      \[\leadsto \color{blue}{\frac{-1}{0 - \frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}} \]

   if 4.79999999999999995e141 < 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. Step-by-step derivation

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

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

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

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

      7. metadata-eval N/A

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

      8. cos-lowering-cos.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   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. *-commutative N/A

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

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

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

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

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

      7. cos-lowering-cos.f64 51.6%

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

   7. Simplified 51.6%

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

   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 18.7%

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

   10. Simplified 18.7%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 910:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}} \cdot \left(1 + th \cdot \left(th \cdot \left(-0.5 + \left(th \cdot th\right) \cdot \left(0.041666666666666664 + \left(th \cdot th\right) \cdot -0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;th \leq 4.8 \cdot 10^{+141}:\\ \;\;\;\;\frac{-1}{\frac{0 - \sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} \mathbf{if}\;th \leq 910:\\ \;\;\;\;\left(1 + th \cdot \left(th \cdot \left(-0.5 + \left(th \cdot th\right) \cdot \left(0.041666666666666664 + \left(th \cdot th\right) \cdot -0.001388888888888889\right)\right)\right)\right) \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 4.8 \cdot 10^{+141}:\\ \;\;\;\;\frac{-1}{\frac{0 - \sqrt{2}}{a1\_m \cdot a1\_m + a2 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (if (<= th 910.0)
   (*
    (+
     1.0
     (*
      th
      (*
       th
       (+
        -0.5
        (*
         (* th th)
         (+ 0.041666666666666664 (* (* th th) -0.001388888888888889)))))))
    (/ (* a2 a2) (sqrt 2.0)))
   (if (<= th 4.8e+141)
     (/ -1.0 (/ (- 0.0 (sqrt 2.0)) (+ (* a1_m a1_m) (* a2 a2))))
     (* a2 (* a2 (sqrt 0.5))))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	double tmp;
	if (th <= 910.0) {
		tmp = (1.0 + (th * (th * (-0.5 + ((th * th) * (0.041666666666666664 + ((th * th) * -0.001388888888888889))))))) * ((a2 * a2) / sqrt(2.0));
	} else if (th <= 4.8e+141) {
		tmp = -1.0 / ((0.0 - sqrt(2.0)) / ((a1_m * a1_m) + (a2 * a2)));
	} else {
		tmp = a2 * (a2 * sqrt(0.5));
	}
	return tmp;
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 910.0d0) then
        tmp = (1.0d0 + (th * (th * ((-0.5d0) + ((th * th) * (0.041666666666666664d0 + ((th * th) * (-0.001388888888888889d0)))))))) * ((a2 * a2) / sqrt(2.0d0))
    else if (th <= 4.8d+141) then
        tmp = (-1.0d0) / ((0.0d0 - sqrt(2.0d0)) / ((a1_m * a1_m) + (a2 * a2)))
    else
        tmp = a2 * (a2 * sqrt(0.5d0))
    end if
    code = tmp
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	double tmp;
	if (th <= 910.0) {
		tmp = (1.0 + (th * (th * (-0.5 + ((th * th) * (0.041666666666666664 + ((th * th) * -0.001388888888888889))))))) * ((a2 * a2) / Math.sqrt(2.0));
	} else if (th <= 4.8e+141) {
		tmp = -1.0 / ((0.0 - Math.sqrt(2.0)) / ((a1_m * a1_m) + (a2 * a2)));
	} else {
		tmp = a2 * (a2 * Math.sqrt(0.5));
	}
	return tmp;
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	tmp = 0
	if th <= 910.0:
		tmp = (1.0 + (th * (th * (-0.5 + ((th * th) * (0.041666666666666664 + ((th * th) * -0.001388888888888889))))))) * ((a2 * a2) / math.sqrt(2.0))
	elif th <= 4.8e+141:
		tmp = -1.0 / ((0.0 - math.sqrt(2.0)) / ((a1_m * a1_m) + (a2 * a2)))
	else:
		tmp = a2 * (a2 * math.sqrt(0.5))
	return tmp

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	tmp = 0.0
	if (th <= 910.0)
		tmp = Float64(Float64(1.0 + Float64(th * Float64(th * Float64(-0.5 + Float64(Float64(th * th) * Float64(0.041666666666666664 + Float64(Float64(th * th) * -0.001388888888888889))))))) * Float64(Float64(a2 * a2) / sqrt(2.0)));
	elseif (th <= 4.8e+141)
		tmp = Float64(-1.0 / Float64(Float64(0.0 - sqrt(2.0)) / Float64(Float64(a1_m * a1_m) + Float64(a2 * a2))));
	else
		tmp = Float64(a2 * Float64(a2 * sqrt(0.5)));
	end
	return tmp
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp_2 = code(a1_m, a2, th)
	tmp = 0.0;
	if (th <= 910.0)
		tmp = (1.0 + (th * (th * (-0.5 + ((th * th) * (0.041666666666666664 + ((th * th) * -0.001388888888888889))))))) * ((a2 * a2) / sqrt(2.0));
	elseif (th <= 4.8e+141)
		tmp = -1.0 / ((0.0 - sqrt(2.0)) / ((a1_m * a1_m) + (a2 * a2)));
	else
		tmp = a2 * (a2 * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := If[LessEqual[th, 910.0], N[(N[(1.0 + N[(th * N[(th * N[(-0.5 + N[(N[(th * th), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(th * th), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 4.8e+141], N[(-1.0 / N[(N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(a1$95$m * a1$95$m), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 910

   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

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {th}^{2} \cdot \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({th}^{2} \cdot \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right)}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(th \cdot th\right) \cdot \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(\color{blue}{a2}, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(th \cdot \left(th \cdot \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \color{blue}{\mathsf{*.f64}\left(a2, a2\right)}\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(th \cdot \left(\left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right) \cdot th\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \left(\left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right) \cdot th\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \color{blue}{\mathsf{*.f64}\left(a2, a2\right)}\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \left(th \cdot \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) + \frac{-1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{-1}{2} + {th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({th}^{2}\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(th \cdot th\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {th}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({th}^{2} \cdot \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({th}^{2}\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(th \cdot th\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      19. *-lowering-*.f64 69.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   7. Simplified 69.5%

      \[\leadsto \color{blue}{\left(1 + th \cdot \left(th \cdot \left(-0.5 + \left(th \cdot th\right) \cdot \left(0.041666666666666664 + \left(th \cdot th\right) \cdot -0.001388888888888889\right)\right)\right)\right)} \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}} \]

   8. Taylor expanded in a1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \color{blue}{\left(\frac{{a2}^{2}}{\sqrt{2}}\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\left({a2}^{2}\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\left(a2 \cdot a2\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 43.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   10. Simplified 43.5%

       \[\leadsto \left(1 + th \cdot \left(th \cdot \left(-0.5 + \left(th \cdot th\right) \cdot \left(0.041666666666666664 + \left(th \cdot th\right) \cdot -0.001388888888888889\right)\right)\right)\right) \cdot \color{blue}{\frac{a2 \cdot a2}{\sqrt{2}}} \]

   if 910 < th < 4.79999999999999995e141

   1. Initial program 99.4%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   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 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   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 17.4%

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

   7. Simplified 17.4%

      \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   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. frac-2neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \color{blue}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\sqrt{2}\right), \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\color{blue}{a1 \cdot a1} + a2 \cdot a2\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\left(a1 \cdot a1\right), \color{blue}{\left(a2 \cdot a2\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(\color{blue}{a2} \cdot a2\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 34.9%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 34.9%

      \[\leadsto \color{blue}{\frac{-1}{0 - \frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}} \]

   if 4.79999999999999995e141 < 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. Step-by-step derivation

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

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

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

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

      7. metadata-eval N/A

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

      8. cos-lowering-cos.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   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. *-commutative N/A

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

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

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

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

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

      7. cos-lowering-cos.f64 51.6%

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

   7. Simplified 51.6%

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

   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 18.7%

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

   10. Simplified 18.7%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 910:\\ \;\;\;\;\left(1 + th \cdot \left(th \cdot \left(-0.5 + \left(th \cdot th\right) \cdot \left(0.041666666666666664 + \left(th \cdot th\right) \cdot -0.001388888888888889\right)\right)\right)\right) \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 4.8 \cdot 10^{+141}:\\ \;\;\;\;\frac{-1}{\frac{0 - \sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := a1\_m \cdot a1\_m + a2 \cdot a2\\ \mathbf{if}\;th \leq 196:\\ \;\;\;\;\frac{t\_1}{\sqrt{2}} \cdot \left(1 + th \cdot \left(th \cdot \left(-0.5 + \left(th \cdot th\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;th \leq 4.8 \cdot 10^{+141}:\\ \;\;\;\;\frac{-1}{\frac{0 - \sqrt{2}}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1_m a1_m) (* a2 a2))))
   (if (<= th 196.0)
     (*
      (/ t_1 (sqrt 2.0))
      (+ 1.0 (* th (* th (+ -0.5 (* (* th th) 0.041666666666666664))))))
     (if (<= th 4.8e+141)
       (/ -1.0 (/ (- 0.0 (sqrt 2.0)) t_1))
       (* a2 (* a2 (sqrt 0.5)))))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	double t_1 = (a1_m * a1_m) + (a2 * a2);
	double tmp;
	if (th <= 196.0) {
		tmp = (t_1 / sqrt(2.0)) * (1.0 + (th * (th * (-0.5 + ((th * th) * 0.041666666666666664)))));
	} else if (th <= 4.8e+141) {
		tmp = -1.0 / ((0.0 - sqrt(2.0)) / t_1);
	} else {
		tmp = a2 * (a2 * sqrt(0.5));
	}
	return tmp;
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a1_m * a1_m) + (a2 * a2)
    if (th <= 196.0d0) then
        tmp = (t_1 / sqrt(2.0d0)) * (1.0d0 + (th * (th * ((-0.5d0) + ((th * th) * 0.041666666666666664d0)))))
    else if (th <= 4.8d+141) then
        tmp = (-1.0d0) / ((0.0d0 - sqrt(2.0d0)) / t_1)
    else
        tmp = a2 * (a2 * sqrt(0.5d0))
    end if
    code = tmp
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	double t_1 = (a1_m * a1_m) + (a2 * a2);
	double tmp;
	if (th <= 196.0) {
		tmp = (t_1 / Math.sqrt(2.0)) * (1.0 + (th * (th * (-0.5 + ((th * th) * 0.041666666666666664)))));
	} else if (th <= 4.8e+141) {
		tmp = -1.0 / ((0.0 - Math.sqrt(2.0)) / t_1);
	} else {
		tmp = a2 * (a2 * Math.sqrt(0.5));
	}
	return tmp;
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	t_1 = (a1_m * a1_m) + (a2 * a2)
	tmp = 0
	if th <= 196.0:
		tmp = (t_1 / math.sqrt(2.0)) * (1.0 + (th * (th * (-0.5 + ((th * th) * 0.041666666666666664)))))
	elif th <= 4.8e+141:
		tmp = -1.0 / ((0.0 - math.sqrt(2.0)) / t_1)
	else:
		tmp = a2 * (a2 * math.sqrt(0.5))
	return tmp

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	t_1 = Float64(Float64(a1_m * a1_m) + Float64(a2 * a2))
	tmp = 0.0
	if (th <= 196.0)
		tmp = Float64(Float64(t_1 / sqrt(2.0)) * Float64(1.0 + Float64(th * Float64(th * Float64(-0.5 + Float64(Float64(th * th) * 0.041666666666666664))))));
	elseif (th <= 4.8e+141)
		tmp = Float64(-1.0 / Float64(Float64(0.0 - sqrt(2.0)) / t_1));
	else
		tmp = Float64(a2 * Float64(a2 * sqrt(0.5)));
	end
	return tmp
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp_2 = code(a1_m, a2, th)
	t_1 = (a1_m * a1_m) + (a2 * a2);
	tmp = 0.0;
	if (th <= 196.0)
		tmp = (t_1 / sqrt(2.0)) * (1.0 + (th * (th * (-0.5 + ((th * th) * 0.041666666666666664)))));
	elseif (th <= 4.8e+141)
		tmp = -1.0 / ((0.0 - sqrt(2.0)) / t_1);
	else
		tmp = a2 * (a2 * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := Block[{t$95$1 = N[(N[(a1$95$m * a1$95$m), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[th, 196.0], N[(N[(t$95$1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(th * N[(th * N[(-0.5 + N[(N[(th * th), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 4.8e+141], N[(-1.0 / N[(N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if th < 196

   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

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {th}^{2} \cdot \left(\frac{1}{24} \cdot {th}^{2} - \frac{1}{2}\right)\right)}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({th}^{2} \cdot \left(\frac{1}{24} \cdot {th}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right)}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(th \cdot th\right) \cdot \left(\frac{1}{24} \cdot {th}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(\color{blue}{a2}, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(th \cdot \left(th \cdot \left(\frac{1}{24} \cdot {th}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \color{blue}{\mathsf{*.f64}\left(a2, a2\right)}\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \left(th \cdot \left(\frac{1}{24} \cdot {th}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \color{blue}{\mathsf{*.f64}\left(a2, a2\right)}\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{1}{24} \cdot {th}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{1}{24} \cdot {th}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{1}{24} \cdot {th}^{2} + \frac{-1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \left(\frac{-1}{2} + \frac{1}{24} \cdot {th}^{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {th}^{2}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \left({th}^{2} \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({th}^{2}\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(th \cdot th\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      13. *-lowering-*.f64 71.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(th, \mathsf{*.f64}\left(th, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(th, th\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   7. Simplified 71.1%

      \[\leadsto \color{blue}{\left(1 + th \cdot \left(th \cdot \left(-0.5 + \left(th \cdot th\right) \cdot 0.041666666666666664\right)\right)\right)} \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}} \]

   if 196 < th < 4.79999999999999995e141

   1. Initial program 99.4%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   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 99.3%

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

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   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 17.2%

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

   7. Simplified 17.2%

      \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   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. frac-2neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \color{blue}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\sqrt{2}\right), \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\color{blue}{a1 \cdot a1} + a2 \cdot a2\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\left(a1 \cdot a1\right), \color{blue}{\left(a2 \cdot a2\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(\color{blue}{a2} \cdot a2\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 34.3%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{\frac{-1}{0 - \frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}} \]

   if 4.79999999999999995e141 < 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. Step-by-step derivation

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

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

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

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

      7. metadata-eval N/A

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

      8. cos-lowering-cos.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   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. *-commutative N/A

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

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

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

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

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

      7. cos-lowering-cos.f64 51.6%

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

   7. Simplified 51.6%

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

   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 18.7%

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

   10. Simplified 18.7%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 196:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}} \cdot \left(1 + th \cdot \left(th \cdot \left(-0.5 + \left(th \cdot th\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;th \leq 4.8 \cdot 10^{+141}:\\ \;\;\;\;\frac{-1}{\frac{0 - \sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} \mathbf{if}\;th \leq 4.8 \cdot 10^{+141}:\\ \;\;\;\;\frac{a1\_m \cdot a1\_m + a2 \cdot a2}{\sqrt{2}} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (if (<= th 4.8e+141)
   (* (/ (+ (* a1_m a1_m) (* a2 a2)) (sqrt 2.0)) (+ 1.0 (* -0.5 (* th th))))
   (* a2 (* a2 (sqrt 0.5)))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	double tmp;
	if (th <= 4.8e+141) {
		tmp = (((a1_m * a1_m) + (a2 * a2)) / sqrt(2.0)) * (1.0 + (-0.5 * (th * th)));
	} else {
		tmp = a2 * (a2 * sqrt(0.5));
	}
	return tmp;
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 4.8d+141) then
        tmp = (((a1_m * a1_m) + (a2 * a2)) / sqrt(2.0d0)) * (1.0d0 + ((-0.5d0) * (th * th)))
    else
        tmp = a2 * (a2 * sqrt(0.5d0))
    end if
    code = tmp
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	double tmp;
	if (th <= 4.8e+141) {
		tmp = (((a1_m * a1_m) + (a2 * a2)) / Math.sqrt(2.0)) * (1.0 + (-0.5 * (th * th)));
	} else {
		tmp = a2 * (a2 * Math.sqrt(0.5));
	}
	return tmp;
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	tmp = 0
	if th <= 4.8e+141:
		tmp = (((a1_m * a1_m) + (a2 * a2)) / math.sqrt(2.0)) * (1.0 + (-0.5 * (th * th)))
	else:
		tmp = a2 * (a2 * math.sqrt(0.5))
	return tmp

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	tmp = 0.0
	if (th <= 4.8e+141)
		tmp = Float64(Float64(Float64(Float64(a1_m * a1_m) + Float64(a2 * a2)) / sqrt(2.0)) * Float64(1.0 + Float64(-0.5 * Float64(th * th))));
	else
		tmp = Float64(a2 * Float64(a2 * sqrt(0.5)));
	end
	return tmp
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp_2 = code(a1_m, a2, th)
	tmp = 0.0;
	if (th <= 4.8e+141)
		tmp = (((a1_m * a1_m) + (a2 * a2)) / sqrt(2.0)) * (1.0 + (-0.5 * (th * th)));
	else
		tmp = a2 * (a2 * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := If[LessEqual[th, 4.8e+141], N[(N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 4.79999999999999995e141

   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

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

         \[\leadsto \left(\frac{-1}{2} \cdot {th}^{2}\right) \cdot \left(\frac{{a1}^{2}}{\sqrt{2}} + \frac{1}{\sqrt{2}} \cdot {a2}^{2}\right) + \left(\frac{{a1}^{2}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}}\right) \]

      10. associate-*l/ N/A

          \[\leadsto \left(\frac{-1}{2} \cdot {th}^{2}\right) \cdot \left(\frac{{a1}^{2}}{\sqrt{2}} + \frac{1 \cdot {a2}^{2}}{\sqrt{2}}\right) + \left(\frac{{a1}^{2}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}}\right) \]

      11. *-lft-identity N/A

          \[\leadsto \left(\frac{-1}{2} \cdot {th}^{2}\right) \cdot \left(\frac{{a1}^{2}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}}\right) + \left(\frac{{a1}^{2}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}}\right) \]

   7. Simplified 62.5%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(th \cdot th\right)\right) \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

   if 4.79999999999999995e141 < 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. Step-by-step derivation

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

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

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

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

      7. metadata-eval N/A

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

      8. cos-lowering-cos.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   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. *-commutative N/A

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

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

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

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

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

      7. cos-lowering-cos.f64 51.6%

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

   7. Simplified 51.6%

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

   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 18.7%

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

   10. Simplified 18.7%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 4.8 \cdot 10^{+141}:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \frac{a1\_m \cdot a1\_m + a2 \cdot a2}{\sqrt{2}} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (/ (+ (* a1_m a1_m) (* a2 a2)) (sqrt 2.0)))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return ((a1_m * a1_m) + (a2 * a2)) / sqrt(2.0);
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = ((a1_m * a1_m) + (a2 * a2)) / sqrt(2.0d0)
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return ((a1_m * a1_m) + (a2 * a2)) / Math.sqrt(2.0);
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return ((a1_m * a1_m) + (a2 * a2)) / math.sqrt(2.0)

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(Float64(a1_m * a1_m) + Float64(a2 * a2)) / sqrt(2.0))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = ((a1_m * a1_m) + (a2 * a2)) / sqrt(2.0);
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] + N[(a2 * a2), $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. 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 60.4%

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

7. Simplified 60.4%

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

Alternative 13: 66.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \left(a1\_m \cdot a1\_m + a2 \cdot a2\right) \cdot \sqrt{0.5} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (* (+ (* a1_m a1_m) (* a2 a2)) (sqrt 0.5)))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return ((a1_m * a1_m) + (a2 * a2)) * sqrt(0.5);
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = ((a1_m * a1_m) + (a2 * a2)) * sqrt(0.5d0)
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return ((a1_m * a1_m) + (a2 * a2)) * Math.sqrt(0.5);
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return ((a1_m * a1_m) + (a2 * a2)) * math.sqrt(0.5)

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(Float64(a1_m * a1_m) + Float64(a2 * a2)) * sqrt(0.5))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = ((a1_m * a1_m) + (a2 * a2)) * sqrt(0.5);
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(N[(a1$95$m * a1$95$m), $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. 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. 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 99.5%

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

6. Applied egg-rr 99.5%

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

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. +-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 60.4%

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

9. Simplified 60.4%

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

10. Final simplification 60.4%

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

Alternative 14: 53.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ a2 \cdot \left(a2 \cdot \sqrt{0.5}\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th) :precision binary64 (* a2 (* a2 (sqrt 0.5))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return a2 * (a2 * sqrt(0.5));
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 * (a2 * sqrt(0.5d0))
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return a2 * (a2 * Math.sqrt(0.5));
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return a2 * (a2 * math.sqrt(0.5))

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(a2 * Float64(a2 * sqrt(0.5)))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = a2 * (a2 * sqrt(0.5));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, 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. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

   4. pow1/2 N/A

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

   5. pow-flip N/A

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

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

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

   7. metadata-eval N/A

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

   8. cos-lowering-cos.f64 99.6%

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

4. Applied egg-rr 99.6%

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

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. *-commutative N/A

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

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

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

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

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

   7. cos-lowering-cos.f64 55.0%

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

7. Simplified 55.0%

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

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 37.4%

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

10. Simplified 37.4%

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

Reproduce

herbie shell --seed 2024163 
(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))))