Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 12.8s

Alternatives: 15

Speedup: 2.0×

• 42.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \frac{\cos th}{\sqrt{2}} \cdot \left(a1\_m \cdot a1\_m\right) + a2\_m \cdot \frac{\cos th \cdot a2\_m}{\sqrt{2}} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (+
  (* (/ (cos th) (sqrt 2.0)) (* a1_m a1_m))
  (* a2_m (/ (* (cos th) a2_m) (sqrt 2.0)))))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return ((cos(th) / sqrt(2.0)) * (a1_m * a1_m)) + (a2_m * ((cos(th) * a2_m) / sqrt(2.0)));
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = ((cos(th) / sqrt(2.0d0)) * (a1_m * a1_m)) + (a2_m * ((cos(th) * a2_m) / sqrt(2.0d0)))
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return ((Math.cos(th) / Math.sqrt(2.0)) * (a1_m * a1_m)) + (a2_m * ((Math.cos(th) * a2_m) / Math.sqrt(2.0)));
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return ((math.cos(th) / math.sqrt(2.0)) * (a1_m * a1_m)) + (a2_m * ((math.cos(th) * a2_m) / math.sqrt(2.0)))

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(Float64(Float64(cos(th) / sqrt(2.0)) * Float64(a1_m * a1_m)) + Float64(a2_m * Float64(Float64(cos(th) * a2_m) / sqrt(2.0))))
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = ((cos(th) / sqrt(2.0)) * (a1_m * a1_m)) + (a2_m * ((cos(th) * a2_m) / sqrt(2.0)));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] + N[(a2$95$m * N[(N[(N[Cos[th], $MachinePrecision] * a2$95$m), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. associate-*r* N/A

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

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

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

   3. associate-*l/ N/A

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

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

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

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

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

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

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

   7. sqrt-lowering-sqrt.f64 99.6%

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \frac{a1\_m \cdot a1\_m + a2\_m \cdot a2\_m}{\frac{\sqrt{2}}{\cos th}} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (/ (+ (* a1_m a1_m) (* a2_m a2_m)) (/ (sqrt 2.0) (cos th))))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return ((a1_m * a1_m) + (a2_m * a2_m)) / (sqrt(2.0) / cos(th));
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = ((a1_m * a1_m) + (a2_m * a2_m)) / (sqrt(2.0d0) / cos(th))
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return ((a1_m * a1_m) + (a2_m * a2_m)) / (Math.sqrt(2.0) / Math.cos(th));
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return ((a1_m * a1_m) + (a2_m * a2_m)) / (math.sqrt(2.0) / math.cos(th))

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(Float64(Float64(a1_m * a1_m) + Float64(a2_m * a2_m)) / Float64(sqrt(2.0) / cos(th)))
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = ((a1_m * a1_m) + (a2_m * a2_m)) / (sqrt(2.0) / cos(th));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. cos-lowering-cos.f64 99.6%

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

4. Applied egg-rr 99.6%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \cos th \cdot \frac{a1\_m \cdot a1\_m + a2\_m \cdot a2\_m}{\sqrt{2}} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (* (cos th) (/ (+ (* a1_m a1_m) (* a2_m a2_m)) (sqrt 2.0))))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return cos(th) * (((a1_m * a1_m) + (a2_m * a2_m)) / sqrt(2.0));
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = cos(th) * (((a1_m * a1_m) + (a2_m * a2_m)) / sqrt(2.0d0))
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return Math.cos(th) * (((a1_m * a1_m) + (a2_m * a2_m)) / Math.sqrt(2.0));
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return math.cos(th) * (((a1_m * a1_m) + (a2_m * a2_m)) / math.sqrt(2.0))

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(cos(th) * Float64(Float64(Float64(a1_m * a1_m) + Float64(a2_m * a2_m)) / sqrt(2.0)))
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = cos(th) * (((a1_m * a1_m) + (a2_m * a2_m)) / sqrt(2.0));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

   13. cos-lowering-cos.f64 99.6%

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \frac{\cos th}{\sqrt{2}} \cdot \left(a1\_m \cdot a1\_m + a2\_m \cdot a2\_m\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (* (/ (cos th) (sqrt 2.0)) (+ (* a1_m a1_m) (* a2_m a2_m))))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return (cos(th) / sqrt(2.0)) * ((a1_m * a1_m) + (a2_m * a2_m));
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = (cos(th) / sqrt(2.0d0)) * ((a1_m * a1_m) + (a2_m * a2_m))
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return (Math.cos(th) / Math.sqrt(2.0)) * ((a1_m * a1_m) + (a2_m * a2_m));
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return (math.cos(th) / math.sqrt(2.0)) * ((a1_m * a1_m) + (a2_m * a2_m))

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(Float64(cos(th) / sqrt(2.0)) * Float64(Float64(a1_m * a1_m) + Float64(a2_m * a2_m)))
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = (cos(th) / sqrt(2.0)) * ((a1_m * a1_m) + (a2_m * a2_m));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(a1$95$m * a1$95$m), $MachinePrecision] + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   8. *-lowering-*.f64 99.5%

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

4. Applied egg-rr 99.5%

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

Alternative 5: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \frac{a2\_m}{\frac{\sqrt{2}}{\cos th \cdot a2\_m}} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (/ a2_m (/ (sqrt 2.0) (* (cos th) a2_m))))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return a2_m / (sqrt(2.0) / (cos(th) * a2_m));
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = a2_m / (sqrt(2.0d0) / (cos(th) * a2_m))
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return a2_m / (Math.sqrt(2.0) / (Math.cos(th) * a2_m));
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return a2_m / (math.sqrt(2.0) / (math.cos(th) * a2_m))

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(a2_m / Float64(sqrt(2.0) / Float64(cos(th) * a2_m)))
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = a2_m / (sqrt(2.0) / (cos(th) * a2_m));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(a2$95$m / N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Cos[th], $MachinePrecision] * a2$95$m), $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. Taylor expanded in a1 around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

   6. sqrt-lowering-sqrt.f64 56.3%

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

5. Simplified 56.3%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*r/ N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   10. cos-lowering-cos.f64 56.2%

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

7. Applied egg-rr 56.2%

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

Alternative 6: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \cos th \cdot \left(\left(a2\_m \cdot a2\_m\right) \cdot \sqrt{0.5}\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (* (cos th) (* (* a2_m a2_m) (sqrt 0.5))))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return cos(th) * ((a2_m * a2_m) * sqrt(0.5));
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = cos(th) * ((a2_m * a2_m) * sqrt(0.5d0))
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return Math.cos(th) * ((a2_m * a2_m) * Math.sqrt(0.5));
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return math.cos(th) * ((a2_m * a2_m) * math.sqrt(0.5))

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(cos(th) * Float64(Float64(a2_m * a2_m) * sqrt(0.5)))
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = cos(th) * ((a2_m * a2_m) * sqrt(0.5));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a2$95$m * a2$95$m), $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. Taylor expanded in a1 around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

   6. sqrt-lowering-sqrt.f64 56.3%

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

5. Simplified 56.3%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. associate-*r* N/A

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

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

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

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

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

   6. pow1/2 N/A

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

   7. pow-flip N/A

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

   8. metadata-eval N/A

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

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

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

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

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

   11. cos-lowering-cos.f64 56.2%

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

7. Applied egg-rr 56.2%

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

8. Taylor expanded in a2 around 0

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

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

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

   2. unpow2 N/A

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

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

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

   4. sqrt-lowering-sqrt.f64 56.2%

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

10. Simplified 56.2%

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

11. Final simplification 56.2%

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

Alternative 7: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \left(a2\_m \cdot a2\_m\right) \cdot \left(\cos th \cdot \sqrt{0.5}\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (* (* a2_m a2_m) (* (cos th) (sqrt 0.5))))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return (a2_m * a2_m) * (cos(th) * sqrt(0.5));
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = (a2_m * a2_m) * (cos(th) * sqrt(0.5d0))
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return (a2_m * a2_m) * (Math.cos(th) * Math.sqrt(0.5));
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return (a2_m * a2_m) * (math.cos(th) * math.sqrt(0.5))

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(Float64(a2_m * a2_m) * Float64(cos(th) * sqrt(0.5)))
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = (a2_m * a2_m) * (cos(th) * sqrt(0.5));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. associate-*l* N/A

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

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

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

   6. pow1/2 N/A

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

   7. pow-flip N/A

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

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

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

   9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

   14. *-lowering-*.f64 99.6%

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

4. Applied egg-rr 99.6%

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

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. *-lowering-*.f64 N/A

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

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

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

   6. sqrt-lowering-sqrt.f64 56.2%

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

7. Simplified 56.2%

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

Alternative 8: 63.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} \mathbf{if}\;th \leq 3 \cdot 10^{+33}:\\ \;\;\;\;{2}^{-0.5} \cdot \left(\left(a1\_m \cdot a1\_m + a2\_m \cdot a2\_m\right) \cdot \left(1 + \left(th \cdot th\right) \cdot \left(-0.5 + \left(th \cdot th\right) \cdot \left(0.041666666666666664 + \left(th \cdot th\right) \cdot -0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;th \leq 2 \cdot 10^{+141}:\\ \;\;\;\;a2\_m \cdot \left(a2\_m \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - -1}{\frac{\sqrt{2}}{a2\_m \cdot a2\_m}}\\ \end{array} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (if (<= th 3e+33)
   (*
    (pow 2.0 -0.5)
    (*
     (+ (* a1_m a1_m) (* a2_m a2_m))
     (+
      1.0
      (*
       (* th th)
       (+
        -0.5
        (*
         (* th th)
         (+ 0.041666666666666664 (* (* th th) -0.001388888888888889))))))))
   (if (<= th 2e+141)
     (* a2_m (* a2_m (sqrt 0.5)))
     (/ (- 0.0 -1.0) (/ (sqrt 2.0) (* a2_m a2_m))))))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	double tmp;
	if (th <= 3e+33) {
		tmp = pow(2.0, -0.5) * (((a1_m * a1_m) + (a2_m * a2_m)) * (1.0 + ((th * th) * (-0.5 + ((th * th) * (0.041666666666666664 + ((th * th) * -0.001388888888888889)))))));
	} else if (th <= 2e+141) {
		tmp = a2_m * (a2_m * sqrt(0.5));
	} else {
		tmp = (0.0 - -1.0) / (sqrt(2.0) / (a2_m * a2_m));
	}
	return tmp;
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 3d+33) then
        tmp = (2.0d0 ** (-0.5d0)) * (((a1_m * a1_m) + (a2_m * a2_m)) * (1.0d0 + ((th * th) * ((-0.5d0) + ((th * th) * (0.041666666666666664d0 + ((th * th) * (-0.001388888888888889d0))))))))
    else if (th <= 2d+141) then
        tmp = a2_m * (a2_m * sqrt(0.5d0))
    else
        tmp = (0.0d0 - (-1.0d0)) / (sqrt(2.0d0) / (a2_m * a2_m))
    end if
    code = tmp
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	double tmp;
	if (th <= 3e+33) {
		tmp = Math.pow(2.0, -0.5) * (((a1_m * a1_m) + (a2_m * a2_m)) * (1.0 + ((th * th) * (-0.5 + ((th * th) * (0.041666666666666664 + ((th * th) * -0.001388888888888889)))))));
	} else if (th <= 2e+141) {
		tmp = a2_m * (a2_m * Math.sqrt(0.5));
	} else {
		tmp = (0.0 - -1.0) / (Math.sqrt(2.0) / (a2_m * a2_m));
	}
	return tmp;
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	tmp = 0
	if th <= 3e+33:
		tmp = math.pow(2.0, -0.5) * (((a1_m * a1_m) + (a2_m * a2_m)) * (1.0 + ((th * th) * (-0.5 + ((th * th) * (0.041666666666666664 + ((th * th) * -0.001388888888888889)))))))
	elif th <= 2e+141:
		tmp = a2_m * (a2_m * math.sqrt(0.5))
	else:
		tmp = (0.0 - -1.0) / (math.sqrt(2.0) / (a2_m * a2_m))
	return tmp

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	tmp = 0.0
	if (th <= 3e+33)
		tmp = Float64((2.0 ^ -0.5) * Float64(Float64(Float64(a1_m * a1_m) + Float64(a2_m * a2_m)) * Float64(1.0 + Float64(Float64(th * th) * Float64(-0.5 + Float64(Float64(th * th) * Float64(0.041666666666666664 + Float64(Float64(th * th) * -0.001388888888888889))))))));
	elseif (th <= 2e+141)
		tmp = Float64(a2_m * Float64(a2_m * sqrt(0.5)));
	else
		tmp = Float64(Float64(0.0 - -1.0) / Float64(sqrt(2.0) / Float64(a2_m * a2_m)));
	end
	return tmp
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp_2 = code(a1_m, a2_m, th)
	tmp = 0.0;
	if (th <= 3e+33)
		tmp = (2.0 ^ -0.5) * (((a1_m * a1_m) + (a2_m * a2_m)) * (1.0 + ((th * th) * (-0.5 + ((th * th) * (0.041666666666666664 + ((th * th) * -0.001388888888888889)))))));
	elseif (th <= 2e+141)
		tmp = a2_m * (a2_m * sqrt(0.5));
	else
		tmp = (0.0 - -1.0) / (sqrt(2.0) / (a2_m * a2_m));
	end
	tmp_2 = tmp;
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := If[LessEqual[th, 3e+33], N[(N[Power[2.0, -0.5], $MachinePrecision] * N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(th * th), $MachinePrecision] * 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, 2e+141], N[(a2$95$m * N[(a2$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0 - -1.0), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 2.99999999999999984e33

   1. Initial program 99.5%

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

      1. distribute-lft-out N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l* N/A

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

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

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

      6. pow1/2 N/A

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

      7. pow-flip N/A

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in th around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 67.0%

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

   7. Simplified 67.0%

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

   if 2.99999999999999984e33 < th < 2.00000000000000003e141

   1. Initial program 99.5%

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

   3. Taylor expanded in a1 around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 44.3%

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

   5. Simplified 44.3%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. pow1/2 N/A

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

      9. pow-flip N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. cos-lowering-cos.f64 44.3%

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

   7. Applied egg-rr 44.3%

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

   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 12.8%

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

   10. Simplified 12.8%

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

   if 2.00000000000000003e141 < th

   1. Initial program 99.7%

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

   3. Taylor expanded in a1 around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 39.4%

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

   5. Simplified 39.4%

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

   6. Taylor expanded in th around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 13.0%

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

   8. Simplified 13.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. neg-sub0 N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 17.2%

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

   10. Applied egg-rr 17.2%

       \[\leadsto \color{blue}{\frac{-1}{0 - \frac{\sqrt{2}}{a2 \cdot a2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 3 \cdot 10^{+33}:\\ \;\;\;\;{2}^{-0.5} \cdot \left(\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(1 + \left(th \cdot th\right) \cdot \left(-0.5 + \left(th \cdot th\right) \cdot \left(0.041666666666666664 + \left(th \cdot th\right) \cdot -0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;th \leq 2 \cdot 10^{+141}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - -1}{\frac{\sqrt{2}}{a2 \cdot a2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} \mathbf{if}\;th \leq 3 \cdot 10^{+33}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(th \cdot th\right)\right) \cdot \left(\left(a1\_m \cdot a1\_m + a2\_m \cdot a2\_m\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;th \leq 2 \cdot 10^{+141}:\\ \;\;\;\;a2\_m \cdot \left(a2\_m \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - -1}{\frac{\sqrt{2}}{a2\_m \cdot a2\_m}}\\ \end{array} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (if (<= th 3e+33)
   (*
    (+ 1.0 (* -0.5 (* th th)))
    (* (+ (* a1_m a1_m) (* a2_m a2_m)) (sqrt 0.5)))
   (if (<= th 2e+141)
     (* a2_m (* a2_m (sqrt 0.5)))
     (/ (- 0.0 -1.0) (/ (sqrt 2.0) (* a2_m a2_m))))))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	double tmp;
	if (th <= 3e+33) {
		tmp = (1.0 + (-0.5 * (th * th))) * (((a1_m * a1_m) + (a2_m * a2_m)) * sqrt(0.5));
	} else if (th <= 2e+141) {
		tmp = a2_m * (a2_m * sqrt(0.5));
	} else {
		tmp = (0.0 - -1.0) / (sqrt(2.0) / (a2_m * a2_m));
	}
	return tmp;
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 3d+33) then
        tmp = (1.0d0 + ((-0.5d0) * (th * th))) * (((a1_m * a1_m) + (a2_m * a2_m)) * sqrt(0.5d0))
    else if (th <= 2d+141) then
        tmp = a2_m * (a2_m * sqrt(0.5d0))
    else
        tmp = (0.0d0 - (-1.0d0)) / (sqrt(2.0d0) / (a2_m * a2_m))
    end if
    code = tmp
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	double tmp;
	if (th <= 3e+33) {
		tmp = (1.0 + (-0.5 * (th * th))) * (((a1_m * a1_m) + (a2_m * a2_m)) * Math.sqrt(0.5));
	} else if (th <= 2e+141) {
		tmp = a2_m * (a2_m * Math.sqrt(0.5));
	} else {
		tmp = (0.0 - -1.0) / (Math.sqrt(2.0) / (a2_m * a2_m));
	}
	return tmp;
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	tmp = 0
	if th <= 3e+33:
		tmp = (1.0 + (-0.5 * (th * th))) * (((a1_m * a1_m) + (a2_m * a2_m)) * math.sqrt(0.5))
	elif th <= 2e+141:
		tmp = a2_m * (a2_m * math.sqrt(0.5))
	else:
		tmp = (0.0 - -1.0) / (math.sqrt(2.0) / (a2_m * a2_m))
	return tmp

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	tmp = 0.0
	if (th <= 3e+33)
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(th * th))) * Float64(Float64(Float64(a1_m * a1_m) + Float64(a2_m * a2_m)) * sqrt(0.5)));
	elseif (th <= 2e+141)
		tmp = Float64(a2_m * Float64(a2_m * sqrt(0.5)));
	else
		tmp = Float64(Float64(0.0 - -1.0) / Float64(sqrt(2.0) / Float64(a2_m * a2_m)));
	end
	return tmp
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp_2 = code(a1_m, a2_m, th)
	tmp = 0.0;
	if (th <= 3e+33)
		tmp = (1.0 + (-0.5 * (th * th))) * (((a1_m * a1_m) + (a2_m * a2_m)) * sqrt(0.5));
	elseif (th <= 2e+141)
		tmp = a2_m * (a2_m * sqrt(0.5));
	else
		tmp = (0.0 - -1.0) / (sqrt(2.0) / (a2_m * a2_m));
	end
	tmp_2 = tmp;
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := If[LessEqual[th, 3e+33], N[(N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 2e+141], N[(a2$95$m * N[(a2$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0 - -1.0), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 2.99999999999999984e33

   1. Initial program 99.5%

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

      1. distribute-lft-out N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l* N/A

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

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

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

      6. pow1/2 N/A

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

      7. pow-flip N/A

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. sqrt-lowering-sqrt.f64 69.3%

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

   7. Simplified 69.3%

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

   if 2.99999999999999984e33 < th < 2.00000000000000003e141

   1. Initial program 99.5%

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

   3. Taylor expanded in a1 around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 44.3%

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

   5. Simplified 44.3%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. pow1/2 N/A

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

      9. pow-flip N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. cos-lowering-cos.f64 44.3%

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

   7. Applied egg-rr 44.3%

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

   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 12.8%

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

   10. Simplified 12.8%

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

   if 2.00000000000000003e141 < th

   1. Initial program 99.7%

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

   3. Taylor expanded in a1 around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 39.4%

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

   5. Simplified 39.4%

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

   6. Taylor expanded in th around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 13.0%

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

   8. Simplified 13.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. neg-sub0 N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 17.2%

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

   10. Applied egg-rr 17.2%

       \[\leadsto \color{blue}{\frac{-1}{0 - \frac{\sqrt{2}}{a2 \cdot a2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 3 \cdot 10^{+33}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(th \cdot th\right)\right) \cdot \left(\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;th \leq 2 \cdot 10^{+141}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - -1}{\frac{\sqrt{2}}{a2 \cdot a2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} \mathbf{if}\;th \leq 3 \cdot 10^{+33}:\\ \;\;\;\;a2\_m \cdot \frac{a2\_m}{\frac{\sqrt{2}}{1 + -0.5 \cdot \left(th \cdot th\right)}}\\ \mathbf{elif}\;th \leq 2 \cdot 10^{+141}:\\ \;\;\;\;a2\_m \cdot \left(a2\_m \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - -1}{\frac{\sqrt{2}}{a2\_m \cdot a2\_m}}\\ \end{array} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (if (<= th 3e+33)
   (* a2_m (/ a2_m (/ (sqrt 2.0) (+ 1.0 (* -0.5 (* th th))))))
   (if (<= th 2e+141)
     (* a2_m (* a2_m (sqrt 0.5)))
     (/ (- 0.0 -1.0) (/ (sqrt 2.0) (* a2_m a2_m))))))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	double tmp;
	if (th <= 3e+33) {
		tmp = a2_m * (a2_m / (sqrt(2.0) / (1.0 + (-0.5 * (th * th)))));
	} else if (th <= 2e+141) {
		tmp = a2_m * (a2_m * sqrt(0.5));
	} else {
		tmp = (0.0 - -1.0) / (sqrt(2.0) / (a2_m * a2_m));
	}
	return tmp;
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 3d+33) then
        tmp = a2_m * (a2_m / (sqrt(2.0d0) / (1.0d0 + ((-0.5d0) * (th * th)))))
    else if (th <= 2d+141) then
        tmp = a2_m * (a2_m * sqrt(0.5d0))
    else
        tmp = (0.0d0 - (-1.0d0)) / (sqrt(2.0d0) / (a2_m * a2_m))
    end if
    code = tmp
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	double tmp;
	if (th <= 3e+33) {
		tmp = a2_m * (a2_m / (Math.sqrt(2.0) / (1.0 + (-0.5 * (th * th)))));
	} else if (th <= 2e+141) {
		tmp = a2_m * (a2_m * Math.sqrt(0.5));
	} else {
		tmp = (0.0 - -1.0) / (Math.sqrt(2.0) / (a2_m * a2_m));
	}
	return tmp;
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	tmp = 0
	if th <= 3e+33:
		tmp = a2_m * (a2_m / (math.sqrt(2.0) / (1.0 + (-0.5 * (th * th)))))
	elif th <= 2e+141:
		tmp = a2_m * (a2_m * math.sqrt(0.5))
	else:
		tmp = (0.0 - -1.0) / (math.sqrt(2.0) / (a2_m * a2_m))
	return tmp

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	tmp = 0.0
	if (th <= 3e+33)
		tmp = Float64(a2_m * Float64(a2_m / Float64(sqrt(2.0) / Float64(1.0 + Float64(-0.5 * Float64(th * th))))));
	elseif (th <= 2e+141)
		tmp = Float64(a2_m * Float64(a2_m * sqrt(0.5)));
	else
		tmp = Float64(Float64(0.0 - -1.0) / Float64(sqrt(2.0) / Float64(a2_m * a2_m)));
	end
	return tmp
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp_2 = code(a1_m, a2_m, th)
	tmp = 0.0;
	if (th <= 3e+33)
		tmp = a2_m * (a2_m / (sqrt(2.0) / (1.0 + (-0.5 * (th * th)))));
	elseif (th <= 2e+141)
		tmp = a2_m * (a2_m * sqrt(0.5));
	else
		tmp = (0.0 - -1.0) / (sqrt(2.0) / (a2_m * a2_m));
	end
	tmp_2 = tmp;
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := If[LessEqual[th, 3e+33], N[(a2$95$m * N[(a2$95$m / N[(N[Sqrt[2.0], $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 2e+141], N[(a2$95$m * N[(a2$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0 - -1.0), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 2.99999999999999984e33

   1. Initial program 99.5%

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

   3. Taylor expanded in a1 around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 59.7%

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

   5. Simplified 59.7%

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

   6. Taylor expanded in th around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 43.5%

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

   8. Simplified 43.5%

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

      1. associate-/l* N/A

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

      2. associate-*l* N/A

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

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

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 42.7%

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

   10. Applied egg-rr 42.7%

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

   if 2.99999999999999984e33 < th < 2.00000000000000003e141

   1. Initial program 99.5%

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

   3. Taylor expanded in a1 around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 44.3%

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

   5. Simplified 44.3%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. pow1/2 N/A

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

      9. pow-flip N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. cos-lowering-cos.f64 44.3%

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

   7. Applied egg-rr 44.3%

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

   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 12.8%

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

   10. Simplified 12.8%

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

   if 2.00000000000000003e141 < th

   1. Initial program 99.7%

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

   3. Taylor expanded in a1 around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 39.4%

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

   5. Simplified 39.4%

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

   6. Taylor expanded in th around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 13.0%

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

   8. Simplified 13.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. neg-sub0 N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 17.2%

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

   10. Applied egg-rr 17.2%

       \[\leadsto \color{blue}{\frac{-1}{0 - \frac{\sqrt{2}}{a2 \cdot a2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 3 \cdot 10^{+33}:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{1 + -0.5 \cdot \left(th \cdot th\right)}}\\ \mathbf{elif}\;th \leq 2 \cdot 10^{+141}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - -1}{\frac{\sqrt{2}}{a2 \cdot a2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} \mathbf{if}\;th \leq 2 \cdot 10^{+141}:\\ \;\;\;\;\left(a1\_m \cdot a1\_m + a2\_m \cdot a2\_m\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - -1}{\frac{\sqrt{2}}{a2\_m \cdot a2\_m}}\\ \end{array} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (if (<= th 2e+141)
   (* (+ (* a1_m a1_m) (* a2_m a2_m)) (sqrt 0.5))
   (/ (- 0.0 -1.0) (/ (sqrt 2.0) (* a2_m a2_m)))))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	double tmp;
	if (th <= 2e+141) {
		tmp = ((a1_m * a1_m) + (a2_m * a2_m)) * sqrt(0.5);
	} else {
		tmp = (0.0 - -1.0) / (sqrt(2.0) / (a2_m * a2_m));
	}
	return tmp;
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 2d+141) then
        tmp = ((a1_m * a1_m) + (a2_m * a2_m)) * sqrt(0.5d0)
    else
        tmp = (0.0d0 - (-1.0d0)) / (sqrt(2.0d0) / (a2_m * a2_m))
    end if
    code = tmp
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	double tmp;
	if (th <= 2e+141) {
		tmp = ((a1_m * a1_m) + (a2_m * a2_m)) * Math.sqrt(0.5);
	} else {
		tmp = (0.0 - -1.0) / (Math.sqrt(2.0) / (a2_m * a2_m));
	}
	return tmp;
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	tmp = 0
	if th <= 2e+141:
		tmp = ((a1_m * a1_m) + (a2_m * a2_m)) * math.sqrt(0.5)
	else:
		tmp = (0.0 - -1.0) / (math.sqrt(2.0) / (a2_m * a2_m))
	return tmp

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	tmp = 0.0
	if (th <= 2e+141)
		tmp = Float64(Float64(Float64(a1_m * a1_m) + Float64(a2_m * a2_m)) * sqrt(0.5));
	else
		tmp = Float64(Float64(0.0 - -1.0) / Float64(sqrt(2.0) / Float64(a2_m * a2_m)));
	end
	return tmp
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp_2 = code(a1_m, a2_m, th)
	tmp = 0.0;
	if (th <= 2e+141)
		tmp = ((a1_m * a1_m) + (a2_m * a2_m)) * sqrt(0.5);
	else
		tmp = (0.0 - -1.0) / (sqrt(2.0) / (a2_m * a2_m));
	end
	tmp_2 = tmp;
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := If[LessEqual[th, 2e+141], N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[(0.0 - -1.0), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 2.00000000000000003e141

   1. Initial program 99.5%

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

      1. distribute-lft-out N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l* N/A

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

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

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

      6. pow1/2 N/A

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

      7. pow-flip N/A

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. sqrt-lowering-sqrt.f64 68.1%

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

   7. Simplified 68.1%

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

   if 2.00000000000000003e141 < th

   1. Initial program 99.7%

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

   3. Taylor expanded in a1 around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 39.4%

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

   5. Simplified 39.4%

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

   6. Taylor expanded in th around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 13.0%

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

   8. Simplified 13.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. neg-sub0 N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 17.2%

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

   10. Applied egg-rr 17.2%

       \[\leadsto \color{blue}{\frac{-1}{0 - \frac{\sqrt{2}}{a2 \cdot a2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.9%

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

Alternative 12: 66.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \left(a1\_m \cdot a1\_m + a2\_m \cdot a2\_m\right) \cdot \sqrt{0.5} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (* (+ (* a1_m a1_m) (* a2_m a2_m)) (sqrt 0.5)))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return ((a1_m * a1_m) + (a2_m * a2_m)) * sqrt(0.5);
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = ((a1_m * a1_m) + (a2_m * a2_m)) * sqrt(0.5d0)
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return ((a1_m * a1_m) + (a2_m * a2_m)) * Math.sqrt(0.5);
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return ((a1_m * a1_m) + (a2_m * a2_m)) * math.sqrt(0.5)

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(Float64(Float64(a1_m * a1_m) + Float64(a2_m * a2_m)) * sqrt(0.5))
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = ((a1_m * a1_m) + (a2_m * a2_m)) * sqrt(0.5);
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. associate-*l* N/A

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

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

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

   6. pow1/2 N/A

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

   7. pow-flip N/A

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

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

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

   9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

   14. *-lowering-*.f64 99.6%

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in th around 0

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

   1. *-commutative N/A

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. unpow2 N/A

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

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

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

   8. sqrt-lowering-sqrt.f64 63.3%

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

7. Simplified 63.3%

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

Alternative 13: 65.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ a2\_m \cdot \frac{a2\_m}{\sqrt{2}} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th) :precision binary64 (* a2_m (/ a2_m (sqrt 2.0))))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return a2_m * (a2_m / sqrt(2.0));
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = a2_m * (a2_m / sqrt(2.0d0))
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return a2_m * (a2_m / Math.sqrt(2.0));
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return a2_m * (a2_m / math.sqrt(2.0))

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(a2_m * Float64(a2_m / sqrt(2.0)))
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = a2_m * (a2_m / sqrt(2.0));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(a2$95$m * N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in a1 around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

   6. sqrt-lowering-sqrt.f64 56.3%

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

5. Simplified 56.3%

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

6. Taylor expanded in th around 0

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

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

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

   2. unpow2 N/A

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

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

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

   4. sqrt-lowering-sqrt.f64 39.2%

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

8. Simplified 39.2%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. sqrt-lowering-sqrt.f64 39.2%

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

10. Applied egg-rr 39.2%

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

11. Final simplification 39.2%

    \[\leadsto a2 \cdot \frac{a2}{\sqrt{2}} \]
12. Add Preprocessing

Alternative 14: 65.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \left(a2\_m \cdot a2\_m\right) \cdot \sqrt{0.5} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th) :precision binary64 (* (* a2_m a2_m) (sqrt 0.5)))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return (a2_m * a2_m) * sqrt(0.5);
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = (a2_m * a2_m) * sqrt(0.5d0)
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return (a2_m * a2_m) * Math.sqrt(0.5);
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return (a2_m * a2_m) * math.sqrt(0.5)

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(Float64(a2_m * a2_m) * sqrt(0.5))
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = (a2_m * a2_m) * sqrt(0.5);
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in a1 around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

   6. sqrt-lowering-sqrt.f64 56.3%

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

5. Simplified 56.3%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

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

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

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

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

   8. pow1/2 N/A

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

   9. pow-flip N/A

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

   10. metadata-eval N/A

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

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

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

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

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

   13. cos-lowering-cos.f64 56.2%

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

7. Applied egg-rr 56.2%

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

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 39.2%

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

10. Simplified 39.2%

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

    1. associate-*r* N/A

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

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

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

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

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

    4. sqrt-lowering-sqrt.f64 39.2%

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

12. Applied egg-rr 39.2%

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

Alternative 15: 65.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ a2\_m \cdot \left(a2\_m \cdot \sqrt{0.5}\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th) :precision binary64 (* a2_m (* a2_m (sqrt 0.5))))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return a2_m * (a2_m * sqrt(0.5));
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = a2_m * (a2_m * sqrt(0.5d0))
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return a2_m * (a2_m * Math.sqrt(0.5));
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return a2_m * (a2_m * math.sqrt(0.5))

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(a2_m * Float64(a2_m * sqrt(0.5)))
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = a2_m * (a2_m * sqrt(0.5));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(a2$95$m * N[(a2$95$m * 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. Taylor expanded in a1 around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

   6. sqrt-lowering-sqrt.f64 56.3%

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

5. Simplified 56.3%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

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

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

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

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

   8. pow1/2 N/A

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

   9. pow-flip N/A

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

   10. metadata-eval N/A

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

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

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

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

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

   13. cos-lowering-cos.f64 56.2%

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

7. Applied egg-rr 56.2%

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

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 39.2%

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

10. Simplified 39.2%

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

Reproduce

herbie shell --seed 2024185 
(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))))