Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 7.9s

Alternatives: 9

Speedup: 1.9×

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a1 * a1), $MachinePrecision] + N[(N[(N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a2$95$m), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 99.6

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-lft-identity N/A

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

   11. associate-*l/ N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

   15. associate-*l/ N/A

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

   16. *-lft-identity N/A

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

   17. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 2: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a2\_m \cdot a2\_m\right) \cdot t\_1 + \left(a1 \cdot a1\right) \cdot t\_1 \leq -5 \cdot 10^{-205}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot a2\_m\right) \cdot \frac{a2\_m}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{a2\_m}{\frac{\sqrt{2}}{a2\_m}}\right)\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a2_m a2_m) t_1) (* (* a1 a1) t_1)) -5e-205)
     (* (* (fma -0.5 (* th th) 1.0) a2_m) (/ a2_m (sqrt 2.0)))
     (fma (/ a1 (sqrt 2.0)) a1 (/ a2_m (/ (sqrt 2.0) a2_m))))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a2_m * a2_m) * t_1) + ((a1 * a1) * t_1)) <= -5e-205) {
		tmp = (fma(-0.5, (th * th), 1.0) * a2_m) * (a2_m / sqrt(2.0));
	} else {
		tmp = fma((a1 / sqrt(2.0)), a1, (a2_m / (sqrt(2.0) / a2_m)));
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a2_m * a2_m) * t_1) + Float64(Float64(a1 * a1) * t_1)) <= -5e-205)
		tmp = Float64(Float64(fma(-0.5, Float64(th * th), 1.0) * a2_m) * Float64(a2_m / sqrt(2.0)));
	else
		tmp = fma(Float64(a1 / sqrt(2.0)), a1, Float64(a2_m / Float64(sqrt(2.0) / a2_m)));
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a1 * a1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -5e-205], N[(N[(N[(-0.5 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * a2$95$m), $MachinePrecision] * N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a1 + N[(a2$95$m / N[(N[Sqrt[2.0], $MachinePrecision] / a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -5.00000000000000001e-205

   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 45.8%

      \[\leadsto \left(\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot a2\right) \cdot \frac{a2}{\sqrt{2}} \]

   if -5.00000000000000001e-205 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   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 th around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{{a2}^{2}}{\sqrt{2}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a1}{\sqrt{2}}}, a1, \frac{{a2}^{2}}{\sqrt{2}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a1}{\color{blue}{\sqrt{2}}}, a1, \frac{{a2}^{2}}{\sqrt{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \color{blue}{\frac{a2}{\sqrt{2}} \cdot a2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \color{blue}{\frac{a2}{\sqrt{2}} \cdot a2}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \color{blue}{\frac{a2}{\sqrt{2}}} \cdot a2\right) \]

      10. lower-sqrt.f64 86.6

          \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{a2}{\color{blue}{\sqrt{2}}} \cdot a2\right) \]

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 86.6%

      \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{a2}{\frac{\sqrt{2}}{a2}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -5 \cdot 10^{-205}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot a2\right) \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{a2}{\frac{\sqrt{2}}{a2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{a2\_m}{\sqrt{2}}\\ t_2 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a2\_m \cdot a2\_m\right) \cdot t\_2 + \left(a1 \cdot a1\right) \cdot t\_2 \leq -5 \cdot 10^{-205}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot a2\_m\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot a2\_m\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (let* ((t_1 (/ a2_m (sqrt 2.0))) (t_2 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a2_m a2_m) t_2) (* (* a1 a1) t_2)) -5e-205)
     (* (* (fma -0.5 (* th th) 1.0) a2_m) t_1)
     (* t_1 a2_m))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	double t_1 = a2_m / sqrt(2.0);
	double t_2 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a2_m * a2_m) * t_2) + ((a1 * a1) * t_2)) <= -5e-205) {
		tmp = (fma(-0.5, (th * th), 1.0) * a2_m) * t_1;
	} else {
		tmp = t_1 * a2_m;
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	t_1 = Float64(a2_m / sqrt(2.0))
	t_2 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a2_m * a2_m) * t_2) + Float64(Float64(a1 * a1) * t_2)) <= -5e-205)
		tmp = Float64(Float64(fma(-0.5, Float64(th * th), 1.0) * a2_m) * t_1);
	else
		tmp = Float64(t_1 * a2_m);
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := Block[{t$95$1 = N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[(a1 * a1), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], -5e-205], N[(N[(N[(-0.5 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * a2$95$m), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * a2$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -5.00000000000000001e-205

   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 45.8%

      \[\leadsto \left(\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot a2\right) \cdot \frac{a2}{\sqrt{2}} \]

   if -5.00000000000000001e-205 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   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 th around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{{a2}^{2}}{\sqrt{2}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a1}{\sqrt{2}}}, a1, \frac{{a2}^{2}}{\sqrt{2}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a1}{\color{blue}{\sqrt{2}}}, a1, \frac{{a2}^{2}}{\sqrt{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \color{blue}{\frac{a2}{\sqrt{2}} \cdot a2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \color{blue}{\frac{a2}{\sqrt{2}} \cdot a2}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \color{blue}{\frac{a2}{\sqrt{2}}} \cdot a2\right) \]

      10. lower-sqrt.f64 86.6

          \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{a2}{\color{blue}{\sqrt{2}}} \cdot a2\right) \]

   5. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{a2}{\sqrt{2}} \cdot a2\right)} \]

   6. Taylor expanded in a1 around 0

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

   8. Applied rewrites 57.3%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -5 \cdot 10^{-205}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot a2\right) \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot a2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{a2\_m}{\sqrt{2}}\\ t_2 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a2\_m \cdot a2\_m\right) \cdot t\_2 + \left(a1 \cdot a1\right) \cdot t\_2 \leq -5 \cdot 10^{-205}:\\ \;\;\;\;\left(\left(\left(th \cdot th\right) \cdot -0.5\right) \cdot a2\_m\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot a2\_m\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (let* ((t_1 (/ a2_m (sqrt 2.0))) (t_2 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a2_m a2_m) t_2) (* (* a1 a1) t_2)) -5e-205)
     (* (* (* (* th th) -0.5) a2_m) t_1)
     (* t_1 a2_m))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	double t_1 = a2_m / sqrt(2.0);
	double t_2 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a2_m * a2_m) * t_2) + ((a1 * a1) * t_2)) <= -5e-205) {
		tmp = (((th * th) * -0.5) * a2_m) * t_1;
	} else {
		tmp = t_1 * a2_m;
	}
	return tmp;
}

Fortran

a2_m = abs(a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1, a2_m, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a2_m / sqrt(2.0d0)
    t_2 = cos(th) / sqrt(2.0d0)
    if ((((a2_m * a2_m) * t_2) + ((a1 * a1) * t_2)) <= (-5d-205)) then
        tmp = (((th * th) * (-0.5d0)) * a2_m) * t_1
    else
        tmp = t_1 * a2_m
    end if
    code = tmp
end function

Java

a2_m = Math.abs(a2);
assert a1 < a2_m && a2_m < th;
public static double code(double a1, double a2_m, double th) {
	double t_1 = a2_m / Math.sqrt(2.0);
	double t_2 = Math.cos(th) / Math.sqrt(2.0);
	double tmp;
	if ((((a2_m * a2_m) * t_2) + ((a1 * a1) * t_2)) <= -5e-205) {
		tmp = (((th * th) * -0.5) * a2_m) * t_1;
	} else {
		tmp = t_1 * a2_m;
	}
	return tmp;
}

Python

a2_m = math.fabs(a2)
[a1, a2_m, th] = sort([a1, a2_m, th])
def code(a1, a2_m, th):
	t_1 = a2_m / math.sqrt(2.0)
	t_2 = math.cos(th) / math.sqrt(2.0)
	tmp = 0
	if (((a2_m * a2_m) * t_2) + ((a1 * a1) * t_2)) <= -5e-205:
		tmp = (((th * th) * -0.5) * a2_m) * t_1
	else:
		tmp = t_1 * a2_m
	return tmp

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	t_1 = Float64(a2_m / sqrt(2.0))
	t_2 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a2_m * a2_m) * t_2) + Float64(Float64(a1 * a1) * t_2)) <= -5e-205)
		tmp = Float64(Float64(Float64(Float64(th * th) * -0.5) * a2_m) * t_1);
	else
		tmp = Float64(t_1 * a2_m);
	end
	return tmp
end

MATLAB

a2_m = abs(a2);
a1, a2_m, th = num2cell(sort([a1, a2_m, th])){:}
function tmp_2 = code(a1, a2_m, th)
	t_1 = a2_m / sqrt(2.0);
	t_2 = cos(th) / sqrt(2.0);
	tmp = 0.0;
	if ((((a2_m * a2_m) * t_2) + ((a1 * a1) * t_2)) <= -5e-205)
		tmp = (((th * th) * -0.5) * a2_m) * t_1;
	else
		tmp = t_1 * a2_m;
	end
	tmp_2 = tmp;
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := Block[{t$95$1 = N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[(a1 * a1), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], -5e-205], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.5), $MachinePrecision] * a2$95$m), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * a2$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -5.00000000000000001e-205

   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 45.8%

      \[\leadsto \left(\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot a2\right) \cdot \frac{a2}{\sqrt{2}} \]

   9. Taylor expanded in th around inf

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

   11. Applied rewrites 45.8%

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

   if -5.00000000000000001e-205 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   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 th around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{{a2}^{2}}{\sqrt{2}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a1}{\sqrt{2}}}, a1, \frac{{a2}^{2}}{\sqrt{2}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a1}{\color{blue}{\sqrt{2}}}, a1, \frac{{a2}^{2}}{\sqrt{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \color{blue}{\frac{a2}{\sqrt{2}} \cdot a2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \color{blue}{\frac{a2}{\sqrt{2}} \cdot a2}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \color{blue}{\frac{a2}{\sqrt{2}}} \cdot a2\right) \]

      10. lower-sqrt.f64 86.6

          \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{a2}{\color{blue}{\sqrt{2}}} \cdot a2\right) \]

   5. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{a2}{\sqrt{2}} \cdot a2\right)} \]

   6. Taylor expanded in a1 around 0

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

   8. Applied rewrites 57.3%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -5 \cdot 10^{-205}:\\ \;\;\;\;\left(\left(\left(th \cdot th\right) \cdot -0.5\right) \cdot a2\right) \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot a2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[(a2$95$m * a2$95$m + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 6: 77.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a2$95$m), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. 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. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 62.1

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

5. Applied rewrites 62.1%

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

7. Applied rewrites 62.1%

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

Alternative 7: 77.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(a2$95$m * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. 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. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 62.1

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

5. Applied rewrites 62.1%

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

6. Final simplification 62.1%

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

Alternative 8: 53.3% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a2$95$m), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-*l/ N/A

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

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{{a2}^{2}}{\sqrt{2}}\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a1}{\sqrt{2}}}, a1, \frac{{a2}^{2}}{\sqrt{2}}\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{a1}{\color{blue}{\sqrt{2}}}, a1, \frac{{a2}^{2}}{\sqrt{2}}\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}}\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \color{blue}{\frac{a2}{\sqrt{2}} \cdot a2}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \color{blue}{\frac{a2}{\sqrt{2}} \cdot a2}\right) \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \color{blue}{\frac{a2}{\sqrt{2}}} \cdot a2\right) \]

   10. lower-sqrt.f64 63.8

       \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{a2}{\color{blue}{\sqrt{2}}} \cdot a2\right) \]

5. Applied rewrites 63.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{a2}{\sqrt{2}} \cdot a2\right)} \]

6. Taylor expanded in a1 around 0

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

8. Applied rewrites 42.4%

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

9. Final simplification 42.4%

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

Alternative 9: 27.6% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a1), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-*l/ N/A

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

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{{a2}^{2}}{\sqrt{2}}\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a1}{\sqrt{2}}}, a1, \frac{{a2}^{2}}{\sqrt{2}}\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{a1}{\color{blue}{\sqrt{2}}}, a1, \frac{{a2}^{2}}{\sqrt{2}}\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}}\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \color{blue}{\frac{a2}{\sqrt{2}} \cdot a2}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \color{blue}{\frac{a2}{\sqrt{2}} \cdot a2}\right) \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \color{blue}{\frac{a2}{\sqrt{2}}} \cdot a2\right) \]

   10. lower-sqrt.f64 63.8

       \[\leadsto \mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{a2}{\color{blue}{\sqrt{2}}} \cdot a2\right) \]

5. Applied rewrites 63.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{a2}{\sqrt{2}} \cdot a2\right)} \]

6. Taylor expanded in a1 around inf

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

8. Applied rewrites 37.0%

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

9. Final simplification 37.0%

   \[\leadsto \frac{a1}{\sqrt{2}} \cdot a1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(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))))