Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 12.5s

Alternatives: 12

Speedup: 1.9×

• 43.5% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2}, \cos th \cdot \left(a2 \cdot a2\right), \sqrt{2} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\right) \cdot 0.5} \]
5. Add Preprocessing

Alternative 2: 63.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1\_m \cdot a1\_m\right) \cdot t\_1 + \left(a2 \cdot a2\right) \cdot t\_1 \leq -2 \cdot 10^{-232}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(a2 \cdot a2\right)\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, 0\right)\\ \end{array} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1_m a1_m) t_1) (* (* a2 a2) t_1)) -2e-232)
     (* (* (sqrt 2.0) (* a2 a2)) (fma -0.25 (* th th) 0.5))
     (fma a2 (/ a2 (sqrt 2.0)) 0.0))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1_m * a1_m) * t_1) + ((a2 * a2) * t_1)) <= -2e-232) {
		tmp = (sqrt(2.0) * (a2 * a2)) * fma(-0.25, (th * th), 0.5);
	} else {
		tmp = fma(a2, (a2 / sqrt(2.0)), 0.0);
	}
	return tmp;
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1_m * a1_m) * t_1) + Float64(Float64(a2 * a2) * t_1)) <= -2e-232)
		tmp = Float64(Float64(sqrt(2.0) * Float64(a2 * a2)) * fma(-0.25, Float64(th * th), 0.5));
	else
		tmp = fma(a2, Float64(a2 / sqrt(2.0)), 0.0);
	end
	return tmp
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -2e-232], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 0.0), $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))) < -2.00000000000000005e-232

   1. Initial program 99.4%

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

      1. associate-*l/ N/A

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

      2. associate-*l/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

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

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

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2}, \cos th \cdot \left(a2 \cdot a2\right), \sqrt{2} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\right) \cdot 0.5} \]

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

   7. Simplified 3.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(th \cdot th, -0.25 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot 0.5\right)\right)} \]

   8. Taylor expanded in a1 around 0

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left({a2}^{2} \cdot \left({th}^{2} \cdot \sqrt{2}\right)\right) + \frac{1}{2} \cdot \left({a2}^{2} \cdot \sqrt{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(\left({th}^{2} \cdot \sqrt{2}\right) \cdot {a2}^{2}\right)} + \frac{1}{2} \cdot \left({a2}^{2} \cdot \sqrt{2}\right) \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \left({th}^{2} \cdot \sqrt{2}\right)\right) \cdot {a2}^{2}} + \frac{1}{2} \cdot \left({a2}^{2} \cdot \sqrt{2}\right) \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} \cdot {th}^{2}\right) \cdot \sqrt{2}\right)} \cdot {a2}^{2} + \frac{1}{2} \cdot \left({a2}^{2} \cdot \sqrt{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {th}^{2}\right) \cdot \left(\sqrt{2} \cdot {a2}^{2}\right)} + \frac{1}{2} \cdot \left({a2}^{2} \cdot \sqrt{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot {th}^{2}\right) \cdot \color{blue}{\left({a2}^{2} \cdot \sqrt{2}\right)} + \frac{1}{2} \cdot \left({a2}^{2} \cdot \sqrt{2}\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\left({a2}^{2} \cdot \sqrt{2}\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right)} \]

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

         \[\leadsto \color{blue}{\left({a2}^{2} \cdot \sqrt{2}\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot {a2}^{2}\right)} \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right) \]

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

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot {a2}^{2}\right)} \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right) \]

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

          \[\leadsto \left(\color{blue}{\sqrt{2}} \cdot {a2}^{2}\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\sqrt{2} \cdot \color{blue}{\left(a2 \cdot a2\right)}\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right) \]

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

          \[\leadsto \left(\sqrt{2} \cdot \color{blue}{\left(a2 \cdot a2\right)}\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right) \]

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

          \[\leadsto \left(\sqrt{2} \cdot \left(a2 \cdot a2\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, {th}^{2}, \frac{1}{2}\right)} \]

      14. unpow2 N/A

          \[\leadsto \left(\sqrt{2} \cdot \left(a2 \cdot a2\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{th \cdot th}, \frac{1}{2}\right) \]

      15. *-lowering-*.f64 36.8

          \[\leadsto \left(\sqrt{2} \cdot \left(a2 \cdot a2\right)\right) \cdot \mathsf{fma}\left(-0.25, \color{blue}{th \cdot th}, 0.5\right) \]

   10. Simplified 36.8%

       \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(a2 \cdot a2\right)\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)} \]

   if -2.00000000000000005e-232 < (+.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 99.6%

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

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

      4. frac-times N/A

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

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

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

   4. Applied egg-rr 12.6%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 9.7%

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

   8. Taylor expanded in a1 around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-rgt-identity N/A

         \[\leadsto {a2}^{2} \cdot \color{blue}{\left(\frac{1}{\sqrt{2}} + 0\right)} \]

      4. distribute-rgt-in N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{{a2}^{2}}}{\sqrt{2}} + 0 \cdot {a2}^{2} \]

      7. mul0-lft N/A

         \[\leadsto \frac{{a2}^{2}}{\sqrt{2}} + \color{blue}{0} \]

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, 0\right)} \]

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

          \[\leadsto \mathsf{fma}\left(a2, \color{blue}{\frac{a2}{\sqrt{2}}}, 0\right) \]

      12. sqrt-lowering-sqrt.f64 53.1

          \[\leadsto \mathsf{fma}\left(a2, \frac{a2}{\color{blue}{\sqrt{2}}}, 0\right) \]

   10. Simplified 53.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -2 \cdot 10^{-232}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(a2 \cdot a2\right)\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1\_m \cdot a1\_m\right) \cdot t\_1 + \left(a2 \cdot a2\right) \cdot t\_1 \leq -2 \cdot 10^{-96}:\\ \;\;\;\;\left(a1\_m \cdot a1\_m\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, 0\right)\\ \end{array} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1_m a1_m) t_1) (* (* a2 a2) t_1)) -2e-96)
     (* (* a1_m a1_m) (* (sqrt 2.0) (fma -0.25 (* th th) 0.5)))
     (fma a2 (/ a2 (sqrt 2.0)) 0.0))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1_m * a1_m) * t_1) + ((a2 * a2) * t_1)) <= -2e-96) {
		tmp = (a1_m * a1_m) * (sqrt(2.0) * fma(-0.25, (th * th), 0.5));
	} else {
		tmp = fma(a2, (a2 / sqrt(2.0)), 0.0);
	}
	return tmp;
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1_m * a1_m) * t_1) + Float64(Float64(a2 * a2) * t_1)) <= -2e-96)
		tmp = Float64(Float64(a1_m * a1_m) * Float64(sqrt(2.0) * fma(-0.25, Float64(th * th), 0.5)));
	else
		tmp = fma(a2, Float64(a2 / sqrt(2.0)), 0.0);
	end
	return tmp
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -2e-96], N[(N[(a1$95$m * a1$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 0.0), $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))) < -1.9999999999999998e-96

   1. Initial program 99.4%

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

      1. associate-*l/ N/A

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

      2. associate-*l/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

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

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

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2}, \cos th \cdot \left(a2 \cdot a2\right), \sqrt{2} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\right) \cdot 0.5} \]

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

   7. Simplified 2.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(th \cdot th, -0.25 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot 0.5\right)\right)} \]

   8. Taylor expanded in a1 around inf

      \[\leadsto \color{blue}{{a1}^{2} \cdot \left(\frac{-1}{4} \cdot \left({th}^{2} \cdot \sqrt{2}\right) + \frac{1}{2} \cdot \sqrt{2}\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{{a1}^{2} \cdot \left(\frac{-1}{4} \cdot \left({th}^{2} \cdot \sqrt{2}\right) + \frac{1}{2} \cdot \sqrt{2}\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(a1 \cdot a1\right)} \cdot \left(\frac{-1}{4} \cdot \left({th}^{2} \cdot \sqrt{2}\right) + \frac{1}{2} \cdot \sqrt{2}\right) \]

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

         \[\leadsto \color{blue}{\left(a1 \cdot a1\right)} \cdot \left(\frac{-1}{4} \cdot \left({th}^{2} \cdot \sqrt{2}\right) + \frac{1}{2} \cdot \sqrt{2}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(a1 \cdot a1\right) \cdot \left(\color{blue}{\left(\frac{-1}{4} \cdot {th}^{2}\right) \cdot \sqrt{2}} + \frac{1}{2} \cdot \sqrt{2}\right) \]

      5. distribute-rgt-out N/A

         \[\leadsto \left(a1 \cdot a1\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right)\right)} \]

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

         \[\leadsto \left(a1 \cdot a1\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right)\right)} \]

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

         \[\leadsto \left(a1 \cdot a1\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right)\right) \]

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

         \[\leadsto \left(a1 \cdot a1\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, {th}^{2}, \frac{1}{2}\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \left(a1 \cdot a1\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{th \cdot th}, \frac{1}{2}\right)\right) \]

      10. *-lowering-*.f64 41.2

          \[\leadsto \left(a1 \cdot a1\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.25, \color{blue}{th \cdot th}, 0.5\right)\right) \]

   10. Simplified 41.2%

       \[\leadsto \color{blue}{\left(a1 \cdot a1\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\right)} \]

   if -1.9999999999999998e-96 < (+.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 99.6%

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

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

      4. frac-times N/A

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

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

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

   4. Applied egg-rr 12.1%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 9.4%

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

   8. Taylor expanded in a1 around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-rgt-identity N/A

         \[\leadsto {a2}^{2} \cdot \color{blue}{\left(\frac{1}{\sqrt{2}} + 0\right)} \]

      4. distribute-rgt-in N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{{a2}^{2}}}{\sqrt{2}} + 0 \cdot {a2}^{2} \]

      7. mul0-lft N/A

         \[\leadsto \frac{{a2}^{2}}{\sqrt{2}} + \color{blue}{0} \]

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, 0\right)} \]

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

          \[\leadsto \mathsf{fma}\left(a2, \color{blue}{\frac{a2}{\sqrt{2}}}, 0\right) \]

      12. sqrt-lowering-sqrt.f64 51.2

          \[\leadsto \mathsf{fma}\left(a2, \frac{a2}{\color{blue}{\sqrt{2}}}, 0\right) \]

   10. Simplified 51.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -2 \cdot 10^{-96}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1\_m \cdot a1\_m\right) \cdot t\_1 + \left(a2 \cdot a2\right) \cdot t\_1 \leq -5 \cdot 10^{-285}:\\ \;\;\;\;th \cdot \left(th \cdot \left(-0.25 \cdot \left(a1\_m \cdot \left(\sqrt{2} \cdot a1\_m\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, 0\right)\\ \end{array} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1_m a1_m) t_1) (* (* a2 a2) t_1)) -5e-285)
     (* th (* th (* -0.25 (* a1_m (* (sqrt 2.0) a1_m)))))
     (fma a2 (/ a2 (sqrt 2.0)) 0.0))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1_m * a1_m) * t_1) + ((a2 * a2) * t_1)) <= -5e-285) {
		tmp = th * (th * (-0.25 * (a1_m * (sqrt(2.0) * a1_m))));
	} else {
		tmp = fma(a2, (a2 / sqrt(2.0)), 0.0);
	}
	return tmp;
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1_m * a1_m) * t_1) + Float64(Float64(a2 * a2) * t_1)) <= -5e-285)
		tmp = Float64(th * Float64(th * Float64(-0.25 * Float64(a1_m * Float64(sqrt(2.0) * a1_m)))));
	else
		tmp = fma(a2, Float64(a2 / sqrt(2.0)), 0.0);
	end
	return tmp
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -5e-285], N[(th * N[(th * N[(-0.25 * N[(a1$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * a1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 0.0), $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))) < -5.00000000000000018e-285

   1. Initial program 99.4%

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

      1. associate-*l/ N/A

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

      2. associate-*l/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

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

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

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2}, \cos th \cdot \left(a2 \cdot a2\right), \sqrt{2} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\right) \cdot 0.5} \]

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

   7. Simplified 3.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(th \cdot th, -0.25 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot 0.5\right)\right)} \]

   8. Taylor expanded in a1 around inf

      \[\leadsto \mathsf{fma}\left(th \cdot th, \color{blue}{\frac{-1}{4} \cdot \left({a1}^{2} \cdot \sqrt{2}\right)}, \sqrt{2} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{1}{2}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(th \cdot th, \color{blue}{\left(\frac{-1}{4} \cdot {a1}^{2}\right) \cdot \sqrt{2}}, \sqrt{2} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{1}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(th \cdot th, \color{blue}{\sqrt{2} \cdot \left(\frac{-1}{4} \cdot {a1}^{2}\right)}, \sqrt{2} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(th \cdot th, \color{blue}{\sqrt{2} \cdot \left(\frac{-1}{4} \cdot {a1}^{2}\right)}, \sqrt{2} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(th \cdot th, \color{blue}{\sqrt{2}} \cdot \left(\frac{-1}{4} \cdot {a1}^{2}\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{1}{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(th \cdot th, \sqrt{2} \cdot \color{blue}{\left({a1}^{2} \cdot \frac{-1}{4}\right)}, \sqrt{2} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(th \cdot th, \sqrt{2} \cdot \color{blue}{\left({a1}^{2} \cdot \frac{-1}{4}\right)}, \sqrt{2} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{1}{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th \cdot th, \sqrt{2} \cdot \left(\color{blue}{\left(a1 \cdot a1\right)} \cdot \frac{-1}{4}\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{1}{2}\right)\right) \]

      8. *-lowering-*.f64 2.3

         \[\leadsto \mathsf{fma}\left(th \cdot th, \sqrt{2} \cdot \left(\color{blue}{\left(a1 \cdot a1\right)} \cdot -0.25\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot 0.5\right)\right) \]

   10. Simplified 2.3%

       \[\leadsto \mathsf{fma}\left(th \cdot th, \color{blue}{\sqrt{2} \cdot \left(\left(a1 \cdot a1\right) \cdot -0.25\right)}, \sqrt{2} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot 0.5\right)\right) \]

   11. Taylor expanded in th around inf

       \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left({a1}^{2} \cdot \left({th}^{2} \cdot \sqrt{2}\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{-1}{4} \cdot \left({a1}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot {th}^{2}\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(\left({a1}^{2} \cdot \sqrt{2}\right) \cdot {th}^{2}\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \left({a1}^{2} \cdot \sqrt{2}\right)\right) \cdot {th}^{2}} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{{th}^{2} \cdot \left(\frac{-1}{4} \cdot \left({a1}^{2} \cdot \sqrt{2}\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto \color{blue}{\left(th \cdot th\right)} \cdot \left(\frac{-1}{4} \cdot \left({a1}^{2} \cdot \sqrt{2}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{th \cdot \left(th \cdot \left(\frac{-1}{4} \cdot \left({a1}^{2} \cdot \sqrt{2}\right)\right)\right)} \]

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

          \[\leadsto \color{blue}{th \cdot \left(th \cdot \left(\frac{-1}{4} \cdot \left({a1}^{2} \cdot \sqrt{2}\right)\right)\right)} \]

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

          \[\leadsto th \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{4} \cdot \left({a1}^{2} \cdot \sqrt{2}\right)\right)\right)} \]

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

          \[\leadsto th \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left({a1}^{2} \cdot \sqrt{2}\right)\right)}\right) \]

       10. unpow2 N/A

           \[\leadsto th \cdot \left(th \cdot \left(\frac{-1}{4} \cdot \left(\color{blue}{\left(a1 \cdot a1\right)} \cdot \sqrt{2}\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto th \cdot \left(th \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(a1 \cdot \left(a1 \cdot \sqrt{2}\right)\right)}\right)\right) \]

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

           \[\leadsto th \cdot \left(th \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(a1 \cdot \left(a1 \cdot \sqrt{2}\right)\right)}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto th \cdot \left(th \cdot \left(\frac{-1}{4} \cdot \left(a1 \cdot \color{blue}{\left(\sqrt{2} \cdot a1\right)}\right)\right)\right) \]

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

           \[\leadsto th \cdot \left(th \cdot \left(\frac{-1}{4} \cdot \left(a1 \cdot \color{blue}{\left(\sqrt{2} \cdot a1\right)}\right)\right)\right) \]

       15. sqrt-lowering-sqrt.f64 33.8

           \[\leadsto th \cdot \left(th \cdot \left(-0.25 \cdot \left(a1 \cdot \left(\color{blue}{\sqrt{2}} \cdot a1\right)\right)\right)\right) \]

   13. Simplified 33.8%

       \[\leadsto \color{blue}{th \cdot \left(th \cdot \left(-0.25 \cdot \left(a1 \cdot \left(\sqrt{2} \cdot a1\right)\right)\right)\right)} \]

   if -5.00000000000000018e-285 < (+.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 99.6%

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

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

      4. frac-times N/A

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

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

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

   4. Applied egg-rr 12.8%

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

   5. Taylor expanded in th around 0

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

   7. Simplified 9.9%

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

   8. Taylor expanded in a1 around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-rgt-identity N/A

         \[\leadsto {a2}^{2} \cdot \color{blue}{\left(\frac{1}{\sqrt{2}} + 0\right)} \]

      4. distribute-rgt-in N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{{a2}^{2}}}{\sqrt{2}} + 0 \cdot {a2}^{2} \]

      7. mul0-lft N/A

         \[\leadsto \frac{{a2}^{2}}{\sqrt{2}} + \color{blue}{0} \]

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, 0\right)} \]

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

          \[\leadsto \mathsf{fma}\left(a2, \color{blue}{\frac{a2}{\sqrt{2}}}, 0\right) \]

      12. sqrt-lowering-sqrt.f64 53.8

          \[\leadsto \mathsf{fma}\left(a2, \frac{a2}{\color{blue}{\sqrt{2}}}, 0\right) \]

   10. Simplified 53.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -5 \cdot 10^{-285}:\\ \;\;\;\;th \cdot \left(th \cdot \left(-0.25 \cdot \left(a1 \cdot \left(\sqrt{2} \cdot a1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2}, \cos th \cdot \left(a2 \cdot a2\right), \sqrt{2} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\right) \cdot 0.5} \]
5. Step-by-step derivation

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. distribute-lft-out N/A

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

   4. +-commutative N/A

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

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

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

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

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

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

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

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

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

   9. +-commutative N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. *-lowering-*.f64 99.6

       \[\leadsto \left(\left(\sqrt{2} \cdot \cos th\right) \cdot \mathsf{fma}\left(a2, a2, \color{blue}{a1 \cdot a1}\right)\right) \cdot 0.5 \]

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \cos th\right) \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right)} \cdot 0.5 \]

7. Final simplification 99.6%

   \[\leadsto 0.5 \cdot \left(\left(\sqrt{2} \cdot \cos th\right) \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \]
8. Add Preprocessing

Alternative 6: 78.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2}, \cos th \cdot \left(a2 \cdot a2\right), \sqrt{2} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\right) \cdot 0.5} \]

5. Taylor expanded in a2 around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

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

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

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

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

   9. cos-lowering-cos.f64 53.5

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

7. Simplified 53.5%

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

   1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   6. cos-lowering-cos.f64 53.5

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

9. Applied egg-rr 53.5%

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

10. Final simplification 53.5%

    \[\leadsto 0.5 \cdot \left(\left(\sqrt{2} \cdot a2\right) \cdot \left(\cos th \cdot a2\right)\right) \]
11. Add Preprocessing

Alternative 7: 78.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2}, \cos th \cdot \left(a2 \cdot a2\right), \sqrt{2} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\right) \cdot 0.5} \]

5. Taylor expanded in a2 around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

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

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

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

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

   9. cos-lowering-cos.f64 53.5

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

7. Simplified 53.5%

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

   1. *-commutative N/A

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

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

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

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

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

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

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

   8. sqrt-lowering-sqrt.f64 53.5

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

9. Applied egg-rr 53.5%

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

10. Final simplification 53.5%

    \[\leadsto 0.5 \cdot \left(\sqrt{2} \cdot \left(\cos th \cdot \left(a2 \cdot a2\right)\right)\right) \]
11. Add Preprocessing

Alternative 8: 78.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2}, \cos th \cdot \left(a2 \cdot a2\right), \sqrt{2} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\right) \cdot 0.5} \]

5. Taylor expanded in a2 around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

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

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

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

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

   9. cos-lowering-cos.f64 53.5

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

7. Simplified 53.5%

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

8. Final simplification 53.5%

   \[\leadsto 0.5 \cdot \left(\sqrt{2} \cdot \left(a2 \cdot \left(\cos th \cdot a2\right)\right)\right) \]
9. Add Preprocessing

Alternative 9: 78.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2}, \cos th \cdot \left(a2 \cdot a2\right), \sqrt{2} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\right) \cdot 0.5} \]
5. Step-by-step derivation

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. distribute-lft-out N/A

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

   4. +-commutative N/A

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

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

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

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

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

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

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

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

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

   9. +-commutative N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. *-lowering-*.f64 99.6

       \[\leadsto \left(\left(\sqrt{2} \cdot \cos th\right) \cdot \mathsf{fma}\left(a2, a2, \color{blue}{a1 \cdot a1}\right)\right) \cdot 0.5 \]

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \cos th\right) \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right)} \cdot 0.5 \]

7. Taylor expanded in a2 around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

   5. unpow2 N/A

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

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

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

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

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

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

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

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

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

   12. cos-lowering-cos.f64 53.5

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

9. Simplified 53.5%

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

10. Final simplification 53.5%

    \[\leadsto \left(a2 \cdot a2\right) \cdot \left(\sqrt{2} \cdot \left(\cos th \cdot 0.5\right)\right) \]
11. Add Preprocessing

Alternative 10: 52.9% accurate, 9.5× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, 0\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th) :precision binary64 (fma a2 (/ a2 (sqrt 2.0)) 0.0))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return fma(a2, (a2 / sqrt(2.0)), 0.0);
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return fma(a2, Float64(a2 / sqrt(2.0)), 0.0)
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 0.0), $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. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. flip3-+ N/A

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

   4. frac-times N/A

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

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

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

4. Applied egg-rr 14.3%

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

5. Taylor expanded in th around 0

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

7. Simplified 7.5%

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

8. Taylor expanded in a1 around 0

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

   1. *-rgt-identity N/A

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

   2. associate-*r/ N/A

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

   3. +-rgt-identity N/A

      \[\leadsto {a2}^{2} \cdot \color{blue}{\left(\frac{1}{\sqrt{2}} + 0\right)} \]

   4. distribute-rgt-in N/A

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

   5. associate-*l/ N/A

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

   6. *-lft-identity N/A

      \[\leadsto \frac{\color{blue}{{a2}^{2}}}{\sqrt{2}} + 0 \cdot {a2}^{2} \]

   7. mul0-lft N/A

      \[\leadsto \frac{{a2}^{2}}{\sqrt{2}} + \color{blue}{0} \]

   8. unpow2 N/A

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

   9. associate-/l* N/A

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

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, 0\right)} \]

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

       \[\leadsto \mathsf{fma}\left(a2, \color{blue}{\frac{a2}{\sqrt{2}}}, 0\right) \]

   12. sqrt-lowering-sqrt.f64 40.6

       \[\leadsto \mathsf{fma}\left(a2, \frac{a2}{\color{blue}{\sqrt{2}}}, 0\right) \]

10. Simplified 40.6%

    \[\leadsto \color{blue}{\mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, 0\right)} \]
11. Add Preprocessing

Alternative 11: 52.9% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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

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

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

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. sqrt-lowering-sqrt.f64 65.7

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

5. Simplified 65.7%

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

6. Taylor expanded in a1 around 0

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

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

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

   2. unpow2 N/A

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

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

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

   4. sqrt-lowering-sqrt.f64 40.6

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

8. Simplified 40.6%

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

Alternative 12: 53.0% accurate, 10.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2}, \cos th \cdot \left(a2 \cdot a2\right), \sqrt{2} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\right) \cdot 0.5} \]

5. Taylor expanded in a2 around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

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

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

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

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

   9. cos-lowering-cos.f64 53.5

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

7. Simplified 53.5%

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

8. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 40.6

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

10. Simplified 40.6%

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

11. Final simplification 40.6%

    \[\leadsto 0.5 \cdot \left(\sqrt{2} \cdot \left(a2 \cdot a2\right)\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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))))