Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 9.7s

Alternatives: 10

Speedup: 1.0×

• 43.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \frac{\cos th}{\sqrt{2}} \cdot \left(a1\_m \cdot a1\_m\right) + \left(\cos th \cdot \left(\sqrt{2} \cdot 0.5\right)\right) \cdot \left(a2\_m \cdot a2\_m\right) \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (+
  (* (/ (cos th) (sqrt 2.0)) (* a1_m a1_m))
  (* (* (cos th) (* (sqrt 2.0) 0.5)) (* a2_m a2_m))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return ((cos(th) / sqrt(2.0)) * (a1_m * a1_m)) + ((cos(th) * (sqrt(2.0) * 0.5)) * (a2_m * a2_m));
}

Fortran

a2_m = abs(a2)
a1_m = abs(a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = ((cos(th) / sqrt(2.0d0)) * (a1_m * a1_m)) + ((cos(th) * (sqrt(2.0d0) * 0.5d0)) * (a2_m * a2_m))
end function

Java

a2_m = Math.abs(a2);
a1_m = Math.abs(a1);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return ((Math.cos(th) / Math.sqrt(2.0)) * (a1_m * a1_m)) + ((Math.cos(th) * (Math.sqrt(2.0) * 0.5)) * (a2_m * a2_m));
}

Python

a2_m = math.fabs(a2)
a1_m = math.fabs(a1)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return ((math.cos(th) / math.sqrt(2.0)) * (a1_m * a1_m)) + ((math.cos(th) * (math.sqrt(2.0) * 0.5)) * (a2_m * a2_m))

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(Float64(Float64(cos(th) / sqrt(2.0)) * Float64(a1_m * a1_m)) + Float64(Float64(cos(th) * Float64(sqrt(2.0) * 0.5)) * Float64(a2_m * a2_m)))
end

MATLAB

a2_m = abs(a2);
a1_m = abs(a1);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = ((cos(th) / sqrt(2.0)) * (a1_m * a1_m)) + ((cos(th) * (sqrt(2.0) * 0.5)) * (a2_m * a2_m));
end

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. div-inv N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. neg-sub0 N/A

      \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(\frac{-1}{\color{blue}{0 - \sqrt{2}}} \cdot \cos th\right) \cdot \left(a2 \cdot a2\right) \]

   9. flip-- N/A

      \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(\frac{-1}{\color{blue}{\frac{0 \cdot 0 - \sqrt{2} \cdot \sqrt{2}}{0 + \sqrt{2}}}} \cdot \cos th\right) \cdot \left(a2 \cdot a2\right) \]

   10. +-lft-identity N/A

       \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(\frac{-1}{\frac{0 \cdot 0 - \sqrt{2} \cdot \sqrt{2}}{\color{blue}{\sqrt{2}}}} \cdot \cos th\right) \cdot \left(a2 \cdot a2\right) \]

   11. associate-/r/ N/A

       \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(\color{blue}{\left(\frac{-1}{0 \cdot 0 - \sqrt{2} \cdot \sqrt{2}} \cdot \sqrt{2}\right)} \cdot \cos th\right) \cdot \left(a2 \cdot a2\right) \]

   12. metadata-eval N/A

       \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(\left(\frac{-1}{\color{blue}{0} - \sqrt{2} \cdot \sqrt{2}} \cdot \sqrt{2}\right) \cdot \cos th\right) \cdot \left(a2 \cdot a2\right) \]

   13. lift-sqrt.f64 N/A

       \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(\left(\frac{-1}{0 - \color{blue}{\sqrt{2}} \cdot \sqrt{2}} \cdot \sqrt{2}\right) \cdot \cos th\right) \cdot \left(a2 \cdot a2\right) \]

   14. lift-sqrt.f64 N/A

       \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(\left(\frac{-1}{0 - \sqrt{2} \cdot \color{blue}{\sqrt{2}}} \cdot \sqrt{2}\right) \cdot \cos th\right) \cdot \left(a2 \cdot a2\right) \]

   15. rem-square-sqrt N/A

       \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(\left(\frac{-1}{0 - \color{blue}{2}} \cdot \sqrt{2}\right) \cdot \cos th\right) \cdot \left(a2 \cdot a2\right) \]

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 99.7

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t_2 := \mathsf{fma}\left(a2\_m, a2\_m, a1\_m \cdot a1\_m\right)\\ \mathbf{if}\;t\_1 \cdot \left(a1\_m \cdot a1\_m\right) + t\_1 \cdot \left(a2\_m \cdot a2\_m\right) \leq -2 \cdot 10^{-136}:\\ \;\;\;\;\frac{-0.5 \cdot \left(th \cdot \left(th \cdot t\_2\right)\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))) (t_2 (fma a2_m a2_m (* a1_m a1_m))))
   (if (<= (+ (* t_1 (* a1_m a1_m)) (* t_1 (* a2_m a2_m))) -2e-136)
     (/ (* -0.5 (* th (* th t_2))) (sqrt 2.0))
     (/ t_2 (sqrt 2.0)))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double t_2 = fma(a2_m, a2_m, (a1_m * a1_m));
	double tmp;
	if (((t_1 * (a1_m * a1_m)) + (t_1 * (a2_m * a2_m))) <= -2e-136) {
		tmp = (-0.5 * (th * (th * t_2))) / sqrt(2.0);
	} else {
		tmp = t_2 / sqrt(2.0);
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	t_2 = fma(a2_m, a2_m, Float64(a1_m * a1_m))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1_m * a1_m)) + Float64(t_1 * Float64(a2_m * a2_m))) <= -2e-136)
		tmp = Float64(Float64(-0.5 * Float64(th * Float64(th * t_2))) / sqrt(2.0));
	else
		tmp = Float64(t_2 / sqrt(2.0));
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a2$95$m * a2$95$m + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-136], N[(N[(-0.5 * N[(th * N[(th * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -2e-136

   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

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l/ N/A

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

      14. lower-/.f64 N/A

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

   5. Simplified 56.3%

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

   6. Taylor expanded in th around 0

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

      1. distribute-lft-out N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. *-rgt-identity N/A

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

   8. Simplified 54.2%

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

   9. Taylor expanded in th around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. unpow2 N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-sqrt.f64 54.2

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

   11. Simplified 54.2%

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

   if -2e-136 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.7%

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

   3. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l/ N/A

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

      14. lower-/.f64 N/A

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

   5. Simplified 68.7%

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

   6. Taylor expanded in th around 0

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

      1. distribute-lft-out N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. *-rgt-identity N/A

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

   8. Simplified 62.1%

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

   9. Taylor expanded in th around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. unpow2 N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-sqrt.f64 85.2

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

   11. Simplified 85.2%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \leq -2 \cdot 10^{-136}:\\ \;\;\;\;\frac{-0.5 \cdot \left(th \cdot \left(th \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right)\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1\_m \cdot a1\_m\right) + t\_1 \cdot \left(a2\_m \cdot a2\_m\right) \leq -5 \cdot 10^{-118}:\\ \;\;\;\;\frac{a2\_m \cdot \mathsf{fma}\left(a2\_m, -0.5 \cdot \left(th \cdot th\right), a2\_m\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2\_m, a2\_m, a1\_m \cdot a1\_m\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1_m a1_m)) (* t_1 (* a2_m a2_m))) -5e-118)
     (/ (* a2_m (fma a2_m (* -0.5 (* th th)) a2_m)) (sqrt 2.0))
     (/ (fma a2_m a2_m (* a1_m a1_m)) (sqrt 2.0)))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1_m * a1_m)) + (t_1 * (a2_m * a2_m))) <= -5e-118) {
		tmp = (a2_m * fma(a2_m, (-0.5 * (th * th)), a2_m)) / sqrt(2.0);
	} else {
		tmp = fma(a2_m, a2_m, (a1_m * a1_m)) / sqrt(2.0);
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1_m * a1_m)) + Float64(t_1 * Float64(a2_m * a2_m))) <= -5e-118)
		tmp = Float64(Float64(a2_m * fma(a2_m, Float64(-0.5 * Float64(th * th)), a2_m)) / sqrt(2.0));
	else
		tmp = Float64(fma(a2_m, a2_m, Float64(a1_m * a1_m)) / sqrt(2.0));
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-118], N[(N[(a2$95$m * N[(a2$95$m * N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision] + a2$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[(a2$95$m * a2$95$m + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -5.00000000000000015e-118

   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

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l/ N/A

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

      14. lower-/.f64 N/A

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

   5. Simplified 57.4%

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

   6. Taylor expanded in a1 around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 46.4

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

   8. Simplified 46.4%

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

   9. Taylor expanded in th around 0

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*r/ N/A

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

       5. distribute-lft1-in N/A

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

       6. +-commutative N/A

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

       7. associate-/l* N/A

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

       8. *-commutative N/A

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

       9. +-commutative N/A

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

       10. distribute-lft-in N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

       13. *-rgt-identity N/A

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

       14. lower-/.f64 N/A

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

   11. Simplified 43.3%

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

   if -5.00000000000000015e-118 < (+.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. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l/ N/A

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

      14. lower-/.f64 N/A

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

   5. Simplified 68.4%

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

   6. Taylor expanded in th around 0

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

      1. distribute-lft-out N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. *-rgt-identity N/A

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

   8. Simplified 61.8%

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

   9. Taylor expanded in th around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. unpow2 N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-sqrt.f64 84.8

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

   11. Simplified 84.8%

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

Alternative 4: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1\_m \cdot a1\_m\right) + t\_1 \cdot \left(a2\_m \cdot a2\_m\right) \leq -5 \cdot 10^{-118}:\\ \;\;\;\;\frac{a1\_m \cdot \left(-0.5 \cdot \left(a1\_m \cdot \left(th \cdot th\right)\right)\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2\_m, a2\_m, a1\_m \cdot a1\_m\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1_m a1_m)) (* t_1 (* a2_m a2_m))) -5e-118)
     (/ (* a1_m (* -0.5 (* a1_m (* th th)))) (sqrt 2.0))
     (/ (fma a2_m a2_m (* a1_m a1_m)) (sqrt 2.0)))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1_m * a1_m)) + (t_1 * (a2_m * a2_m))) <= -5e-118) {
		tmp = (a1_m * (-0.5 * (a1_m * (th * th)))) / sqrt(2.0);
	} else {
		tmp = fma(a2_m, a2_m, (a1_m * a1_m)) / sqrt(2.0);
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1_m * a1_m)) + Float64(t_1 * Float64(a2_m * a2_m))) <= -5e-118)
		tmp = Float64(Float64(a1_m * Float64(-0.5 * Float64(a1_m * Float64(th * th)))) / sqrt(2.0));
	else
		tmp = Float64(fma(a2_m, a2_m, Float64(a1_m * a1_m)) / sqrt(2.0));
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-118], N[(N[(a1$95$m * N[(-0.5 * N[(a1$95$m * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[(a2$95$m * a2$95$m + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -5.00000000000000015e-118

   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

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l/ N/A

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

      14. lower-/.f64 N/A

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

   5. Simplified 57.4%

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

   6. Taylor expanded in th around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{a1}^{2} \cdot {th}^{2}}{\sqrt{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 49.2

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

   8. Simplified 49.2%

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

   if -5.00000000000000015e-118 < (+.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. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l/ N/A

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

      14. lower-/.f64 N/A

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

   5. Simplified 68.4%

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

   6. Taylor expanded in th around 0

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

      1. distribute-lft-out N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. *-rgt-identity N/A

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

   8. Simplified 61.8%

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

   9. Taylor expanded in th around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. unpow2 N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-sqrt.f64 84.8

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

   11. Simplified 84.8%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \leq -5 \cdot 10^{-118}:\\ \;\;\;\;\frac{a1 \cdot \left(-0.5 \cdot \left(a1 \cdot \left(th \cdot th\right)\right)\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1\_m \cdot a1\_m\right) + t\_1 \cdot \left(a2\_m \cdot a2\_m\right) \leq -2 \cdot 10^{-136}:\\ \;\;\;\;a1\_m \cdot \frac{th \cdot \left(-0.5 \cdot \left(th \cdot a1\_m\right)\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2\_m, a2\_m, a1\_m \cdot a1\_m\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1_m a1_m)) (* t_1 (* a2_m a2_m))) -2e-136)
     (* a1_m (/ (* th (* -0.5 (* th a1_m))) (sqrt 2.0)))
     (/ (fma a2_m a2_m (* a1_m a1_m)) (sqrt 2.0)))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1_m * a1_m)) + (t_1 * (a2_m * a2_m))) <= -2e-136) {
		tmp = a1_m * ((th * (-0.5 * (th * a1_m))) / sqrt(2.0));
	} else {
		tmp = fma(a2_m, a2_m, (a1_m * a1_m)) / sqrt(2.0);
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1_m * a1_m)) + Float64(t_1 * Float64(a2_m * a2_m))) <= -2e-136)
		tmp = Float64(a1_m * Float64(Float64(th * Float64(-0.5 * Float64(th * a1_m))) / sqrt(2.0)));
	else
		tmp = Float64(fma(a2_m, a2_m, Float64(a1_m * a1_m)) / sqrt(2.0));
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-136], N[(a1$95$m * N[(N[(th * N[(-0.5 * N[(th * a1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a2$95$m * a2$95$m + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -2e-136

   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

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l/ N/A

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

      14. lower-/.f64 N/A

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

   5. Simplified 56.3%

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

   6. Taylor expanded in th around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{a1}^{2} \cdot {th}^{2}}{\sqrt{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 48.2

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

   8. Simplified 48.2%

      \[\leadsto \color{blue}{\frac{a1 \cdot \left(-0.5 \cdot \left(\left(th \cdot th\right) \cdot a1\right)\right)}{\sqrt{2}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 48.2

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 42.3

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

   10. Applied egg-rr 42.3%

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

   if -2e-136 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.7%

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

   3. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l/ N/A

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

      14. lower-/.f64 N/A

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

   5. Simplified 68.7%

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

   6. Taylor expanded in th around 0

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

      1. distribute-lft-out N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. *-rgt-identity N/A

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

   8. Simplified 62.1%

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

   9. Taylor expanded in th around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. unpow2 N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-sqrt.f64 85.2

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

   11. Simplified 85.2%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \leq -2 \cdot 10^{-136}:\\ \;\;\;\;a1 \cdot \frac{th \cdot \left(-0.5 \cdot \left(th \cdot a1\right)\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \mathsf{fma}\left(\frac{a1\_m}{\sqrt{2}}, a1\_m, \left(a2\_m \cdot \left(\cos th \cdot a2\_m\right)\right) \cdot \left(\sqrt{2} \cdot 0.5\right)\right) \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (fma
  (/ a1_m (sqrt 2.0))
  a1_m
  (* (* a2_m (* (cos th) a2_m)) (* (sqrt 2.0) 0.5))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return fma((a1_m / sqrt(2.0)), a1_m, ((a2_m * (cos(th) * a2_m)) * (sqrt(2.0) * 0.5)));
}

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return fma(Float64(a1_m / sqrt(2.0)), a1_m, Float64(Float64(a2_m * Float64(cos(th) * a2_m)) * Float64(sqrt(2.0) * 0.5)))
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(a1$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a1$95$m + N[(N[(a2$95$m * N[(N[Cos[th], $MachinePrecision] * a2$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $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. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l/ N/A

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

   5. div-inv N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. neg-sub0 N/A

       \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot \left(\cos th \cdot a2\right)\right) \cdot \frac{-1}{\color{blue}{0 - \sqrt{2}}} \]

   15. flip-- N/A

       \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot \left(\cos th \cdot a2\right)\right) \cdot \frac{-1}{\color{blue}{\frac{0 \cdot 0 - \sqrt{2} \cdot \sqrt{2}}{0 + \sqrt{2}}}} \]

   16. +-lft-identity N/A

       \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot \left(\cos th \cdot a2\right)\right) \cdot \frac{-1}{\frac{0 \cdot 0 - \sqrt{2} \cdot \sqrt{2}}{\color{blue}{\sqrt{2}}}} \]

   17. associate-/r/ N/A

       \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot \left(\cos th \cdot a2\right)\right) \cdot \color{blue}{\left(\frac{-1}{0 \cdot 0 - \sqrt{2} \cdot \sqrt{2}} \cdot \sqrt{2}\right)} \]

4. Applied egg-rr 99.7%

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

5. Taylor expanded in th around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 83.1

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

7. Simplified 83.1%

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

   1. lift-sqrt.f64 N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 83.9

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 83.9

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

9. Applied egg-rr 83.9%

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

Alternative 7: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \left(a2\_m \cdot a2\_m\right) \cdot \left(\sqrt{2} \cdot \left(\cos th \cdot 0.5\right)\right) \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (* (* a2_m a2_m) (* (sqrt 2.0) (* (cos th) 0.5))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return (a2_m * a2_m) * (sqrt(2.0) * (cos(th) * 0.5));
}

Fortran

a2_m = abs(a2)
a1_m = abs(a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = (a2_m * a2_m) * (sqrt(2.0d0) * (cos(th) * 0.5d0))
end function

Java

a2_m = Math.abs(a2);
a1_m = Math.abs(a1);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return (a2_m * a2_m) * (Math.sqrt(2.0) * (Math.cos(th) * 0.5));
}

Python

a2_m = math.fabs(a2)
a1_m = math.fabs(a1)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return (a2_m * a2_m) * (math.sqrt(2.0) * (math.cos(th) * 0.5))

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(Float64(a2_m * a2_m) * Float64(sqrt(2.0) * Float64(cos(th) * 0.5)))
end

MATLAB

a2_m = abs(a2);
a1_m = abs(a1);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = (a2_m * a2_m) * (sqrt(2.0) * (cos(th) * 0.5));
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * N[(N[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. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l/ N/A

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

   5. div-inv N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. neg-sub0 N/A

       \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot \left(\cos th \cdot a2\right)\right) \cdot \frac{-1}{\color{blue}{0 - \sqrt{2}}} \]

   15. flip-- N/A

       \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot \left(\cos th \cdot a2\right)\right) \cdot \frac{-1}{\color{blue}{\frac{0 \cdot 0 - \sqrt{2} \cdot \sqrt{2}}{0 + \sqrt{2}}}} \]

   16. +-lft-identity N/A

       \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot \left(\cos th \cdot a2\right)\right) \cdot \frac{-1}{\frac{0 \cdot 0 - \sqrt{2} \cdot \sqrt{2}}{\color{blue}{\sqrt{2}}}} \]

   17. associate-/r/ N/A

       \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot \left(\cos th \cdot a2\right)\right) \cdot \color{blue}{\left(\frac{-1}{0 \cdot 0 - \sqrt{2} \cdot \sqrt{2}} \cdot \sqrt{2}\right)} \]

4. Applied egg-rr 99.7%

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

5. Taylor expanded in th around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 83.1

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

7. Simplified 83.1%

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

8. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a2}^{2} \cdot \left(\cos th \cdot \sqrt{2}\right)\right)} \]
9. 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. lower-*.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. lower-*.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. lower-*.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. lower-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. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 53.7

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

10. Simplified 53.7%

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

Alternative 8: 66.0% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. associate-*l/ N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-/l* N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-/l* N/A

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

   11. associate-*r* N/A

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

   12. distribute-rgt-out N/A

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

   13. associate-*l/ N/A

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

   14. lower-/.f64 N/A

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

5. Simplified 66.4%

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

6. Taylor expanded in th around 0

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

   1. distribute-lft-out N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. +-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. *-rgt-identity N/A

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

   12. associate-*r/ N/A

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

   13. *-rgt-identity N/A

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

8. Simplified 60.6%

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

9. Taylor expanded in th around 0

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

    1. lower-/.f64 N/A

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

    2. +-commutative N/A

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

    3. unpow2 N/A

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

    4. lower-fma.f64 N/A

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

    5. unpow2 N/A

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

    6. lower-*.f64 N/A

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

    7. lower-sqrt.f64 69.3

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

11. Simplified 69.3%

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

Alternative 9: 65.7% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 85.1

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

5. Simplified 85.1%

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

6. Taylor expanded in a1 around 0

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

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

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

   4. lower-sqrt.f64 38.9

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

8. Simplified 38.9%

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

Alternative 10: 65.7% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in th around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 85.1

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

5. Simplified 85.1%

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

6. Taylor expanded in a1 around 0

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

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

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

   4. lower-sqrt.f64 38.9

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

8. Simplified 38.9%

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

   1. lift-sqrt.f64 N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 38.9

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

10. Applied egg-rr 38.9%

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

11. Final simplification 38.9%

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

Reproduce

herbie shell --seed 2024219 
(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))))