Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 13.0s

Alternatives: 10

Speedup: 1.9×

• 44.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[(N[(N[Cos[th], $MachinePrecision] * a2$95$m), $MachinePrecision] * a2$95$m + N[(N[Cos[th], $MachinePrecision] * N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.7%

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

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

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

   4. distribute-lft-out N/A

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

   5. +-commutative N/A

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

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

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

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

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

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

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

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

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

   10. sqrt-lowering-sqrt.f64 99.7

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

6. Applied egg-rr 99.7%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

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

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

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

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

   8. *-commutative N/A

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

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

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

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

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

   11. *-lowering-*.f64 99.7

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

8. Applied egg-rr 99.7%

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

Alternative 2: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1 \cdot a1\right) \cdot t\_1 + \left(a2\_m \cdot a2\_m\right) \cdot t\_1 \leq -2 \cdot 10^{-114}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left(a2\_m \cdot \left(a2\_m \cdot \mathsf{fma}\left(th \cdot th, -0.5, 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{1}{\frac{\sqrt{2}}{a2\_m \cdot a2\_m}}\right)\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1 a1) t_1) (* (* a2_m a2_m) t_1)) -2e-114)
     (* 0.5 (* (sqrt 2.0) (* a2_m (* a2_m (fma (* th th) -0.5 1.0)))))
     (fma a1 (/ a1 (sqrt 2.0)) (/ 1.0 (/ (sqrt 2.0) (* a2_m a2_m)))))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_1) + ((a2_m * a2_m) * t_1)) <= -2e-114) {
		tmp = 0.5 * (sqrt(2.0) * (a2_m * (a2_m * fma((th * th), -0.5, 1.0))));
	} else {
		tmp = fma(a1, (a1 / sqrt(2.0)), (1.0 / (sqrt(2.0) / (a2_m * a2_m))));
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1 * a1) * t_1) + Float64(Float64(a2_m * a2_m) * t_1)) <= -2e-114)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * Float64(a2_m * Float64(a2_m * fma(Float64(th * th), -0.5, 1.0)))));
	else
		tmp = fma(a1, Float64(a1 / sqrt(2.0)), Float64(1.0 / Float64(sqrt(2.0) / Float64(a2_m * a2_m))));
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1 * a1), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -2e-114], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a2$95$m * N[(a2$95$m * N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a1 * N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -2.0000000000000001e-114

   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. Step-by-step derivation

      1. associate-*l/ N/A

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

      2. associate-*l/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

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

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

   4. Applied egg-rr 99.6%

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

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

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

      4. distribute-lft-out N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 57.0

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

   9. Simplified 57.0%

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

   10. Taylor expanded in a1 around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 50.6

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

   12. Simplified 50.6%

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

   if -2.0000000000000001e-114 < (+.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 99.5%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. sqrt-lowering-sqrt.f64 85.3

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

   5. Simplified 85.3%

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

      1. clear-num N/A

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

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

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

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

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

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

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

      5. *-lowering-*.f64 85.3

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

   7. Applied egg-rr 85.3%

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \color{blue}{\frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -2 \cdot 10^{-114}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left(a2 \cdot \left(a2 \cdot \mathsf{fma}\left(th \cdot th, -0.5, 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1 \cdot a1\right) \cdot t\_1 + \left(a2\_m \cdot a2\_m\right) \cdot t\_1 \leq -2 \cdot 10^{-114}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left(a2\_m \cdot \left(a2\_m \cdot \mathsf{fma}\left(th \cdot th, -0.5, 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1, a1, a2\_m \cdot a2\_m\right) \cdot \left(\sqrt{2} \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1 a1) t_1) (* (* a2_m a2_m) t_1)) -2e-114)
     (* 0.5 (* (sqrt 2.0) (* a2_m (* a2_m (fma (* th th) -0.5 1.0)))))
     (* (fma a1 a1 (* a2_m a2_m)) (* (sqrt 2.0) 0.5)))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_1) + ((a2_m * a2_m) * t_1)) <= -2e-114) {
		tmp = 0.5 * (sqrt(2.0) * (a2_m * (a2_m * fma((th * th), -0.5, 1.0))));
	} else {
		tmp = fma(a1, a1, (a2_m * a2_m)) * (sqrt(2.0) * 0.5);
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1 * a1) * t_1) + Float64(Float64(a2_m * a2_m) * t_1)) <= -2e-114)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * Float64(a2_m * Float64(a2_m * fma(Float64(th * th), -0.5, 1.0)))));
	else
		tmp = Float64(fma(a1, a1, Float64(a2_m * a2_m)) * Float64(sqrt(2.0) * 0.5));
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1 * a1), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -2e-114], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a2$95$m * N[(a2$95$m * N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a1 * a1 + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 0.5), $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))) < -2.0000000000000001e-114

   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. Step-by-step derivation

      1. associate-*l/ N/A

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

      2. associate-*l/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

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

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

   4. Applied egg-rr 99.6%

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

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

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

      4. distribute-lft-out N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 57.0

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

   9. Simplified 57.0%

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

   10. Taylor expanded in a1 around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 50.6

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

   12. Simplified 50.6%

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

   if -2.0000000000000001e-114 < (+.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 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-*l/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

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

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

   4. Applied egg-rr 99.7%

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

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

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

      4. distribute-lft-out N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 99.7

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

   6. Applied egg-rr 99.7%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. *-lowering-*.f64 99.7

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in th around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

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

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

       4. unpow2 N/A

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

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

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

       6. unpow2 N/A

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

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

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

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

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

       9. sqrt-lowering-sqrt.f64 85.3

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

   11. Simplified 85.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \left(0.5 \cdot \sqrt{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -2 \cdot 10^{-114}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left(a2 \cdot \left(a2 \cdot \mathsf{fma}\left(th \cdot th, -0.5, 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \left(\sqrt{2} \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * N[(a1 * a1 + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.7%

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

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

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

   4. distribute-lft-out N/A

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

   5. +-commutative N/A

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

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

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

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

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

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

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

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

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

   10. sqrt-lowering-sqrt.f64 99.7

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 77.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a2$95$m * N[(N[Cos[th], $MachinePrecision] * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.7%

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

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

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

   4. distribute-lft-out N/A

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

   5. +-commutative N/A

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

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

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

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

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

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

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

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

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

   10. sqrt-lowering-sqrt.f64 99.7

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

6. Applied egg-rr 99.7%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

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

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

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

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

   8. *-commutative N/A

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

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

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

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

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

   11. *-lowering-*.f64 99.7

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

8. Applied egg-rr 99.7%

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

9. Taylor expanded in a2 around inf

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

    1. unpow2 N/A

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

    2. associate-*l* N/A

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

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

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

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

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

    5. cos-lowering-cos.f64 56.2

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

11. Simplified 56.2%

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

12. Final simplification 56.2%

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

Alternative 6: 77.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[Cos[th], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.7%

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

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

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

   4. distribute-lft-out N/A

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

   5. +-commutative N/A

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

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

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

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

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

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

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

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

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

   10. sqrt-lowering-sqrt.f64 99.7

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in a1 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

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

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 56.2

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

9. Simplified 56.2%

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

10. Final simplification 56.2%

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

Alternative 7: 66.9% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(a1 * a1 + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.7%

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

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

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

   4. distribute-lft-out N/A

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

   5. +-commutative N/A

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

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

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

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

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

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

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

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

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

   10. sqrt-lowering-sqrt.f64 99.7

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

6. Applied egg-rr 99.7%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

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

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

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

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

   8. *-commutative N/A

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

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

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

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

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

   11. *-lowering-*.f64 99.7

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

8. Applied egg-rr 99.7%

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

9. Taylor expanded in th around 0

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

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

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

    4. unpow2 N/A

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

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

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

    6. unpow2 N/A

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

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

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

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

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

    9. sqrt-lowering-sqrt.f64 69.4

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

11. Simplified 69.4%

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

12. Final simplification 69.4%

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

Alternative 8: 66.9% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a1 * a1 + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in th around 0

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

   1. distribute-rgt-out N/A

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. unpow2 N/A

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

   7. *-lowering-*.f64 69.4

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

7. Simplified 69.4%

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

8. Final simplification 69.4%

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

Alternative 9: 53.1% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. sqrt-lowering-sqrt.f64 69.4

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

5. Simplified 69.4%

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

6. Taylor expanded in a1 around 0

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

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

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

   2. unpow2 N/A

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

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

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

   4. sqrt-lowering-sqrt.f64 39.8

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

8. Simplified 39.8%

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

Alternative 10: 27.2% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. sqrt-lowering-sqrt.f64 69.4

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

5. Simplified 69.4%

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

6. Taylor expanded in a1 around inf

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

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

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

   2. unpow2 N/A

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

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

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

   4. sqrt-lowering-sqrt.f64 43.6

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

8. Simplified 43.6%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. sqrt-lowering-sqrt.f64 43.6

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

10. Applied egg-rr 43.6%

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

11. Final simplification 43.6%

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

Reproduce

herbie shell --seed 2024196 
(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))))