Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 9.9s

Alternatives: 9

Speedup: 1.9×

• 43.9% 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: 99.5% 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

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

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
  (* (cos th) (/ a2_m (sqrt 2.0)))
  a2_m
  (* (cos th) (* 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((cos(th) * (a2_m / sqrt(2.0))), a2_m, (cos(th) * (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 fma(Float64(cos(th) * Float64(a2_m / sqrt(2.0))), a2_m, Float64(cos(th) * Float64(a1_m * Float64(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[(N[Cos[th], $MachinePrecision] * N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a2$95$m + N[(N[Cos[th], $MachinePrecision] * N[(a1$95$m * N[(a1$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. associate-*l/ N/A

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

   12. *-lft-identity N/A

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

   13. lower-/.f64 99.7

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

   14. lift-*.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. associate-*l/ N/A

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

   17. associate-/l* N/A

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

   18. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

Alternative 2: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1\_m \cdot a1\_m\right) \cdot t\_1 + \left(a2\_m \cdot a2\_m\right) \cdot t\_1 \leq -2 \cdot 10^{-186}:\\ \;\;\;\;\left(\sqrt{2} \cdot \mathsf{fma}\left(a2\_m, a2\_m, a1\_m \cdot a1\_m\right)\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a2\_m, \frac{a2\_m}{\sqrt{2}}, \frac{a1\_m \cdot a1\_m}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

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 (<= (+ (* (* a1_m a1_m) t_1) (* (* a2_m a2_m) t_1)) -2e-186)
     (* (* (sqrt 2.0) (fma a2_m a2_m (* a1_m a1_m))) (fma -0.25 (* th th) 0.5))
     (fma a2_m (/ a2_m (sqrt 2.0)) (/ (* 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 ((((a1_m * a1_m) * t_1) + ((a2_m * a2_m) * t_1)) <= -2e-186) {
		tmp = (sqrt(2.0) * fma(a2_m, a2_m, (a1_m * a1_m))) * fma(-0.25, (th * th), 0.5);
	} else {
		tmp = fma(a2_m, (a2_m / sqrt(2.0)), ((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(Float64(a1_m * a1_m) * t_1) + Float64(Float64(a2_m * a2_m) * t_1)) <= -2e-186)
		tmp = Float64(Float64(sqrt(2.0) * fma(a2_m, a2_m, Float64(a1_m * a1_m))) * fma(-0.25, Float64(th * th), 0.5));
	else
		tmp = fma(a2_m, Float64(a2_m / sqrt(2.0)), Float64(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[(N[(a1$95$m * a1$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -2e-186], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(a2$95$m * a2$95$m + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(a2$95$m * N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a1$95$m * a1$95$m), $MachinePrecision] / N[Sqrt[2.0], $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))) < -1.9999999999999998e-186

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 99.5

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-lft-out N/A

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

      12. lift-fma.f64 N/A

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

      13. lower-*.f64 99.5

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 58.6

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

   9. Applied rewrites 58.6%

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

   if -1.9999999999999998e-186 < (+.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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 82.2

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

   5. Applied rewrites 82.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -2 \cdot 10^{-186}:\\ \;\;\;\;\left(\sqrt{2} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, \frac{a1 \cdot a1}{\sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a2\_m, a2\_m, a1\_m \cdot a1\_m\right)\\ t_2 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1\_m \cdot a1\_m\right) \cdot t\_2 + \left(a2\_m \cdot a2\_m\right) \cdot t\_2 \leq -2 \cdot 10^{-186}:\\ \;\;\;\;\left(\sqrt{2} \cdot t\_1\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.5 \cdot t\_1\right)\\ \end{array} \end{array} \]

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 (fma a2_m a2_m (* a1_m a1_m))) (t_2 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1_m a1_m) t_2) (* (* a2_m a2_m) t_2)) -2e-186)
     (* (* (sqrt 2.0) t_1) (fma -0.25 (* th th) 0.5))
     (* (sqrt 2.0) (* 0.5 t_1)))))

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 = fma(a2_m, a2_m, (a1_m * a1_m));
	double t_2 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1_m * a1_m) * t_2) + ((a2_m * a2_m) * t_2)) <= -2e-186) {
		tmp = (sqrt(2.0) * t_1) * fma(-0.25, (th * th), 0.5);
	} else {
		tmp = sqrt(2.0) * (0.5 * t_1);
	}
	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 = fma(a2_m, a2_m, Float64(a1_m * a1_m))
	t_2 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1_m * a1_m) * t_2) + Float64(Float64(a2_m * a2_m) * t_2)) <= -2e-186)
		tmp = Float64(Float64(sqrt(2.0) * t_1) * fma(-0.25, Float64(th * th), 0.5));
	else
		tmp = Float64(sqrt(2.0) * Float64(0.5 * t_1));
	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[(a2$95$m * a2$95$m + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], -2e-186], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.5 * t$95$1), $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))) < -1.9999999999999998e-186

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 99.5

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-lft-out N/A

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

      12. lift-fma.f64 N/A

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

      13. lower-*.f64 99.5

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 58.6

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

   9. Applied rewrites 58.6%

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

   if -1.9999999999999998e-186 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 99.7

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-lft-out N/A

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

      12. lift-fma.f64 N/A

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

      13. lower-*.f64 99.7

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in th around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 82.2

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

   9. Applied rewrites 82.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -2 \cdot 10^{-186}:\\ \;\;\;\;\left(\sqrt{2} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.5 \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.9× speedup

Math

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

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
 (* (sqrt 2.0) (* (* (cos th) 0.5) (fma a2_m a2_m (* a1_m a1_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 sqrt(2.0) * ((cos(th) * 0.5) * fma(a2_m, a2_m, (a1_m * a1_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(sqrt(2.0) * Float64(Float64(cos(th) * 0.5) * fma(a2_m, a2_m, Float64(a1_m * a1_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[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[th], $MachinePrecision] * 0.5), $MachinePrecision] * N[(a2$95$m * a2$95$m + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. associate-*l/ N/A

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

   12. *-lft-identity N/A

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

   13. lower-/.f64 99.7

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

   14. lift-*.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. associate-*l/ N/A

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

   17. associate-/l* N/A

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

   18. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*r/ N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. distribute-lft-out N/A

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

   12. lift-fma.f64 N/A

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

   13. lower-*.f64 99.6

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

   14. lift-fma.f64 N/A

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

   15. +-commutative N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in a1 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-*r* N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. distribute-rgt-out N/A

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

9. Applied rewrites 99.6%

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

10. Final simplification 99.6%

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

Alternative 5: 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-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. associate-*l/ N/A

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

   12. *-lft-identity N/A

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

   13. lower-/.f64 99.7

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

   14. lift-*.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. associate-*l/ N/A

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

   17. associate-/l* N/A

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

   18. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*r/ N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. distribute-lft-out N/A

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

   12. lift-fma.f64 N/A

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

   13. lower-*.f64 99.6

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

   14. lift-fma.f64 N/A

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

   15. +-commutative N/A

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

6. Applied rewrites 99.6%

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

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

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

   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. lower-*.f64 N/A

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

   12. lower-cos.f64 53.4

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

9. Applied rewrites 53.4%

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

10. Final simplification 53.4%

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

Alternative 6: 66.8% accurate, 8.3× speedup

Math

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

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
 (* (sqrt 2.0) (* 0.5 (fma a2_m a2_m (* a1_m a1_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 sqrt(2.0) * (0.5 * fma(a2_m, a2_m, (a1_m * a1_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(sqrt(2.0) * Float64(0.5 * fma(a2_m, a2_m, Float64(a1_m * a1_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[Sqrt[2.0], $MachinePrecision] * N[(0.5 * N[(a2$95$m * a2$95$m + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. associate-*l/ N/A

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

   12. *-lft-identity N/A

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

   13. lower-/.f64 99.7

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

   14. lift-*.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. associate-*l/ N/A

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

   17. associate-/l* N/A

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

   18. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*r/ N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. distribute-lft-out N/A

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

   12. lift-fma.f64 N/A

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

   13. lower-*.f64 99.6

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

   14. lift-fma.f64 N/A

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

   15. +-commutative N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in th around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. distribute-lft-out N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. unpow2 N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 63.3

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

9. Applied rewrites 63.3%

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

Alternative 7: 66.5% accurate, 9.5× speedup

Math

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

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)))) -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))) * -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))) * (-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))) * -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))) * -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(-Float64(a2_m * Float64(a2_m * sqrt(2.0)))) * -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))) * -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 * N[(a2$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) * -0.5), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. 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. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 63.4

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

5. Applied rewrites 63.4%

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

7. Applied rewrites 63.2%

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

8. Taylor expanded in a1 around 0

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

10. Applied rewrites 38.6%

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

12. Applied rewrites 38.7%

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

13. Final simplification 38.7%

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

Alternative 8: 66.5% 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

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 63.4

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

5. Applied rewrites 63.4%

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

6. Taylor expanded in a1 around 0

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

8. Applied rewrites 38.7%

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

Alternative 9: 13.2% accurate, 9.9× speedup

Math

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

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 (* 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 a1_m * (a1_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 = a1_m * (a1_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 a1_m * (a1_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 a1_m * (a1_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(a1_m * Float64(a1_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 = 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[(a1$95$m * N[(a1$95$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

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 63.4

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

5. Applied rewrites 63.4%

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

6. Taylor expanded in a1 around inf

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

8. Applied rewrites 36.5%

   \[\leadsto a1 \cdot \color{blue}{\frac{a1}{\sqrt{2}}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(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))))