Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 9.6s

Alternatives: 12

Speedup: 1.7×

• 43.1% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. +-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. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. associate-*l/ N/A

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

   13. *-lft-identity N/A

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

   14. lower-/.f64 99.7

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

   15. lift-*.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. associate-*l/ N/A

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

   18. associate-/l* N/A

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

   19. *-commutative N/A

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a2\_m \cdot a2\_m\right) + \left(a1 \cdot a1\right) \cdot t\_1 \leq -5 \cdot 10^{-105}:\\ \;\;\;\;\left(\mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot \sqrt{0.5}\right) \cdot \left(a2\_m \cdot a2\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{a2\_m}{\sqrt{2}} \cdot a2\_m\right)\\ \end{array} \end{array} \]

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 (<= (+ (* t_1 (* a2_m a2_m)) (* (* a1 a1) t_1)) -5e-105)
     (* (* (fma (* th th) -0.5 1.0) (sqrt 0.5)) (* a2_m a2_m))
     (fma (/ a1 (sqrt 2.0)) a1 (* (/ a2_m (sqrt 2.0)) a2_m)))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a2_m * a2_m)) + ((a1 * a1) * t_1)) <= -5e-105) {
		tmp = (fma((th * th), -0.5, 1.0) * sqrt(0.5)) * (a2_m * a2_m);
	} else {
		tmp = fma((a1 / sqrt(2.0)), a1, ((a2_m / sqrt(2.0)) * a2_m));
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a2_m * a2_m)) + Float64(Float64(a1 * a1) * t_1)) <= -5e-105)
		tmp = Float64(Float64(fma(Float64(th * th), -0.5, 1.0) * sqrt(0.5)) * Float64(a2_m * a2_m));
	else
		tmp = fma(Float64(a1 / sqrt(2.0)), a1, Float64(Float64(a2_m / sqrt(2.0)) * a2_m));
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(a1 * a1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -5e-105], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a1 + N[(N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a2$95$m), $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))) < -4.99999999999999963e-105

   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

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. pow-flip N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.7

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 60.3

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

   7. Applied rewrites 60.3%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 49.7%

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

   if -4.99999999999999963e-105 < (+.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. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 85.9

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

   5. Applied rewrites 85.9%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) + \left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -5 \cdot 10^{-105}:\\ \;\;\;\;\left(\mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot \sqrt{0.5}\right) \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a1}{\sqrt{2}}, a1, \frac{a2}{\sqrt{2}} \cdot a2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a2\_m \cdot a2\_m\right) + \left(a1 \cdot a1\right) \cdot t\_1 \leq -5 \cdot 10^{-105}:\\ \;\;\;\;\left(\mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot \sqrt{0.5}\right) \cdot \left(a2\_m \cdot a2\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{0.5} \cdot a2\_m, a2\_m, \frac{a1}{\sqrt{2}} \cdot a1\right)\\ \end{array} \end{array} \]

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 (<= (+ (* t_1 (* a2_m a2_m)) (* (* a1 a1) t_1)) -5e-105)
     (* (* (fma (* th th) -0.5 1.0) (sqrt 0.5)) (* a2_m a2_m))
     (fma (* (sqrt 0.5) a2_m) a2_m (* (/ a1 (sqrt 2.0)) a1)))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a2_m * a2_m)) + ((a1 * a1) * t_1)) <= -5e-105) {
		tmp = (fma((th * th), -0.5, 1.0) * sqrt(0.5)) * (a2_m * a2_m);
	} else {
		tmp = fma((sqrt(0.5) * a2_m), a2_m, ((a1 / sqrt(2.0)) * a1));
	}
	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(t_1 * Float64(a2_m * a2_m)) + Float64(Float64(a1 * a1) * t_1)) <= -5e-105)
		tmp = Float64(Float64(fma(Float64(th * th), -0.5, 1.0) * sqrt(0.5)) * Float64(a2_m * a2_m));
	else
		tmp = fma(Float64(sqrt(0.5) * a2_m), a2_m, Float64(Float64(a1 / sqrt(2.0)) * a1));
	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[(t$95$1 * N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(a1 * a1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -5e-105], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[0.5], $MachinePrecision] * a2$95$m), $MachinePrecision] * a2$95$m + N[(N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a1), $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))) < -4.99999999999999963e-105

   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

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. pow-flip N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.7

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 60.3

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

   7. Applied rewrites 60.3%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 49.7%

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

   if -4.99999999999999963e-105 < (+.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

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. pow-flip N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.6

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 85.8

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

   7. Applied rewrites 85.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) + \left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -5 \cdot 10^{-105}:\\ \;\;\;\;\left(\mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot \sqrt{0.5}\right) \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{0.5} \cdot a2, a2, \frac{a1}{\sqrt{2}} \cdot a1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a2\_m \cdot a2\_m\right) + \left(a1 \cdot a1\right) \cdot t\_1 \leq -5 \cdot 10^{-105}:\\ \;\;\;\;\left(\mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot \sqrt{0.5}\right) \cdot \left(a2\_m \cdot a2\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1 \cdot a1, \sqrt{2}, \left(a2\_m \cdot a2\_m\right) \cdot \sqrt{2}\right) \cdot 0.5\\ \end{array} \end{array} \]

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 (<= (+ (* t_1 (* a2_m a2_m)) (* (* a1 a1) t_1)) -5e-105)
     (* (* (fma (* th th) -0.5 1.0) (sqrt 0.5)) (* a2_m a2_m))
     (* (fma (* a1 a1) (sqrt 2.0) (* (* 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 (((t_1 * (a2_m * a2_m)) + ((a1 * a1) * t_1)) <= -5e-105) {
		tmp = (fma((th * th), -0.5, 1.0) * sqrt(0.5)) * (a2_m * a2_m);
	} else {
		tmp = fma((a1 * a1), sqrt(2.0), ((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(t_1 * Float64(a2_m * a2_m)) + Float64(Float64(a1 * a1) * t_1)) <= -5e-105)
		tmp = Float64(Float64(fma(Float64(th * th), -0.5, 1.0) * sqrt(0.5)) * Float64(a2_m * a2_m));
	else
		tmp = Float64(fma(Float64(a1 * a1), sqrt(2.0), Float64(Float64(a2_m * a2_m) * 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[(t$95$1 * N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(a1 * a1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -5e-105], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $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))) < -4.99999999999999963e-105

   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

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. pow-flip N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.7

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 60.3

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

   7. Applied rewrites 60.3%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 49.7%

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

   if -4.99999999999999963e-105 < (+.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. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 85.9

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

   5. Applied rewrites 85.9%

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

   7. Applied rewrites 85.9%

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

   9. Applied rewrites 85.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) + \left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -5 \cdot 10^{-105}:\\ \;\;\;\;\left(\mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot \sqrt{0.5}\right) \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1 \cdot a1, \sqrt{2}, \left(a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a2\_m \cdot a2\_m\right) + \left(a1 \cdot a1\right) \cdot t\_1 \leq -5 \cdot 10^{-105}:\\ \;\;\;\;\left(\mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot \sqrt{0.5}\right) \cdot \left(a2\_m \cdot a2\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2\_m}{\sqrt{2}} \cdot a2\_m\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a2_m a2_m)) (* (* a1 a1) t_1)) -5e-105)
     (* (* (fma (* th th) -0.5 1.0) (sqrt 0.5)) (* a2_m a2_m))
     (* (/ a2_m (sqrt 2.0)) a2_m))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a2_m * a2_m)) + ((a1 * a1) * t_1)) <= -5e-105) {
		tmp = (fma((th * th), -0.5, 1.0) * sqrt(0.5)) * (a2_m * a2_m);
	} else {
		tmp = (a2_m / sqrt(2.0)) * a2_m;
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a2_m * a2_m)) + Float64(Float64(a1 * a1) * t_1)) <= -5e-105)
		tmp = Float64(Float64(fma(Float64(th * th), -0.5, 1.0) * sqrt(0.5)) * Float64(a2_m * a2_m));
	else
		tmp = Float64(Float64(a2_m / sqrt(2.0)) * a2_m);
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(a1 * a1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -5e-105], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a2$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -4.99999999999999963e-105

   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

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. pow-flip N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.7

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

      19. lift-*.f64 N/A

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

      20. lift-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 60.3

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

   7. Applied rewrites 60.3%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 49.7%

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

   if -4.99999999999999963e-105 < (+.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. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 85.9

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in a1 around 0

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

   8. Applied rewrites 49.4%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) + \left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -5 \cdot 10^{-105}:\\ \;\;\;\;\left(\mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot \sqrt{0.5}\right) \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot a2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. +-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

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

   5. clear-num N/A

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

   6. associate-/r/ N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. pow1/2 N/A

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

   11. pow-flip N/A

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

   12. lower-pow.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.6

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

   19. lift-*.f64 N/A

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

   20. lift-/.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. distribute-lft-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. *-lft-identity N/A

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

   16. associate-*l/ N/A

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

   17. unpow2 N/A

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

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

Alternative 7: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. +-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

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

   5. clear-num N/A

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

   6. associate-/r/ N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. pow1/2 N/A

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

   11. pow-flip N/A

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

   12. lower-pow.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.6

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

   19. lift-*.f64 N/A

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

   20. lift-/.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. distribute-lft-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. *-lft-identity N/A

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

   16. associate-*l/ N/A

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

   17. unpow2 N/A

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

7. Applied rewrites 99.7%

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

9. Applied rewrites 99.7%

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

10. Final simplification 99.7%

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

Alternative 8: 78.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in a1 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 56.8

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

5. Applied rewrites 56.8%

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

7. Applied rewrites 56.8%

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

8. Final simplification 56.8%

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

Alternative 9: 78.2% accurate, 2.0× speedup

Math

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

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (* (* (cos th) 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 (cos(th) * 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 = (cos(th) * 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 (Math.cos(th) * 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 (math.cos(th) * 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(cos(th) * a2_m) * Float64(a2_m / sqrt(2.0)))
end

MATLAB

a2_m = abs(a2);
a1, a2_m, th = num2cell(sort([a1, a2_m, th])){:}
function tmp = code(a1, a2_m, th)
	tmp = (cos(th) * 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[(N[Cos[th], $MachinePrecision] * a2$95$m), $MachinePrecision] * N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in a1 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 56.8

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

5. Applied rewrites 56.8%

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

Alternative 10: 78.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

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

   5. clear-num N/A

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

   6. associate-/r/ N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. pow1/2 N/A

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

   11. pow-flip N/A

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

   12. lower-pow.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.6

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

   19. lift-*.f64 N/A

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

   20. lift-/.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. distribute-lft-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. *-lft-identity N/A

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

   16. associate-*l/ N/A

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

   17. unpow2 N/A

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in a1 around 0

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

10. Applied rewrites 56.8%

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

11. Final simplification 56.8%

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

Alternative 11: 52.9% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-*l/ N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*l/ N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sqrt.f64 66.8

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

5. Applied rewrites 66.8%

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

6. Taylor expanded in a1 around 0

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

8. Applied rewrites 38.6%

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

Alternative 12: 52.9% accurate, 12.7× speedup

Math

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

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th) :precision binary64 (* (* (sqrt 0.5) 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 (sqrt(0.5) * 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 = (sqrt(0.5d0) * 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.sqrt(0.5) * 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.sqrt(0.5) * 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(sqrt(0.5) * 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 = (sqrt(0.5) * 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[Sqrt[0.5], $MachinePrecision] * a2$95$m), $MachinePrecision] * a2$95$m), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

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

   5. clear-num N/A

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

   6. associate-/r/ N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. pow1/2 N/A

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

   11. pow-flip N/A

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

   12. lower-pow.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.6

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

   19. lift-*.f64 N/A

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

   20. lift-/.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 56.8

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

7. Applied rewrites 56.8%

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

8. Taylor expanded in th around 0

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

10. Applied rewrites 38.5%

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

Reproduce

herbie shell --seed 2024250 
(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))))