Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 12.9s

Alternatives: 11

Speedup: 1.9×

• 44.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (*
  (fma (sqrt 2.0) (* (cos th) (* a2 a2)) (* (sqrt 2.0) (* (cos th) (* a1 a1))))
  0.5))

C

double code(double a1, double a2, double th) {
	return fma(sqrt(2.0), (cos(th) * (a2 * a2)), (sqrt(2.0) * (cos(th) * (a1 * a1)))) * 0.5;
}

Julia

function code(a1, a2, th)
	return Float64(fma(sqrt(2.0), Float64(cos(th) * Float64(a2 * a2)), Float64(sqrt(2.0) * Float64(cos(th) * Float64(a1 * a1)))) * 0.5)
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.6%

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

Alternative 2: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a2}{\sqrt{2}}\\ t_2 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1 \cdot a1\right) \cdot t\_2 + \left(a2 \cdot a2\right) \cdot t\_2 \leq -1 \cdot 10^{-180}:\\ \;\;\;\;\mathsf{fma}\left(a2, t\_1, \mathsf{fma}\left(a2, a2, 0\right) \cdot \left(\left(th \cdot \frac{th}{\sqrt{2}}\right) \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, a2, \frac{a1 \cdot a1}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ a2 (sqrt 2.0))) (t_2 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1 a1) t_2) (* (* a2 a2) t_2)) -1e-180)
     (fma a2 t_1 (* (fma a2 a2 0.0) (* (* th (/ th (sqrt 2.0))) -0.5)))
     (fma t_1 a2 (/ (* a1 a1) (sqrt 2.0))))))

C

double code(double a1, double a2, double th) {
	double t_1 = a2 / sqrt(2.0);
	double t_2 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_2) + ((a2 * a2) * t_2)) <= -1e-180) {
		tmp = fma(a2, t_1, (fma(a2, a2, 0.0) * ((th * (th / sqrt(2.0))) * -0.5)));
	} else {
		tmp = fma(t_1, a2, ((a1 * a1) / sqrt(2.0)));
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = Float64(a2 / sqrt(2.0))
	t_2 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1 * a1) * t_2) + Float64(Float64(a2 * a2) * t_2)) <= -1e-180)
		tmp = fma(a2, t_1, Float64(fma(a2, a2, 0.0) * Float64(Float64(th * Float64(th / sqrt(2.0))) * -0.5)));
	else
		tmp = fma(t_1, a2, Float64(Float64(a1 * a1) / sqrt(2.0)));
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1 * a1), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], -1e-180], N[(a2 * t$95$1 + N[(N[(a2 * a2 + 0.0), $MachinePrecision] * N[(N[(th * N[(th / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * a2 + N[(N[(a1 * a1), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -1e-180

   1. Initial program 99.7%

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. div-inv N/A

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

      6. associate-/r* N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

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

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

      16. /-lowering-/.f64 N/A

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

      17. cos-lowering-cos.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 0.9

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

   7. Simplified 0.9%

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

   8. Taylor expanded in a1 around 0

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

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

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

      2. unpow2 N/A

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 1.4

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

   10. Simplified 1.4%

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

   11. Taylor expanded in th around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-/l* N/A

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

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

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

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

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

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

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

       7. *-commutative N/A

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

       8. associate-/l* N/A

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

       9. associate-*l* N/A

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

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

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

       11. +-rgt-identity N/A

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

       12. unpow2 N/A

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

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

           \[\leadsto \mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, \color{blue}{\mathsf{fma}\left(a2, a2, 0\right)} \cdot \left(\frac{{th}^{2}}{\sqrt{2}} \cdot \frac{-1}{2}\right)\right) \]

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

           \[\leadsto \mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, \mathsf{fma}\left(a2, a2, 0\right) \cdot \color{blue}{\left(\frac{{th}^{2}}{\sqrt{2}} \cdot \frac{-1}{2}\right)}\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, \mathsf{fma}\left(a2, a2, 0\right) \cdot \left(\frac{\color{blue}{th \cdot th}}{\sqrt{2}} \cdot \frac{-1}{2}\right)\right) \]

       16. associate-/l* N/A

           \[\leadsto \mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, \mathsf{fma}\left(a2, a2, 0\right) \cdot \left(\color{blue}{\left(th \cdot \frac{th}{\sqrt{2}}\right)} \cdot \frac{-1}{2}\right)\right) \]

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

           \[\leadsto \mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, \mathsf{fma}\left(a2, a2, 0\right) \cdot \left(\color{blue}{\left(th \cdot \frac{th}{\sqrt{2}}\right)} \cdot \frac{-1}{2}\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, \mathsf{fma}\left(a2, a2, 0\right) \cdot \left(\left(th \cdot \color{blue}{\frac{th}{\sqrt{2}}}\right) \cdot \frac{-1}{2}\right)\right) \]

       19. sqrt-lowering-sqrt.f64 39.8

           \[\leadsto \mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, \mathsf{fma}\left(a2, a2, 0\right) \cdot \left(\left(th \cdot \frac{th}{\color{blue}{\sqrt{2}}}\right) \cdot -0.5\right)\right) \]

   13. Simplified 39.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, \mathsf{fma}\left(a2, a2, 0\right) \cdot \left(\left(th \cdot \frac{th}{\sqrt{2}}\right) \cdot -0.5\right)\right)} \]

   if -1e-180 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. sqrt-lowering-sqrt.f64 83.8

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

   5. Simplified 83.8%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. associate-*r/ N/A

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 83.8

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

   7. Applied egg-rr 83.8%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -1 \cdot 10^{-180}:\\ \;\;\;\;\mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, \mathsf{fma}\left(a2, a2, 0\right) \cdot \left(\left(th \cdot \frac{th}{\sqrt{2}}\right) \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a2}{\sqrt{2}}, a2, \frac{a1 \cdot a1}{\sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1 \cdot a1\right) \cdot t\_1 + \left(a2 \cdot a2\right) \cdot t\_1 \leq -1 \cdot 10^{-180}:\\ \;\;\;\;\left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a2}{\sqrt{2}}, a2, \frac{a1 \cdot a1}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1 a1) t_1) (* (* a2 a2) t_1)) -1e-180)
     (* (* (sqrt 2.0) (fma a1 a1 (* a2 a2))) (fma -0.25 (* th th) 0.5))
     (fma (/ a2 (sqrt 2.0)) a2 (/ (* a1 a1) (sqrt 2.0))))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -1e-180) {
		tmp = (sqrt(2.0) * fma(a1, a1, (a2 * a2))) * fma(-0.25, (th * th), 0.5);
	} else {
		tmp = fma((a2 / sqrt(2.0)), a2, ((a1 * a1) / sqrt(2.0)));
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1 * a1) * t_1) + Float64(Float64(a2 * a2) * t_1)) <= -1e-180)
		tmp = Float64(Float64(sqrt(2.0) * fma(a1, a1, Float64(a2 * a2))) * fma(-0.25, Float64(th * th), 0.5));
	else
		tmp = fma(Float64(a2 / sqrt(2.0)), a2, Float64(Float64(a1 * a1) / sqrt(2.0)));
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1 * a1), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -1e-180], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a2 + N[(N[(a1 * a1), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -1e-180

   1. Initial program 99.7%

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

      1. associate-*l/ N/A

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

      2. associate-*l/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

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

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

   4. Applied egg-rr 99.7%

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

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

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

      4. distribute-lft-out N/A

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

      5. +-commutative N/A

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

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

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

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

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

      8. +-commutative N/A

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

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

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

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

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

      11. sqrt-lowering-sqrt.f64 99.7

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 58.0

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

   9. Simplified 58.0%

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

   if -1e-180 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. sqrt-lowering-sqrt.f64 83.8

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

   5. Simplified 83.8%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. associate-*r/ N/A

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 83.8

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

   7. Applied egg-rr 83.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -1 \cdot 10^{-180}:\\ \;\;\;\;\left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a2}{\sqrt{2}}, a2, \frac{a1 \cdot a1}{\sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\\ t_2 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1 \cdot a1\right) \cdot t\_2 + \left(a2 \cdot a2\right) \cdot t\_2 \leq -1 \cdot 10^{-180}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (* (sqrt 2.0) (fma a1 a1 (* a2 a2))))
        (t_2 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1 a1) t_2) (* (* a2 a2) t_2)) -1e-180)
     (* t_1 (fma -0.25 (* th th) 0.5))
     (* 0.5 t_1))))

C

double code(double a1, double a2, double th) {
	double t_1 = sqrt(2.0) * fma(a1, a1, (a2 * a2));
	double t_2 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_2) + ((a2 * a2) * t_2)) <= -1e-180) {
		tmp = t_1 * fma(-0.25, (th * th), 0.5);
	} else {
		tmp = 0.5 * t_1;
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = Float64(sqrt(2.0) * fma(a1, a1, Float64(a2 * a2)))
	t_2 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1 * a1) * t_2) + Float64(Float64(a2 * a2) * t_2)) <= -1e-180)
		tmp = Float64(t_1 * fma(-0.25, Float64(th * th), 0.5));
	else
		tmp = Float64(0.5 * t_1);
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1 * a1), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], -1e-180], N[(t$95$1 * N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * t$95$1), $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))) < -1e-180

   1. Initial program 99.7%

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

      1. associate-*l/ N/A

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

      2. associate-*l/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

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

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

   4. Applied egg-rr 99.7%

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

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

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

      4. distribute-lft-out N/A

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

      5. +-commutative N/A

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

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

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

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

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

      8. +-commutative N/A

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

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

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

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

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

      11. sqrt-lowering-sqrt.f64 99.7

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 58.0

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

   9. Simplified 58.0%

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

   if -1e-180 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-*l/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

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

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in th around 0

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

      1. distribute-rgt-out N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 83.8

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

   7. Simplified 83.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -1 \cdot 10^{-180}:\\ \;\;\;\;\left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* 0.5 (* (sqrt 2.0) (* (cos th) (fma a2 a2 (* a1 a1))))))

C

double code(double a1, double a2, double th) {
	return 0.5 * (sqrt(2.0) * (cos(th) * fma(a2, a2, (a1 * a1))));
}

Julia

function code(a1, a2, th)
	return Float64(0.5 * Float64(sqrt(2.0) * Float64(cos(th) * fma(a2, a2, Float64(a1 * a1)))))
end

Wolfram

code[a1_, a2_, th_] := N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * N[(a2 * a2 + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.6%

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

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

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

   4. distribute-lft-out N/A

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

   5. +-commutative N/A

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

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

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

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

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

   8. +-commutative N/A

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

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

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

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

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

   11. sqrt-lowering-sqrt.f64 99.6

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 6: 56.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\sqrt{2} \cdot \left(a2 \cdot \left(\cos th \cdot a2\right)\right)\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* 0.5 (* (sqrt 2.0) (* a2 (* (cos th) a2)))))

C

double code(double a1, double a2, double th) {
	return 0.5 * (sqrt(2.0) * (a2 * (cos(th) * a2)));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = 0.5d0 * (sqrt(2.0d0) * (a2 * (cos(th) * a2)))
end function

Java

public static double code(double a1, double a2, double th) {
	return 0.5 * (Math.sqrt(2.0) * (a2 * (Math.cos(th) * a2)));
}

Python

def code(a1, a2, th):
	return 0.5 * (math.sqrt(2.0) * (a2 * (math.cos(th) * a2)))

Julia

function code(a1, a2, th)
	return Float64(0.5 * Float64(sqrt(2.0) * Float64(a2 * Float64(cos(th) * a2))))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = 0.5 * (sqrt(2.0) * (a2 * (cos(th) * a2)));
end

Wolfram

code[a1_, a2_, th_] := N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a2 * N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.6%

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

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

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

   4. distribute-lft-out N/A

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

   5. +-commutative N/A

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

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

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

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

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

   8. +-commutative N/A

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

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

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

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

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

   11. sqrt-lowering-sqrt.f64 99.6

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in a2 around inf

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. cos-lowering-cos.f64 56.4

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

9. Simplified 56.4%

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

10. Final simplification 56.4%

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

Alternative 7: 56.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\left(a2 \cdot a2\right) \cdot \left(\sqrt{2} \cdot \cos th\right)\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* 0.5 (* (* a2 a2) (* (sqrt 2.0) (cos th)))))

C

double code(double a1, double a2, double th) {
	return 0.5 * ((a2 * a2) * (sqrt(2.0) * cos(th)));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = 0.5d0 * ((a2 * a2) * (sqrt(2.0d0) * cos(th)))
end function

Java

public static double code(double a1, double a2, double th) {
	return 0.5 * ((a2 * a2) * (Math.sqrt(2.0) * Math.cos(th)));
}

Python

def code(a1, a2, th):
	return 0.5 * ((a2 * a2) * (math.sqrt(2.0) * math.cos(th)))

Julia

function code(a1, a2, th)
	return Float64(0.5 * Float64(Float64(a2 * a2) * Float64(sqrt(2.0) * cos(th))))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = 0.5 * ((a2 * a2) * (sqrt(2.0) * cos(th)));
end

Wolfram

code[a1_, a2_, th_] := N[(0.5 * N[(N[(a2 * a2), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in a2 around inf

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

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

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

   2. unpow2 N/A

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

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

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

   4. *-commutative N/A

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

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

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

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

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

   7. cos-lowering-cos.f64 56.4

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

7. Simplified 56.4%

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

8. Final simplification 56.4%

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

Alternative 8: 65.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* 0.5 (* (sqrt 2.0) (fma a1 a1 (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	return 0.5 * (sqrt(2.0) * fma(a1, a1, (a2 * a2)));
}

Julia

function code(a1, a2, th)
	return Float64(0.5 * Float64(sqrt(2.0) * fma(a1, a1, Float64(a2 * a2))))
end

Wolfram

code[a1_, a2_, th_] := N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in th around 0

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

   1. distribute-rgt-out N/A

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. unpow2 N/A

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

   7. *-lowering-*.f64 66.2

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

7. Simplified 66.2%

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

8. Final simplification 66.2%

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

Alternative 9: 39.5% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ a2 \cdot \frac{a2}{\sqrt{2}} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* a2 (/ a2 (sqrt 2.0))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a1, a2, th)
	return Float64(a2 * Float64(a2 / sqrt(2.0)))
end

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. sqrt-lowering-sqrt.f64 66.2

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

5. Simplified 66.2%

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

   1. associate-*r/ N/A

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

   2. clear-num N/A

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

   3. clear-num N/A

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

   4. frac-add N/A

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

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

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

7. Applied egg-rr 27.5%

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

8. Taylor expanded in a2 around inf

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

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

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

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

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

   5. sqrt-lowering-sqrt.f64 40.0

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

10. Simplified 40.0%

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

Alternative 10: 39.5% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \left(a2 \cdot a2\right) \cdot \left(\sqrt{2} \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* (* a2 a2) (* (sqrt 2.0) 0.5)))

C

double code(double a1, double a2, double th) {
	return (a2 * a2) * (sqrt(2.0) * 0.5);
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (a2 * a2) * (sqrt(2.0d0) * 0.5d0)
end function

Java

public static double code(double a1, double a2, double th) {
	return (a2 * a2) * (Math.sqrt(2.0) * 0.5);
}

Python

def code(a1, a2, th):
	return (a2 * a2) * (math.sqrt(2.0) * 0.5)

Julia

function code(a1, a2, th)
	return Float64(Float64(a2 * a2) * Float64(sqrt(2.0) * 0.5))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (a2 * a2) * (sqrt(2.0) * 0.5);
end

Wolfram

code[a1_, a2_, th_] := N[(N[(a2 * a2), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. sqrt-lowering-sqrt.f64 66.2

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

5. Simplified 66.2%

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

   1. associate-*r/ N/A

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

   2. clear-num N/A

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

   3. clear-num N/A

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

   4. frac-add N/A

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

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

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

7. Applied egg-rr 27.5%

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

8. Taylor expanded in a2 around inf

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

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

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

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

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

   5. sqrt-lowering-sqrt.f64 40.0

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

10. Simplified 40.0%

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

    1. associate-*r/ N/A

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

    2. div-inv N/A

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

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

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

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

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

    5. pow1/2 N/A

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

    6. metadata-eval N/A

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

    7. metadata-eval N/A

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

    8. pow-plus N/A

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

    9. pow-flip N/A

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

    10. pow1/2 N/A

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

    11. metadata-eval N/A

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

    12. div-inv N/A

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

    13. clear-num N/A

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

    14. div-inv N/A

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

    15. clear-num N/A

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

    16. div-inv N/A

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

    17. metadata-eval N/A

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

    18. *-rgt-identity N/A

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

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

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

    20. sqrt-lowering-sqrt.f64 40.0

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

12. Applied egg-rr 40.0%

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

13. Final simplification 40.0%

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

Alternative 11: 39.6% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(a1 \cdot \left(\sqrt{2} \cdot a1\right)\right) \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* 0.5 (* a1 (* (sqrt 2.0) a1))))

C

double code(double a1, double a2, double th) {
	return 0.5 * (a1 * (sqrt(2.0) * a1));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = 0.5d0 * (a1 * (sqrt(2.0d0) * a1))
end function

Java

public static double code(double a1, double a2, double th) {
	return 0.5 * (a1 * (Math.sqrt(2.0) * a1));
}

Python

def code(a1, a2, th):
	return 0.5 * (a1 * (math.sqrt(2.0) * a1))

Julia

function code(a1, a2, th)
	return Float64(0.5 * Float64(a1 * Float64(sqrt(2.0) * a1)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = 0.5 * (a1 * (sqrt(2.0) * a1));
end

Wolfram

code[a1_, a2_, th_] := N[(0.5 * N[(a1 * N[(N[Sqrt[2.0], $MachinePrecision] * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. sqrt-lowering-sqrt.f64 66.2

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

5. Simplified 66.2%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

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

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

7. Applied egg-rr 18.4%

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

8. Taylor expanded in a2 around 0

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

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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

   5. *-commutative N/A

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

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

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

   7. sqrt-lowering-sqrt.f64 39.3

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

10. Simplified 39.3%

    \[\leadsto \color{blue}{0.5 \cdot \left(a1 \cdot \left(\sqrt{2} \cdot a1\right)\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024194 
(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))))