Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 11.2s

Alternatives: 12

Speedup: 1.9×

• 43.4% 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

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

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 99.7%

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

Alternative 2: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1 \cdot a1\right) \cdot t\_1 + \left(a2 \cdot a2\right) \cdot t\_1 \leq -2 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, th \cdot th, 0.5\right) \cdot \left(a2 \cdot \left(\sqrt{2} \cdot a2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{2}}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1 a1) t_1) (* (* a2 a2) t_1)) -2e-107)
     (* (fma -0.25 (* th th) 0.5) (* a2 (* (sqrt 2.0) a2)))
     (/ 1.0 (/ (sqrt 2.0) (fma a1 a1 (* a2 a2)))))))

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)) <= -2e-107) {
		tmp = fma(-0.25, (th * th), 0.5) * (a2 * (sqrt(2.0) * a2));
	} else {
		tmp = 1.0 / (sqrt(2.0) / fma(a1, a1, (a2 * a2)));
	}
	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)) <= -2e-107)
		tmp = Float64(fma(-0.25, Float64(th * th), 0.5) * Float64(a2 * Float64(sqrt(2.0) * a2)));
	else
		tmp = Float64(1.0 / Float64(sqrt(2.0) / fma(a1, a1, Float64(a2 * a2))));
	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], -2e-107], N[(N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $MachinePrecision] * N[(a2 * N[(N[Sqrt[2.0], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -2e-107

   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. 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. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. cos-lowering-cos.f64 50.7

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

   7. Simplified 50.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. cos-lowering-cos.f64 50.8

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

   9. Applied egg-rr 50.8%

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

   10. Taylor expanded in th around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. distribute-lft-out N/A

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

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

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 42.8

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

   12. Simplified 42.8%

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

   if -2e-107 < (+.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. distribute-lft-out N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

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

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

      6. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in th around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 84.9

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

   7. Simplified 84.9%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -2 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, th \cdot th, 0.5\right) \cdot \left(a2 \cdot \left(\sqrt{2} \cdot a2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{2}}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1 \cdot a1\right) \cdot t\_1 + \left(a2 \cdot a2\right) \cdot t\_1 \leq -2 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, th \cdot th, 0.5\right) \cdot \left(a2 \cdot \left(\sqrt{2} \cdot a2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1 a1) t_1) (* (* a2 a2) t_1)) -2e-107)
     (* (fma -0.25 (* th th) 0.5) (* a2 (* (sqrt 2.0) a2)))
     (* 0.5 (* (sqrt 2.0) (fma a1 a1 (* a2 a2)))))))

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)) <= -2e-107) {
		tmp = fma(-0.25, (th * th), 0.5) * (a2 * (sqrt(2.0) * a2));
	} else {
		tmp = 0.5 * (sqrt(2.0) * fma(a1, a1, (a2 * a2)));
	}
	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)) <= -2e-107)
		tmp = Float64(fma(-0.25, Float64(th * th), 0.5) * Float64(a2 * Float64(sqrt(2.0) * a2)));
	else
		tmp = Float64(0.5 * Float64(sqrt(2.0) * fma(a1, a1, Float64(a2 * a2))));
	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], -2e-107], N[(N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $MachinePrecision] * N[(a2 * N[(N[Sqrt[2.0], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -2e-107

   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. 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. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. cos-lowering-cos.f64 50.7

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

   7. Simplified 50.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. cos-lowering-cos.f64 50.8

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

   9. Applied egg-rr 50.8%

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

   10. Taylor expanded in th around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. distribute-lft-out N/A

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

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

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 42.8

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

   12. Simplified 42.8%

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

   if -2e-107 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

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

      1. associate-*l/ N/A

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

      2. associate-*l/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

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

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

   4. Applied egg-rr 99.7%

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

   5. 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 84.8

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

   7. Simplified 84.8%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -2 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, th \cdot th, 0.5\right) \cdot \left(a2 \cdot \left(\sqrt{2} \cdot a2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{\sqrt{2}}{\cos th}}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (/ 1.0 (/ (/ (sqrt 2.0) (cos th)) (fma a1 a1 (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	return 1.0 / ((sqrt(2.0) / cos(th)) / fma(a1, a1, (a2 * a2)));
}

Julia

function code(a1, a2, th)
	return Float64(1.0 / Float64(Float64(sqrt(2.0) / cos(th)) / fma(a1, a1, Float64(a2 * a2))))
end

Wolfram

code[a1_, a2_, th_] := N[(1.0 / N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $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. distribute-lft-out N/A

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

   2. clear-num N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

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

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

   6. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

   12. *-lowering-*.f64 99.6

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

4. Applied egg-rr 99.6%

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

Alternative 5: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / 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. 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. div-inv N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

   12. cos-lowering-cos.f64 99.6

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 6: 57.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return 0.5 * (a2 * (sqrt(2.0) * (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 * (a2 * (sqrt(2.0d0) * (cos(th) * a2)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(0.5 * N[(a2 * N[(N[Sqrt[2.0], $MachinePrecision] * 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 99.7%

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

5. Taylor expanded in 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. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

   6. unpow2 N/A

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

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

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

   8. cos-lowering-cos.f64 59.2

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

7. Simplified 59.2%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

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

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

   11. cos-lowering-cos.f64 59.3

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

9. Applied egg-rr 59.3%

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

10. Final simplification 59.3%

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

Alternative 7: 57.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return 0.5 * ((cos(th) * a2) * (sqrt(2.0) * 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 * ((cos(th) * a2) * (sqrt(2.0d0) * a2))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in 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. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

   6. unpow2 N/A

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

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

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

   8. cos-lowering-cos.f64 59.2

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

7. Simplified 59.2%

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

   1. associate-*l* N/A

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

   6. *-commutative N/A

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

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

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

   8. cos-lowering-cos.f64 59.3

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

9. Applied egg-rr 59.3%

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

10. Final simplification 59.3%

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

Alternative 8: 57.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return 0.5 * (sqrt(2.0) * (cos(th) * (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
    code = 0.5d0 * (sqrt(2.0d0) * (cos(th) * (a2 * a2)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * 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 99.7%

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

5. Taylor expanded in 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. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

   6. unpow2 N/A

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

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

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

   8. cos-lowering-cos.f64 59.2

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

7. Simplified 59.2%

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

8. Final simplification 59.2%

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

Alternative 9: 57.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return (a2 * a2) * (sqrt(2.0) * (cos(th) * 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) * (cos(th) * 0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in a2 around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

   5. unpow2 N/A

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

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

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

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

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

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

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

   11. *-commutative N/A

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

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

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

   13. cos-lowering-cos.f64 59.2

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

7. Simplified 59.2%

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

Alternative 10: 67.0% 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 99.7%

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

5. Taylor expanded in th around 0

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

   1. distribute-rgt-out N/A

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. unpow2 N/A

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

   7. *-lowering-*.f64 66.0

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

7. Simplified 66.0%

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

8. Final simplification 66.0%

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

Alternative 11: 40.4% accurate, 10.2× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return 0.5 * (a2 * (sqrt(2.0) * 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 * (a2 * (sqrt(2.0d0) * a2))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in 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. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

   6. unpow2 N/A

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

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

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

   8. cos-lowering-cos.f64 59.2

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

7. Simplified 59.2%

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

8. Taylor expanded in th around 0

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

   1. *-commutative N/A

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

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

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 41.2

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

10. Simplified 41.2%

    \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(a2 \cdot a2\right)\right)} \cdot 0.5 \]
11. Step-by-step derivation

    1. associate-*r* N/A

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

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

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

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

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

    4. sqrt-lowering-sqrt.f64 41.2

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

12. Applied egg-rr 41.2%

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

13. Final simplification 41.2%

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

Alternative 12: 40.4% accurate, 10.2× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return 0.5 * (sqrt(2.0) * (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
    code = 0.5d0 * (sqrt(2.0d0) * (a2 * a2))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in 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. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

   6. unpow2 N/A

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

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

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

   8. cos-lowering-cos.f64 59.2

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

7. Simplified 59.2%

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

8. Taylor expanded in th around 0

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

   1. *-commutative N/A

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

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

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 41.2

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

10. Simplified 41.2%

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

11. Final simplification 41.2%

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

Reproduce

herbie shell --seed 2024205 
(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))))