Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 11.4s

Alternatives: 7

Speedup: 1.9×

• 42.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(a1, a2, th)
	return Float64(Float64(sqrt(2.0) * Float64(cos(th) * fma(a2, a2, 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 + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l/ N/A

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

   9. frac-add N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. rem-square-sqrt N/A

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

   13. div-inv N/A

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

   14. metadata-eval N/A

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

   15. lower-*.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%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. distribute-rgt-out N/A

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

   11. lift-*.f64 N/A

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

   12. distribute-lft-in N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-fma.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 99.7

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

6. Applied egg-rr 99.7%

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

Alternative 2: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1 \cdot a1\right) \cdot t\_1 + \left(a2 \cdot a2\right) \cdot t\_1 \leq -5 \cdot 10^{-134}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1 a1) t_1) (* (* a2 a2) t_1)) -5e-134)
     (* (* a2 a2) (* (sqrt 2.0) (fma -0.25 (* th th) 0.5)))
     (* 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)) <= -5e-134) {
		tmp = (a2 * a2) * (sqrt(2.0) * fma(-0.25, (th * th), 0.5));
	} 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)) <= -5e-134)
		tmp = Float64(Float64(a2 * a2) * Float64(sqrt(2.0) * fma(-0.25, Float64(th * th), 0.5)));
	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], -5e-134], N[(N[(a2 * a2), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $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))) < -5.0000000000000003e-134

   1. Initial program 99.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-add N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lower-*.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 99.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in a2 around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 55.5

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

   9. Simplified 55.5%

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

   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

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. distribute-lft-out N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

       13. distribute-rgt-out N/A

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

       14. lower-*.f64 N/A

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

       15. lower-sqrt.f64 N/A

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

       16. *-commutative N/A

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

       17. lower-fma.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 40.4

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

   12. Simplified 40.4%

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

   if -5.0000000000000003e-134 < (+.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. lift-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-add N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lower-*.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%

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

   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. lower-*.f64 N/A

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

      3. lower-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. lower-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. lower-*.f64 84.1

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

   7. Simplified 84.1%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -5 \cdot 10^{-134}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024218 
(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))))