Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 13.4s

Alternatives: 16

Speedup: 1.9×

• 43.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. associate-/l* N/A

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

   9. *-lft-identity N/A

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

   10. associate-*l/ N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. associate-*l/ N/A

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

   14. *-lft-identity N/A

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

   15. lower-/.f64 99.6

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

   16. lift-*.f64 N/A

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

   17. lift-/.f64 N/A

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

4. Applied rewrites 99.6%

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

Alternative 2: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + \left(a2\_m \cdot a2\_m\right) \cdot t\_1 \leq -5 \cdot 10^{-130}:\\ \;\;\;\;\frac{a2\_m \cdot \mathsf{fma}\left(th, a2\_m \cdot \left(th \cdot -0.5\right), a2\_m\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{1}{\frac{\sqrt{2}}{a2\_m \cdot a2\_m}}\right)\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* (* a2_m a2_m) t_1)) -5e-130)
     (/ (* a2_m (fma th (* a2_m (* th -0.5)) a2_m)) (sqrt 2.0))
     (fma a1 (/ a1 (sqrt 2.0)) (/ 1.0 (/ (sqrt 2.0) (* a2_m a2_m)))))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + ((a2_m * a2_m) * t_1)) <= -5e-130) {
		tmp = (a2_m * fma(th, (a2_m * (th * -0.5)), a2_m)) / sqrt(2.0);
	} else {
		tmp = fma(a1, (a1 / sqrt(2.0)), (1.0 / (sqrt(2.0) / (a2_m * a2_m))));
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(Float64(a2_m * a2_m) * t_1)) <= -5e-130)
		tmp = Float64(Float64(a2_m * fma(th, Float64(a2_m * Float64(th * -0.5)), a2_m)) / sqrt(2.0));
	else
		tmp = fma(a1, Float64(a1 / sqrt(2.0)), Float64(1.0 / Float64(sqrt(2.0) / Float64(a2_m * a2_m))));
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -5e-130], N[(N[(a2$95$m * N[(th * N[(a2$95$m * N[(th * -0.5), $MachinePrecision]), $MachinePrecision] + a2$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(a1 * N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a2$95$m * a2$95$m), $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))) < -4.9999999999999996e-130

   1. Initial program 99.6%

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

   3. Taylor expanded in th around 0

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

      1. distribute-lft-out N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 56.7%

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

   6. Taylor expanded in a1 around 0

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

   8. Applied rewrites 52.4%

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

   10. Applied rewrites 50.3%

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

   if -4.9999999999999996e-130 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 83.3

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

   5. Applied rewrites 83.3%

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

   7. Applied rewrites 83.2%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -5 \cdot 10^{-130}:\\ \;\;\;\;\frac{a2 \cdot \mathsf{fma}\left(th, a2 \cdot \left(th \cdot -0.5\right), a2\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(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))))