Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 11.5s

Alternatives: 8

Speedup: 1.9×

• 43.3% 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, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a1_, a2_, th_] := N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $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. 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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 74.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot t\_1 \leq -5 \cdot 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(th, th \cdot 0.5, -1\right) \cdot \frac{a2 \cdot a2}{-\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* (* a2 a2) t_1)) -5e-127)
     (* (fma th (* th 0.5) -1.0) (/ (* a2 a2) (- (sqrt 2.0))))
     (/ (fma a1 a1 (* a2 a2)) (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + ((a2 * a2) * t_1)) <= -5e-127) {
		tmp = fma(th, (th * 0.5), -1.0) * ((a2 * a2) / -sqrt(2.0));
	} else {
		tmp = fma(a1, a1, (a2 * a2)) / sqrt(2.0);
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(Float64(a2 * a2) * t_1)) <= -5e-127)
		tmp = Float64(fma(th, Float64(th * 0.5), -1.0) * Float64(Float64(a2 * a2) / Float64(-sqrt(2.0))));
	else
		tmp = Float64(fma(a1, a1, Float64(a2 * a2)) / sqrt(2.0));
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -5e-127], N[(N[(th * N[(th * 0.5), $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] / (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $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))) < -4.9999999999999997e-127

   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-+.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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in th around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 54.0

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

   7. Applied rewrites 54.0%

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

   8. Taylor expanded in a1 around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sqrt.f64 42.0

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

   10. Applied rewrites 42.0%

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

   if -4.9999999999999997e-127 < (+.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-+.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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

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

      1. lower-sqrt.f64 84.2

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

   7. Applied rewrites 84.2%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -5 \cdot 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(th, th \cdot 0.5, -1\right) \cdot \frac{a2 \cdot a2}{-\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(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))))