Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 11.1s

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, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(N[(N[(0.5 * N[(a2 * a2 + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $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-+.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

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

   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-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. 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 rewrites 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 inf

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

   1. *-commutative N/A

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

   2. distribute-rgt-out N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

9. Applied rewrites 99.7%

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

Alternative 2: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [a1, a2, th] = \mathsf{sort}([a1, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1 \cdot a1\right) \cdot t\_1 + \left(a2 \cdot a2\right) \cdot t\_1 \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th, th \cdot -0.5, 1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{a2 \cdot a2}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1 a1) t_1) (* (* a2 a2) t_1)) -2e-102)
     (* (* a2 a2) (/ (fma th (* th -0.5) 1.0) (sqrt 2.0)))
     (fma a1 (/ a1 (sqrt 2.0)) (/ (* a2 a2) (sqrt 2.0))))))

C

assert(a1 < a2 && a2 < th);
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-102) {
		tmp = (a2 * a2) * (fma(th, (th * -0.5), 1.0) / sqrt(2.0));
	} else {
		tmp = fma(a1, (a1 / sqrt(2.0)), ((a2 * a2) / sqrt(2.0)));
	}
	return tmp;
}

Julia

a1, a2, th = sort([a1, a2, th])
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-102)
		tmp = Float64(Float64(a2 * a2) * Float64(fma(th, Float64(th * -0.5), 1.0) / sqrt(2.0)));
	else
		tmp = fma(a1, Float64(a1 / sqrt(2.0)), Float64(Float64(a2 * a2) / sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
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-102], N[(N[(a2 * a2), $MachinePrecision] * N[(N[(th * N[(th * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a1 * N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -1.99999999999999987e-102

   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. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

   8. Taylor expanded in a1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 41.5

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

   10. Applied rewrites 41.5%

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

   if -1.99999999999999987e-102 < (+.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 83.9

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

   5. Applied rewrites 83.9%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th, th \cdot -0.5, 1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{a2 \cdot a2}{\sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024225 
(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))))