Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 13.1s

Alternatives: 11

Speedup: 1.0×

• 43.5% 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: 99.6% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. associate-*l/ N/A

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

   12. *-lft-identity N/A

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

   13. lower-/.f64 99.5

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

   14. lift-*.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. associate-*l/ N/A

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

   17. associate-/l* N/A

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

   18. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 99.5

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. associate-*r/ N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r/ N/A

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

   12. lower-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 74.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [a1, a2, th] = \mathsf{sort}([a1, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot t\_1 \leq -4 \cdot 10^{-88}:\\ \;\;\;\;\mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{2}}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}\\ \end{array} \end{array} \]

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 (<= (+ (* t_1 (* a1 a1)) (* (* a2 a2) t_1)) -4e-88)
     (* (fma (* th th) -0.5 1.0) (/ (* a2 a2) (sqrt 2.0)))
     (/ 1.0 (/ (sqrt 2.0) (fma a1 a1 (* a2 a2)))))))

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 (((t_1 * (a1 * a1)) + ((a2 * a2) * t_1)) <= -4e-88) {
		tmp = fma((th * th), -0.5, 1.0) * ((a2 * a2) / sqrt(2.0));
	} else {
		tmp = 1.0 / (sqrt(2.0) / fma(a1, a1, (a2 * a2)));
	}
	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(t_1 * Float64(a1 * a1)) + Float64(Float64(a2 * a2) * t_1)) <= -4e-88)
		tmp = Float64(fma(Float64(th * th), -0.5, 1.0) * Float64(Float64(a2 * a2) / sqrt(2.0)));
	else
		tmp = Float64(1.0 / Float64(sqrt(2.0) / fma(a1, a1, Float64(a2 * a2))));
	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[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -4e-88], N[(N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $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))) < -3.99999999999999974e-88

   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. +-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

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 99.6

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.6

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in th around 0

      \[\leadsto \color{blue}{{th}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{a1}^{2}}{\sqrt{2}} + \frac{-1}{2} \cdot \frac{{a2}^{2}}{\sqrt{2}}\right) + \left(\frac{{a1}^{2}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}}\right)} \]
   8. 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. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

   9. Applied rewrites 55.9%

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

   10. Taylor expanded in a1 around 0

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

   12. Applied rewrites 42.2%

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

   if -3.99999999999999974e-88 < (+.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 99.6%

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

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

      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. clear-num N/A

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

      7. associate-*l/ N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. *-lft-identity N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-sqrt.f64 83.1

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

   7. Applied rewrites 83.1%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -4 \cdot 10^{-88}:\\ \;\;\;\;\mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{2}}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024222 
(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))))