Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 11.0s

Alternatives: 10

Speedup: 2.0×

• 43.6% 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, 1.3× speedup

Math

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

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[th], $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. Step-by-step derivation

   1. distribute-lft-out 99.5%

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

   2. cos-neg 99.5%

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

   3. associate-*l/ 99.6%

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

   4. *-commutative 99.6%

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

   5. associate-*l/ 99.6%

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

   6. fma-def 99.6%

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

   7. cos-neg 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a2 * a2 + N[(a1 * a1), $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. Step-by-step derivation

   1. +-commutative 99.5%

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

   2. distribute-lft-out 99.5%

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

   3. fma-def 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = ((a2 * a2) + (a1 * a1)) * (cos(th) * sqrt(0.5d0))
end function

Java

public static double code(double a1, double a2, double th) {
	return ((a2 * a2) + (a1 * a1)) * (Math.cos(th) * Math.sqrt(0.5));
}

Python

def code(a1, a2, th):
	return ((a2 * a2) + (a1 * a1)) * (math.cos(th) * math.sqrt(0.5))

Julia

function code(a1, a2, th)
	return Float64(Float64(Float64(a2 * a2) + Float64(a1 * a1)) * Float64(cos(th) * sqrt(0.5)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = ((a2 * a2) + (a1 * a1)) * (cos(th) * sqrt(0.5));
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * N[Sqrt[0.5], $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. Step-by-step derivation

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   2. inv-pow 99.5%

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

5. Applied egg-rr 99.5%

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

   1. unpow-1 99.5%

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

   2. associate-/r/ 99.5%

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

   3. add-sqr-sqrt 99.5%

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

   4. sqrt-unprod 99.5%

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

   5. frac-times 99.5%

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

   6. metadata-eval 99.5%

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

   7. rem-square-sqrt 99.5%

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

   8. metadata-eval 99.5%

      \[\leadsto \left(\sqrt{\color{blue}{0.5}} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

7. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \cos th\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

8. Final simplification 99.5%

   \[\leadsto \left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \left(\cos th \cdot \sqrt{0.5}\right) \]

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $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. Step-by-step derivation

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 5: 57.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{a2}{\frac{\frac{{2}^{0.5}}{a2}}{\cos th}} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (/ a2 (/ (/ (pow 2.0 0.5) a2) (cos th))))

C

double code(double a1, double a2, double th) {
	return a2 / ((pow(2.0, 0.5) / a2) / cos(th));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 / (((2.0d0 ** 0.5d0) / a2) / cos(th))
end function

Java

public static double code(double a1, double a2, double th) {
	return a2 / ((Math.pow(2.0, 0.5) / a2) / Math.cos(th));
}

Python

def code(a1, a2, th):
	return a2 / ((math.pow(2.0, 0.5) / a2) / math.cos(th))

Julia

function code(a1, a2, th)
	return Float64(a2 / Float64(Float64((2.0 ^ 0.5) / a2) / cos(th)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = a2 / (((2.0 ^ 0.5) / a2) / cos(th));
end

Wolfram

code[a1_, a2_, th_] := N[(a2 / N[(N[(N[Power[2.0, 0.5], $MachinePrecision] / a2), $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. Step-by-step derivation

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in a1 around 0 49.7%

   \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. pow2 49.7%

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

   2. associate-/l* 49.7%

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

   3. div-inv 49.7%

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

   4. times-frac 49.7%

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

6. Applied egg-rr 49.7%

   \[\leadsto \color{blue}{\frac{a2}{\sqrt{2}} \cdot \frac{a2}{\frac{1}{\cos th}}} \]
7. Step-by-step derivation

   1. associate-*l/ 49.7%

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

   2. associate-/l* 49.7%

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

   3. associate-/r/ 49.7%

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

   4. /-rgt-identity 49.7%

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

   5. *-commutative 49.7%

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

8. Applied egg-rr 49.7%

   \[\leadsto \color{blue}{\frac{a2}{\frac{\sqrt{2}}{\cos th \cdot a2}}} \]
9. Step-by-step derivation

   1. div-inv 49.7%

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

   2. add-sqr-sqrt 49.7%

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

   3. associate-*l* 49.7%

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

   4. pow1/2 49.7%

      \[\leadsto \frac{a2}{\sqrt{\color{blue}{{2}^{0.5}}} \cdot \left(\sqrt{\sqrt{2}} \cdot \frac{1}{\cos th \cdot a2}\right)} \]

   5. sqrt-pow1 49.7%

      \[\leadsto \frac{a2}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot \left(\sqrt{\sqrt{2}} \cdot \frac{1}{\cos th \cdot a2}\right)} \]

   6. metadata-eval 49.7%

      \[\leadsto \frac{a2}{{2}^{\color{blue}{0.25}} \cdot \left(\sqrt{\sqrt{2}} \cdot \frac{1}{\cos th \cdot a2}\right)} \]

   7. pow1/2 49.7%

      \[\leadsto \frac{a2}{{2}^{0.25} \cdot \left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot \frac{1}{\cos th \cdot a2}\right)} \]

   8. sqrt-pow1 49.7%

      \[\leadsto \frac{a2}{{2}^{0.25} \cdot \left(\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot \frac{1}{\cos th \cdot a2}\right)} \]

   9. metadata-eval 49.7%

      \[\leadsto \frac{a2}{{2}^{0.25} \cdot \left({2}^{\color{blue}{0.25}} \cdot \frac{1}{\cos th \cdot a2}\right)} \]

   10. *-commutative 49.7%

       \[\leadsto \frac{a2}{{2}^{0.25} \cdot \left({2}^{0.25} \cdot \frac{1}{\color{blue}{a2 \cdot \cos th}}\right)} \]

   11. associate-/r* 49.7%

       \[\leadsto \frac{a2}{{2}^{0.25} \cdot \left({2}^{0.25} \cdot \color{blue}{\frac{\frac{1}{a2}}{\cos th}}\right)} \]

10. Applied egg-rr 49.7%

    \[\leadsto \frac{a2}{\color{blue}{{2}^{0.25} \cdot \left({2}^{0.25} \cdot \frac{\frac{1}{a2}}{\cos th}\right)}} \]
11. Step-by-step derivation

    1. associate-/l/ 49.7%

       \[\leadsto \frac{a2}{{2}^{0.25} \cdot \left({2}^{0.25} \cdot \color{blue}{\frac{1}{\cos th \cdot a2}}\right)} \]

    2. associate-*r/ 49.7%

       \[\leadsto \frac{a2}{{2}^{0.25} \cdot \color{blue}{\frac{{2}^{0.25} \cdot 1}{\cos th \cdot a2}}} \]

    3. *-rgt-identity 49.7%

       \[\leadsto \frac{a2}{{2}^{0.25} \cdot \frac{\color{blue}{{2}^{0.25}}}{\cos th \cdot a2}} \]

    4. associate-*r/ 49.7%

       \[\leadsto \frac{a2}{\color{blue}{\frac{{2}^{0.25} \cdot {2}^{0.25}}{\cos th \cdot a2}}} \]

    5. associate-/l/ 49.7%

       \[\leadsto \frac{a2}{\color{blue}{\frac{\frac{{2}^{0.25} \cdot {2}^{0.25}}{a2}}{\cos th}}} \]

    6. pow-sqr 49.7%

       \[\leadsto \frac{a2}{\frac{\frac{\color{blue}{{2}^{\left(2 \cdot 0.25\right)}}}{a2}}{\cos th}} \]

    7. metadata-eval 49.7%

       \[\leadsto \frac{a2}{\frac{\frac{{2}^{\color{blue}{0.5}}}{a2}}{\cos th}} \]

12. Simplified 49.7%

    \[\leadsto \frac{a2}{\color{blue}{\frac{\frac{{2}^{0.5}}{a2}}{\cos th}}} \]

13. Final simplification 49.7%

    \[\leadsto \frac{a2}{\frac{\frac{{2}^{0.5}}{a2}}{\cos th}} \]

Alternative 6: 57.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ a2 \cdot \left(\sqrt{0.5} \cdot \left(a2 \cdot \cos th\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 * (sqrt(0.5d0) * (a2 * cos(th)))
end function

Java

public static double code(double a1, double a2, double th) {
	return a2 * (Math.sqrt(0.5) * (a2 * Math.cos(th)));
}

Python

def code(a1, a2, th):
	return a2 * (math.sqrt(0.5) * (a2 * math.cos(th)))

Julia

function code(a1, a2, th)
	return Float64(a2 * Float64(sqrt(0.5) * Float64(a2 * cos(th))))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = a2 * (sqrt(0.5) * (a2 * cos(th)));
end

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(N[Sqrt[0.5], $MachinePrecision] * N[(a2 * N[Cos[th], $MachinePrecision]), $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. Step-by-step derivation

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in a1 around 0 49.7%

   \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. pow2 49.7%

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

   2. div-inv 49.7%

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

   3. *-commutative 49.7%

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

   4. *-commutative 49.7%

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

   5. associate-*l* 49.6%

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

   6. inv-pow 49.6%

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

   7. sqrt-pow2 49.7%

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

   8. metadata-eval 49.7%

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

   9. associate-*r* 49.7%

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

6. Applied egg-rr 49.7%

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

7. Taylor expanded in th around inf 49.7%

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

   1. associate-*r* 49.7%

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

   2. *-commutative 49.7%

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

   3. *-commutative 49.7%

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

9. Simplified 49.7%

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

10. Final simplification 49.7%

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

Alternative 7: 57.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{a2}{\frac{\sqrt{2}}{a2 \cdot \cos th}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a2 * N[Cos[th], $MachinePrecision]), $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. Step-by-step derivation

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in a1 around 0 49.7%

   \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. pow2 49.7%

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

   2. associate-/l* 49.7%

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

   3. div-inv 49.7%

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

   4. times-frac 49.7%

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

6. Applied egg-rr 49.7%

   \[\leadsto \color{blue}{\frac{a2}{\sqrt{2}} \cdot \frac{a2}{\frac{1}{\cos th}}} \]
7. Step-by-step derivation

   1. associate-*l/ 49.7%

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

   2. associate-/l* 49.7%

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

   3. associate-/r/ 49.7%

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

   4. /-rgt-identity 49.7%

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

   5. *-commutative 49.7%

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

8. Applied egg-rr 49.7%

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

9. Final simplification 49.7%

   \[\leadsto \frac{a2}{\frac{\sqrt{2}}{a2 \cdot \cos th}} \]

Alternative 8: 66.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* (+ (* a2 a2) (* a1 a1)) (sqrt 0.5)))

C

double code(double a1, double a2, double th) {
	return ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = ((a2 * a2) + (a1 * a1)) * sqrt(0.5d0)
end function

Java

public static double code(double a1, double a2, double th) {
	return ((a2 * a2) + (a1 * a1)) * Math.sqrt(0.5);
}

Python

def code(a1, a2, th):
	return ((a2 * a2) + (a1 * a1)) * math.sqrt(0.5)

Julia

function code(a1, a2, th)
	return Float64(Float64(Float64(a2 * a2) + Float64(a1 * a1)) * sqrt(0.5))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $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. Step-by-step derivation

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

      \[\leadsto \left(\frac{1}{\color{blue}{{2}^{0.5}}} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   4. pow-flip 99.5%

      \[\leadsto \left(\color{blue}{{2}^{\left(-0.5\right)}} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   5. metadata-eval 99.5%

      \[\leadsto \left({2}^{\color{blue}{-0.5}} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

5. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\left({2}^{-0.5} \cdot \cos th\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

6. Taylor expanded in th around 0 65.1%

   \[\leadsto \color{blue}{\sqrt{0.5}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

7. Final simplification 65.1%

   \[\leadsto \left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5} \]

Alternative 9: 39.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ a2 \cdot \left(a2 \cdot \sqrt{0.5}\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 * (a2 * sqrt(0.5d0))
end function

Java

public static double code(double a1, double a2, double th) {
	return a2 * (a2 * Math.sqrt(0.5));
}

Python

def code(a1, a2, th):
	return a2 * (a2 * math.sqrt(0.5))

Julia

function code(a1, a2, th)
	return Float64(a2 * Float64(a2 * sqrt(0.5)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = a2 * (a2 * sqrt(0.5));
end

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 * N[Sqrt[0.5], $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. Step-by-step derivation

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in a1 around 0 49.7%

   \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. pow2 49.7%

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

   2. div-inv 49.7%

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

   3. *-commutative 49.7%

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

   4. *-commutative 49.7%

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

   5. associate-*l* 49.6%

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

   6. inv-pow 49.6%

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

   7. sqrt-pow2 49.7%

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

   8. metadata-eval 49.7%

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

   9. associate-*r* 49.7%

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

6. Applied egg-rr 49.7%

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

7. Taylor expanded in th around 0 32.9%

   \[\leadsto \color{blue}{\left(a2 \cdot \sqrt{0.5}\right)} \cdot a2 \]
8. Step-by-step derivation

   1. *-commutative 32.9%

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

9. Simplified 32.9%

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

10. Final simplification 32.9%

    \[\leadsto a2 \cdot \left(a2 \cdot \sqrt{0.5}\right) \]

Alternative 10: 39.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ a2 \cdot \frac{a2}{\sqrt{2}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 / N[Sqrt[2.0], $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. Step-by-step derivation

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in a1 around 0 49.7%

   \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. pow2 49.7%

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

   2. associate-/l* 49.7%

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

   3. div-inv 49.7%

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

   4. times-frac 49.7%

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

6. Applied egg-rr 49.7%

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

7. Taylor expanded in th around 0 32.9%

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

8. Final simplification 32.9%

   \[\leadsto a2 \cdot \frac{a2}{\sqrt{2}} \]

Reproduce

herbie shell --seed 2023301 
(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))))