Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 9.0s

Alternatives: 8

Speedup: 2.0×

• 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, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return ((a1 * a1) + (a2 * a2)) * (sqrt(0.5) * 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 = ((a1 * a1) + (a2 * a2)) * (sqrt(0.5d0) * cos(th))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.6%

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

   3. pow1/2 99.6%

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

   4. pow-flip 99.7%

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

   5. metadata-eval 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in th around inf 99.7%

   \[\leadsto \color{blue}{\left(\cos th \cdot \sqrt{0.5}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
7. Step-by-step derivation

   1. *-commutative 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 57.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return a2 * (a2 * (sqrt(0.5) * 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 * (a2 * (sqrt(0.5d0) * cos(th)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. *-commutative 99.6%

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

   2. clear-num 99.6%

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

   3. un-div-inv 99.7%

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

   4. add-sqr-sqrt 99.7%

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

   5. pow2 99.7%

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

   6. hypot-def 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in a1 around 0 60.8%

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

   1. associate-/l* 60.8%

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

8. Simplified 60.8%

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

   1. div-inv 60.8%

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

   2. unpow2 60.8%

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

   3. clear-num 60.8%

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

   4. associate-*l* 60.9%

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

   5. div-inv 60.8%

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

   6. pow1/2 60.8%

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

   7. pow-flip 60.8%

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

   8. metadata-eval 60.8%

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

   9. add-sqr-sqrt 60.6%

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

   10. sqrt-unprod 60.8%

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

   11. pow-prod-up 60.8%

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

   12. metadata-eval 60.8%

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

   13. metadata-eval 60.8%

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

10. Applied egg-rr 60.8%

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

11. Final simplification 60.8%

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

Alternative 3: 57.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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 * (cos(th) * (sqrt(0.5d0) * a2))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. *-commutative 99.6%

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

   2. clear-num 99.6%

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

   3. un-div-inv 99.7%

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

   4. add-sqr-sqrt 99.7%

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

   5. pow2 99.7%

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

   6. hypot-def 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in a1 around 0 60.8%

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

   1. associate-/l* 60.8%

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

8. Simplified 60.8%

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

   1. associate-/r/ 60.8%

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

   2. unpow2 60.8%

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

   3. associate-*r/ 60.8%

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

   4. associate-*l* 60.8%

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

   5. div-inv 60.8%

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

   6. pow1/2 60.8%

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

   7. pow-flip 60.8%

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

   8. metadata-eval 60.8%

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

   9. add-sqr-sqrt 60.6%

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

   10. sqrt-unprod 60.8%

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

   11. pow-prod-up 60.8%

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

   12. metadata-eval 60.8%

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

   13. metadata-eval 60.8%

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

10. Applied egg-rr 60.8%

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

11. Final simplification 60.8%

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

Alternative 4: 66.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return sqrt(0.5) * ((a1 * a1) + (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
    code = sqrt(0.5d0) * ((a1 * a1) + (a2 * a2))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.6%

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

   3. pow1/2 99.6%

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

   4. pow-flip 99.7%

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

   5. metadata-eval 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in th around 0 70.4%

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

7. Final simplification 70.4%

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

Alternative 5: 39.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{0.5} \cdot a2}{\frac{1}{a2}} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (/ (* (sqrt 0.5) a2) (/ 1.0 a2)))

C

double code(double a1, double a2, double th) {
	return (sqrt(0.5) * a2) / (1.0 / a2);
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (sqrt(0.5d0) * a2) / (1.0d0 / a2)
end function

Java

public static double code(double a1, double a2, double th) {
	return (Math.sqrt(0.5) * a2) / (1.0 / a2);
}

Python

def code(a1, a2, th):
	return (math.sqrt(0.5) * a2) / (1.0 / a2)

Julia

function code(a1, a2, th)
	return Float64(Float64(sqrt(0.5) * a2) / Float64(1.0 / a2))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (sqrt(0.5) * a2) / (1.0 / a2);
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Sqrt[0.5], $MachinePrecision] * a2), $MachinePrecision] / N[(1.0 / a2), $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. Step-by-step derivation

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 70.3%

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

5. Taylor expanded in a1 around 0 44.4%

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

   1. pow2 44.4%

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

   2. div-inv 44.4%

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

   3. associate-*l* 44.4%

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

   4. pow1/2 44.4%

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

   5. pow-flip 44.4%

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

   6. metadata-eval 44.4%

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

7. Applied egg-rr 44.4%

   \[\leadsto \color{blue}{a2 \cdot \left(a2 \cdot {2}^{-0.5}\right)} \]
8. Step-by-step derivation

   1. associate-*r* 44.4%

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

   2. unpow2 44.4%

      \[\leadsto \color{blue}{{a2}^{2}} \cdot {2}^{-0.5} \]

   3. metadata-eval 44.4%

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

   4. pow-flip 44.4%

      \[\leadsto {a2}^{2} \cdot \color{blue}{\frac{1}{{2}^{0.5}}} \]

   5. pow1/2 44.4%

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

   6. div-inv 44.4%

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

   7. unpow2 44.4%

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

   8. associate-*l/ 44.4%

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

   9. /-rgt-identity 44.4%

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

   10. associate-*r/ 44.4%

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

   11. associate-/l* 44.4%

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

   12. div-inv 44.4%

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

   13. pow1/2 44.4%

       \[\leadsto \frac{a2 \cdot \frac{1}{\color{blue}{{2}^{0.5}}}}{\frac{1}{a2}} \]

   14. pow-flip 44.4%

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

   15. metadata-eval 44.4%

       \[\leadsto \frac{a2 \cdot {2}^{\color{blue}{-0.5}}}{\frac{1}{a2}} \]

   16. add-sqr-sqrt 44.3%

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

   17. sqrt-unprod 44.4%

       \[\leadsto \frac{a2 \cdot \color{blue}{\sqrt{{2}^{-0.5} \cdot {2}^{-0.5}}}}{\frac{1}{a2}} \]

   18. pow-prod-up 44.4%

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

   19. metadata-eval 44.4%

       \[\leadsto \frac{a2 \cdot \sqrt{{2}^{\color{blue}{-1}}}}{\frac{1}{a2}} \]

   20. metadata-eval 44.4%

       \[\leadsto \frac{a2 \cdot \sqrt{\color{blue}{0.5}}}{\frac{1}{a2}} \]

9. Applied egg-rr 44.4%

   \[\leadsto \color{blue}{\frac{a2 \cdot \sqrt{0.5}}{\frac{1}{a2}}} \]

10. Final simplification 44.4%

    \[\leadsto \frac{\sqrt{0.5} \cdot a2}{\frac{1}{a2}} \]

Alternative 6: 39.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return a1 * (sqrt(0.5) * 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 = a1 * (sqrt(0.5d0) * a1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 70.3%

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

5. Taylor expanded in a1 around inf 42.0%

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

   1. div-inv 42.0%

      \[\leadsto \color{blue}{{a1}^{2} \cdot \frac{1}{\sqrt{2}}} \]

   2. unpow2 42.0%

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

   3. pow1/2 42.0%

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

   4. pow-flip 42.0%

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

   5. metadata-eval 42.0%

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

   6. associate-*l* 42.0%

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

   7. add-sqr-sqrt 41.9%

      \[\leadsto a1 \cdot \left(a1 \cdot \color{blue}{\left(\sqrt{{2}^{-0.5}} \cdot \sqrt{{2}^{-0.5}}\right)}\right) \]

   8. sqrt-unprod 42.0%

      \[\leadsto a1 \cdot \left(a1 \cdot \color{blue}{\sqrt{{2}^{-0.5} \cdot {2}^{-0.5}}}\right) \]

   9. pow-prod-up 42.0%

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

   10. metadata-eval 42.0%

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

   11. metadata-eval 42.0%

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

7. Applied egg-rr 42.0%

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

8. Final simplification 42.0%

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

Alternative 7: 39.9% 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 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 70.3%

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

5. Taylor expanded in a1 around 0 44.4%

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

   1. pow2 44.4%

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

   2. div-inv 44.4%

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

   3. associate-*l* 44.4%

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

   4. pow1/2 44.4%

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

   5. pow-flip 44.4%

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

   6. metadata-eval 44.4%

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

7. Applied egg-rr 44.4%

   \[\leadsto \color{blue}{a2 \cdot \left(a2 \cdot {2}^{-0.5}\right)} \]
8. Step-by-step derivation

   1. metadata-eval 44.4%

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

   2. pow-flip 44.4%

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

   3. pow1/2 44.4%

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

   4. div-inv 44.4%

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

9. Applied egg-rr 44.4%

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

10. Final simplification 44.4%

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

Alternative 8: 39.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 70.3%

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

5. Taylor expanded in a1 around 0 44.4%

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

   1. pow2 44.4%

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

   2. add-sqr-sqrt 44.3%

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

   3. sqrt-unprod 41.5%

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

   4. frac-times 41.5%

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

   5. pow2 41.5%

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

   6. pow2 41.5%

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

   7. pow-prod-up 41.5%

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

   8. metadata-eval 41.5%

      \[\leadsto \sqrt{\frac{{a2}^{\color{blue}{4}}}{\sqrt{2} \cdot \sqrt{2}}} \]

   9. rem-square-sqrt 41.5%

      \[\leadsto \sqrt{\frac{{a2}^{4}}{\color{blue}{2}}} \]

7. Applied egg-rr 41.5%

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

   1. sqrt-div 41.5%

      \[\leadsto \color{blue}{\frac{\sqrt{{a2}^{4}}}{\sqrt{2}}} \]

   2. sqrt-pow1 44.4%

      \[\leadsto \frac{\color{blue}{{a2}^{\left(\frac{4}{2}\right)}}}{\sqrt{2}} \]

   3. metadata-eval 44.4%

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

   4. unpow2 44.4%

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

   5. associate-*l/ 44.4%

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

   6. div-inv 44.4%

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

   7. pow1/2 44.4%

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

   8. pow-flip 44.4%

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

   9. metadata-eval 44.4%

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

   10. add-sqr-sqrt 44.3%

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

   11. sqrt-unprod 44.4%

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

   12. pow-prod-up 44.4%

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

   13. metadata-eval 44.4%

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

   14. metadata-eval 44.4%

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

9. Applied egg-rr 44.4%

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

10. Final simplification 44.4%

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

Reproduce

herbie shell --seed 2023333 
(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))))