Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 16.4s

Alternatives: 9

Speedup: 2.0×

• 43.7% 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

\[\begin{array}{l} \\ \frac{a2 \cdot \cos th}{\frac{{2}^{0.5}}{a2}} + a1 \cdot \left(\cos th \cdot \frac{a1}{\sqrt{2}}\right) \end{array} \]

FPCore

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

C

double code(double a1, double a2, double th) {
	return ((a2 * cos(th)) / (pow(2.0, 0.5) / a2)) + (a1 * (cos(th) * (a1 / 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 * cos(th)) / ((2.0d0 ** 0.5d0) / a2)) + (a1 * (cos(th) * (a1 / sqrt(2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a2 * N[Cos[th], $MachinePrecision]), $MachinePrecision] / N[(N[Power[2.0, 0.5], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision] + N[(a1 * N[(N[Cos[th], $MachinePrecision] * N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $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. +-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. Simplified 99.5%

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

   1. distribute-lft-in 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. associate-*l/ 99.5%

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

   3. add-sqr-sqrt 99.5%

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

   4. times-frac 99.5%

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

   5. fma-def 99.5%

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

   6. pow1/2 99.5%

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

   7. sqrt-pow1 99.5%

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

   8. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\cos th}{{2}^{\color{blue}{0.25}}}, \frac{a2 \cdot a2}{\sqrt{\sqrt{2}}}, \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right) \]

   9. pow1/2 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\cos th}{{2}^{0.25}}, \frac{a2 \cdot a2}{\sqrt{\color{blue}{{2}^{0.5}}}}, \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right) \]

   10. sqrt-pow1 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{\cos th}{{2}^{0.25}}, \frac{a2 \cdot a2}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}, \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right) \]

   11. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{\cos th}{{2}^{0.25}}, \frac{a2 \cdot a2}{{2}^{\color{blue}{0.25}}}, \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right) \]

   12. associate-*l/ 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{\cos th}{{2}^{0.25}}, \frac{a2 \cdot a2}{{2}^{0.25}}, \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}}}\right) \]

   13. associate-/l* 99.2%

       \[\leadsto \mathsf{fma}\left(\frac{\cos th}{{2}^{0.25}}, \frac{a2 \cdot a2}{{2}^{0.25}}, \color{blue}{\frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1}}}\right) \]

5. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\cos th}{{2}^{0.25}}, \frac{a2 \cdot a2}{{2}^{0.25}}, \frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1}}\right)} \]
6. Step-by-step derivation

   1. fma-udef 99.2%

      \[\leadsto \color{blue}{\frac{\cos th}{{2}^{0.25}} \cdot \frac{a2 \cdot a2}{{2}^{0.25}} + \frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1}}} \]

   2. times-frac 99.3%

      \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a2 \cdot a2\right)}{{2}^{0.25} \cdot {2}^{0.25}}} + \frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1}} \]

   3. associate-*r* 99.2%

      \[\leadsto \frac{\color{blue}{\left(\cos th \cdot a2\right) \cdot a2}}{{2}^{0.25} \cdot {2}^{0.25}} + \frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1}} \]

   4. associate-/l* 99.2%

      \[\leadsto \color{blue}{\frac{\cos th \cdot a2}{\frac{{2}^{0.25} \cdot {2}^{0.25}}{a2}}} + \frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1}} \]

   5. *-commutative 99.2%

      \[\leadsto \frac{\color{blue}{a2 \cdot \cos th}}{\frac{{2}^{0.25} \cdot {2}^{0.25}}{a2}} + \frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1}} \]

   6. pow-sqr 99.3%

      \[\leadsto \frac{a2 \cdot \cos th}{\frac{\color{blue}{{2}^{\left(2 \cdot 0.25\right)}}}{a2}} + \frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1}} \]

   7. metadata-eval 99.3%

      \[\leadsto \frac{a2 \cdot \cos th}{\frac{{2}^{\color{blue}{0.5}}}{a2}} + \frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1}} \]

   8. associate-/l* 99.6%

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

   9. *-commutative 99.6%

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

   10. associate-/l* 99.6%

       \[\leadsto \frac{a2 \cdot \cos th}{\frac{{2}^{0.5}}{a2}} + \color{blue}{\frac{a1 \cdot a1}{\frac{\sqrt{2}}{\cos th}}} \]

   11. associate-*r/ 99.6%

       \[\leadsto \frac{a2 \cdot \cos th}{\frac{{2}^{0.5}}{a2}} + \color{blue}{a1 \cdot \frac{a1}{\frac{\sqrt{2}}{\cos th}}} \]

   12. associate-/r/ 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (/ (/ (cos th) (pow 2.0 0.25)) (pow 2.0 0.25)) (+ (* a2 a2) (* a1 a1))))

C

double code(double a1, double a2, double th) {
	return ((cos(th) / pow(2.0, 0.25)) / pow(2.0, 0.25)) * ((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) / (2.0d0 ** 0.25d0)) / (2.0d0 ** 0.25d0)) * ((a2 * a2) + (a1 * a1))
end function

Java

public static double code(double a1, double a2, double th) {
	return ((Math.cos(th) / Math.pow(2.0, 0.25)) / Math.pow(2.0, 0.25)) * ((a2 * a2) + (a1 * a1));
}

Python

def code(a1, a2, th):
	return ((math.cos(th) / math.pow(2.0, 0.25)) / math.pow(2.0, 0.25)) * ((a2 * a2) + (a1 * a1))

Julia

function code(a1, a2, th)
	return Float64(Float64(Float64(cos(th) / (2.0 ^ 0.25)) / (2.0 ^ 0.25)) * Float64(Float64(a2 * a2) + Float64(a1 * a1)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = ((cos(th) / (2.0 ^ 0.25)) / (2.0 ^ 0.25)) * ((a2 * a2) + (a1 * a1));
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(N[Cos[th], $MachinePrecision] / N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision] / N[Power[2.0, 0.25], $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. +-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. Simplified 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. add-sqr-sqrt 99.6%

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

   3. times-frac 99.2%

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

   4. pow1/2 99.2%

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

   5. sqrt-pow1 99.2%

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

   6. metadata-eval 99.2%

      \[\leadsto \left(\frac{1}{{2}^{\color{blue}{0.25}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

   7. pow1/2 99.2%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\sqrt{\color{blue}{{2}^{0.5}}}}\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

   8. sqrt-pow1 99.2%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

   9. metadata-eval 99.2%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{\color{blue}{0.25}}}\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

5. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{0.25}}\right)} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]
6. Step-by-step derivation

   1. associate-*l/ 99.6%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}}} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

   2. *-lft-identity 99.6%

      \[\leadsto \frac{\color{blue}{\frac{\cos th}{{2}^{0.25}}}}{{2}^{0.25}} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

7. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}}} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

8. Final simplification 99.6%

   \[\leadsto \frac{\frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

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. +-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. Simplified 99.5%

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

   1. fma-def 99.5%

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

5. Applied egg-rr 99.5%

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

   1. fma-def 99.5%

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

   2. distribute-lft-in 99.5%

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

   3. div-inv 99.4%

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

   4. add-sqr-sqrt 99.4%

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

   5. sqrt-unprod 99.4%

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

   6. frac-times 99.4%

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

   7. metadata-eval 99.4%

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

   8. add-sqr-sqrt 99.6%

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

   9. metadata-eval 99.6%

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

7. Applied egg-rr 99.6%

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

   1. distribute-lft-out 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 4: 56.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Sqrt[0.5], $MachinePrecision] * N[(a2 * 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. +-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. Simplified 99.5%

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

4. Taylor expanded in a2 around inf 56.6%

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

   1. unpow2 56.6%

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

6. Simplified 56.6%

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

   1. associate-*l/ 56.7%

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

   2. associate-/l* 56.7%

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

8. Applied egg-rr 56.7%

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

   1. associate-/r/ 56.6%

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

   2. div-inv 56.6%

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

   3. pow1/2 56.6%

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

   4. pow-flip 56.7%

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

   5. metadata-eval 56.7%

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

   6. *-commutative 56.7%

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

   7. associate-*l* 56.7%

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

   8. metadata-eval 56.7%

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

   9. pow-flip 56.6%

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

   10. pow1/2 56.6%

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

   11. *-commutative 56.6%

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

   12. add-sqr-sqrt 56.6%

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

   13. sqrt-unprod 56.6%

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

   14. frac-times 56.6%

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

   15. metadata-eval 56.6%

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

   16. add-sqr-sqrt 56.7%

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

   17. metadata-eval 56.7%

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

10. Applied egg-rr 56.7%

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

11. Taylor expanded in th around inf 56.7%

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

    1. unpow2 56.7%

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

    2. associate-*r* 56.7%

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

13. Simplified 56.7%

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

14. Final simplification 56.7%

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

Alternative 5: 56.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * N[(a2 * a2), $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. Simplified 99.5%

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

4. Taylor expanded in a2 around inf 56.6%

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

   1. unpow2 56.6%

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

6. Simplified 56.6%

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

   1. associate-*l/ 56.7%

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

   2. associate-/l* 56.7%

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

8. Applied egg-rr 56.7%

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

   1. associate-/r/ 56.6%

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

   2. div-inv 56.6%

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

   3. pow1/2 56.6%

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

   4. pow-flip 56.7%

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

   5. metadata-eval 56.7%

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

   6. *-commutative 56.7%

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

   7. associate-*l* 56.7%

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

   8. metadata-eval 56.7%

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

   9. pow-flip 56.6%

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

   10. pow1/2 56.6%

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

   11. *-commutative 56.6%

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

   12. add-sqr-sqrt 56.6%

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

   13. sqrt-unprod 56.6%

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

   14. frac-times 56.6%

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

   15. metadata-eval 56.6%

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

   16. add-sqr-sqrt 56.7%

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

   17. metadata-eval 56.7%

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

10. Applied egg-rr 56.7%

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

11. Final simplification 56.7%

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

Alternative 6: 65.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 10^{+195}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a2 \cdot a2\right) \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 1e+195)
   (* (+ (* a2 a2) (* a1 a1)) (sqrt 0.5))
   (* (sqrt 0.5) (* (* a2 a2) (+ 1.0 (* -0.5 (* th th)))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 1e+195) {
		tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
	} else {
		tmp = sqrt(0.5) * ((a2 * a2) * (1.0 + (-0.5 * (th * th))));
	}
	return tmp;
}

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) :: tmp
    if (a2 <= 1d+195) then
        tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5d0)
    else
        tmp = sqrt(0.5d0) * ((a2 * a2) * (1.0d0 + ((-0.5d0) * (th * th))))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 1e+195) {
		tmp = ((a2 * a2) + (a1 * a1)) * Math.sqrt(0.5);
	} else {
		tmp = Math.sqrt(0.5) * ((a2 * a2) * (1.0 + (-0.5 * (th * th))));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 1e+195:
		tmp = ((a2 * a2) + (a1 * a1)) * math.sqrt(0.5)
	else:
		tmp = math.sqrt(0.5) * ((a2 * a2) * (1.0 + (-0.5 * (th * th))))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 1e+195)
		tmp = Float64(Float64(Float64(a2 * a2) + Float64(a1 * a1)) * sqrt(0.5));
	else
		tmp = Float64(sqrt(0.5) * Float64(Float64(a2 * a2) * Float64(1.0 + Float64(-0.5 * Float64(th * th)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 1e+195)
		tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
	else
		tmp = sqrt(0.5) * ((a2 * a2) * (1.0 + (-0.5 * (th * th))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 1e+195], N[(N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 9.99999999999999977e194

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in th around 0 61.8%

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

      1. expm1-log1p-u 61.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{2}}\right)\right)} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

      2. expm1-udef 61.8%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2}}\right)} - 1\right)} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

      3. add-sqr-sqrt 61.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{1}{\sqrt{2}}} \cdot \sqrt{\frac{1}{\sqrt{2}}}}\right)} - 1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

      4. sqrt-unprod 61.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}}}\right)} - 1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

      5. frac-times 61.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1 \cdot 1}{\sqrt{2} \cdot \sqrt{2}}}}\right)} - 1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

      6. metadata-eval 61.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{1}}{\sqrt{2} \cdot \sqrt{2}}}\right)} - 1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

      7. add-sqr-sqrt 61.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\color{blue}{2}}}\right)} - 1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

      8. metadata-eval 61.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{0.5}}\right)} - 1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

   6. Applied egg-rr 61.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{0.5}\right)} - 1\right)} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]
   7. Step-by-step derivation

      1. expm1-def 61.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5}\right)\right)} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

      2. expm1-log1p 61.9%

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

   8. Simplified 61.9%

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

   if 9.99999999999999977e194 < a2

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a2 around inf 100.0%

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

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. associate-*l/ 100.0%

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

      2. associate-/l* 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. associate-/r/ 100.0%

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

      2. div-inv 100.0%

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

      3. pow1/2 100.0%

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

      4. pow-flip 100.0%

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

      5. metadata-eval 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

      8. metadata-eval 100.0%

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

      9. pow-flip 100.0%

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

      10. pow1/2 100.0%

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

      11. *-commutative 100.0%

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

      12. add-sqr-sqrt 100.0%

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

      13. sqrt-unprod 100.0%

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

      14. frac-times 100.0%

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

      15. metadata-eval 100.0%

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

      16. add-sqr-sqrt 100.0%

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

      17. metadata-eval 100.0%

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in th around 0 83.3%

       \[\leadsto \left(\color{blue}{\left(1 + -0.5 \cdot {th}^{2}\right)} \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{0.5} \]
   12. Step-by-step derivation

       1. unpow2 83.3%

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

   13. Simplified 83.3%

       \[\leadsto \left(\color{blue}{\left(1 + -0.5 \cdot \left(th \cdot th\right)\right)} \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 10^{+195}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a2 \cdot a2\right) \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \end{array} \]

Alternative 7: 65.7% 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. +-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. Simplified 99.5%

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

4. Taylor expanded in th around 0 63.1%

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

   1. expm1-log1p-u 63.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{2}}\right)\right)} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

   2. expm1-udef 63.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2}}\right)} - 1\right)} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

   3. add-sqr-sqrt 63.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{1}{\sqrt{2}}} \cdot \sqrt{\frac{1}{\sqrt{2}}}}\right)} - 1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

   4. sqrt-unprod 63.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}}}\right)} - 1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

   5. frac-times 63.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1 \cdot 1}{\sqrt{2} \cdot \sqrt{2}}}}\right)} - 1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

   6. metadata-eval 63.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{1}}{\sqrt{2} \cdot \sqrt{2}}}\right)} - 1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

   7. add-sqr-sqrt 62.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\color{blue}{2}}}\right)} - 1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

   8. metadata-eval 62.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{0.5}}\right)} - 1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

6. Applied egg-rr 62.8%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{0.5}\right)} - 1\right)} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]
7. Step-by-step derivation

   1. expm1-def 62.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5}\right)\right)} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

   2. expm1-log1p 63.1%

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

8. Simplified 63.1%

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

9. Final simplification 63.1%

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

Alternative 8: 39.0% accurate, 4.0× speedup

Math

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

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(Float64(a2 * 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[(N[(a2 * a2), $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. +-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. Simplified 99.5%

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

4. Taylor expanded in a2 around inf 56.6%

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

   1. unpow2 56.6%

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

6. Simplified 56.6%

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

   1. clear-num 56.6%

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

   2. associate-/r/ 56.6%

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

   3. pow1/2 56.6%

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

   4. pow-flip 56.7%

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

   5. metadata-eval 56.7%

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

8. Applied egg-rr 56.7%

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

9. Taylor expanded in th around 0 39.3%

   \[\leadsto \color{blue}{{a2}^{2} \cdot \sqrt{0.5}} \]
10. Step-by-step derivation

    1. unpow2 39.3%

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

    2. *-commutative 39.3%

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

11. Simplified 39.3%

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

12. Final simplification 39.3%

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

Alternative 9: 39.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return a2 / (sqrt(2.0) / 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 = a2 / (sqrt(2.0d0) / a2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $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. Simplified 99.5%

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

4. Taylor expanded in th around 0 63.1%

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

5. Taylor expanded in a2 around inf 39.3%

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

   1. unpow2 56.6%

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

7. Simplified 39.3%

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

   1. associate-*l/ 39.3%

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

   2. *-un-lft-identity 39.3%

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

   3. associate-/l* 39.3%

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

9. Applied egg-rr 39.3%

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

10. Final simplification 39.3%

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

Reproduce

herbie shell --seed 2023275 
(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))))