Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 8.0s

Alternatives: 8

Speedup: 2.0×

• 44.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t_1 \cdot \left(a1 \cdot a1\right) + t_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    t_1 = cos(th) / sqrt(2.0d0)
    code = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	return Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t_1 \cdot \left(a1 \cdot a1\right) + t_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    t_1 = cos(th) / sqrt(2.0d0)
    code = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	return Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return ((cos(th) / pow(2.0, 0.25)) / pow(2.0, 0.25)) * ((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 = ((cos(th) / (2.0d0 ** 0.25d0)) / (2.0d0 ** 0.25d0)) * ((a1 * a1) + (a2 * a2))
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)) * ((a1 * a1) + (a2 * a2));
}

Python

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

Julia

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

MATLAB

function tmp = code(a1, a2, th)
	tmp = ((cos(th) / (2.0 ^ 0.25)) / (2.0 ^ 0.25)) * ((a1 * a1) + (a2 * a2));
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[(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. *-un-lft-identity 99.6%

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

   2. add-sqr-sqrt 99.6%

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

   3. times-frac 99.2%

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

   4. pow1/2 99.2%

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

   5. sqrt-pow1 99.2%

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

   6. metadata-eval 99.2%

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

   7. pow1/2 99.2%

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

   8. sqrt-pow1 99.2%

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

   9. metadata-eval 99.2%

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

5. Applied egg-rr 99.2%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * N[Power[2.0, -0.5], $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 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] / 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. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $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%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around inf 99.6%

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

   1. associate-/l* 99.6%

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

   2. unpow2 99.6%

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

   3. unpow2 99.6%

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

   4. +-commutative 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 5: 57.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 / N[(N[Sqrt[2.0], $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%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 58.7%

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

   1. unpow2 58.7%

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

   2. associate-*l* 58.7%

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

   3. associate-*r/ 58.7%

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

   4. associate-/l* 58.7%

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

6. Simplified 58.7%

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

7. Final simplification 58.7%

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

Alternative 6: 41.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a1 \cdot a1 \leq 2 \cdot 10^{+100}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;{0.25}^{0.25} \cdot \left(\left(a2 \cdot a2\right) \cdot \left(th \cdot \left(th \cdot -0.5\right) + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (* a1 a1) 2e+100)
   (* a2 (/ a2 (sqrt 2.0)))
   (* (pow 0.25 0.25) (* (* a2 a2) (+ (* th (* th -0.5)) 1.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((a1 * a1) <= 2e+100) {
		tmp = a2 * (a2 / sqrt(2.0));
	} else {
		tmp = pow(0.25, 0.25) * ((a2 * a2) * ((th * (th * -0.5)) + 1.0));
	}
	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 ((a1 * a1) <= 2d+100) then
        tmp = a2 * (a2 / sqrt(2.0d0))
    else
        tmp = (0.25d0 ** 0.25d0) * ((a2 * a2) * ((th * (th * (-0.5d0))) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((a1 * a1) <= 2e+100) {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	} else {
		tmp = Math.pow(0.25, 0.25) * ((a2 * a2) * ((th * (th * -0.5)) + 1.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (a1 * a1) <= 2e+100:
		tmp = a2 * (a2 / math.sqrt(2.0))
	else:
		tmp = math.pow(0.25, 0.25) * ((a2 * a2) * ((th * (th * -0.5)) + 1.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (Float64(a1 * a1) <= 2e+100)
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	else
		tmp = Float64((0.25 ^ 0.25) * Float64(Float64(a2 * a2) * Float64(Float64(th * Float64(th * -0.5)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((a1 * a1) <= 2e+100)
		tmp = a2 * (a2 / sqrt(2.0));
	else
		tmp = (0.25 ^ 0.25) * ((a2 * a2) * ((th * (th * -0.5)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[(a1 * a1), $MachinePrecision], 2e+100], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[0.25, 0.25], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] * N[(N[(th * N[(th * -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a1 a1) < 2.00000000000000003e100

   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. associate-*l/ 99.5%

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

      3. associate-*r/ 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a1 around 0 76.5%

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

      1. unpow2 76.5%

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

      2. associate-*l* 76.4%

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

      3. associate-*r/ 76.5%

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

      4. associate-/l* 76.5%

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

   6. Simplified 76.5%

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

   7. Taylor expanded in th around 0 54.0%

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

   if 2.00000000000000003e100 < (*.f64 a1 a1)

   1. Initial program 99.8%

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

      1. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. add-sqr-sqrt 99.8%

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

      3. times-frac 99.5%

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

      4. pow1/2 99.5%

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

      5. sqrt-pow1 99.5%

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

      6. metadata-eval 99.5%

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

      7. pow1/2 99.5%

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

      8. sqrt-pow1 99.5%

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

      9. metadata-eval 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in a1 around 0 35.1%

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

      1. unpow2 35.1%

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

      2. *-commutative 35.1%

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

   10. Simplified 35.1%

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

   11. Taylor expanded in th around 0 8.2%

       \[\leadsto {0.25}^{0.25} \cdot \color{blue}{\left({a2}^{2} + -0.5 \cdot \left({th}^{2} \cdot {a2}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. unpow2 8.2%

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

       2. unpow2 8.2%

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

       3. associate-*r* 8.2%

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

       4. distribute-rgt1-in 34.5%

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

       5. unpow2 34.5%

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

       6. associate-*r* 34.5%

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

   13. Simplified 34.5%

       \[\leadsto {0.25}^{0.25} \cdot \color{blue}{\left(\left(\left(-0.5 \cdot th\right) \cdot th + 1\right) \cdot \left(a2 \cdot a2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a1 \leq 2 \cdot 10^{+100}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;{0.25}^{0.25} \cdot \left(\left(a2 \cdot a2\right) \cdot \left(th \cdot \left(th \cdot -0.5\right) + 1\right)\right)\\ \end{array} \]

Alternative 7: 65.8% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $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 67.9%

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

   1. expm1-log1p-u 67.9%

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

   2. expm1-udef 67.9%

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

   3. add-sqr-sqrt 67.9%

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

   4. sqrt-unprod 67.9%

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

   5. frac-times 67.9%

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

   6. metadata-eval 67.9%

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

   7. add-sqr-sqrt 67.6%

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

   8. metadata-eval 67.6%

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

6. Applied egg-rr 67.6%

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

   1. expm1-def 67.6%

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

   2. expm1-log1p 68.0%

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

8. Simplified 68.0%

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

9. Final simplification 68.0%

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

Alternative 8: 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 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 58.7%

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

   1. unpow2 58.7%

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

   2. associate-*l* 58.7%

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

   3. associate-*r/ 58.7%

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

   4. associate-/l* 58.7%

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

6. Simplified 58.7%

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

7. Taylor expanded in th around 0 41.2%

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

8. Final simplification 41.2%

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

Reproduce

herbie shell --seed 2023207 
(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))))