Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 10.5s

Alternatives: 9

Speedup: 2.0×

• 44.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. frac-add 99.3%

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

   5. fma-def 99.3%

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

   6. add-sqr-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

   1. fma-udef 99.7%

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

   2. *-commutative 99.7%

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

   3. distribute-rgt-out 99.7%

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

   4. distribute-lft-out 99.7%

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

   5. +-commutative 99.7%

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

   6. unpow2 99.7%

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

   7. unpow2 99.7%

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

   8. *-commutative 99.7%

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

   9. unpow2 99.7%

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

   10. unpow2 99.7%

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

   11. fma-def 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] * N[Sqrt[0.5], $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. clear-num 99.5%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.7%

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

   5. metadata-eval 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in th around inf 99.7%

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

   1. *-commutative 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 56.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 * N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $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 50.2%

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

   1. unpow2 50.2%

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

   2. associate-*l/ 50.2%

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

6. Simplified 50.2%

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

7. Taylor expanded in a2 around 0 50.2%

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

   1. unpow2 50.2%

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

   2. associate-*r/ 50.2%

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

   3. associate-*l* 50.1%

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

9. Simplified 50.1%

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

10. Final simplification 50.1%

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

Alternative 4: 56.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.7%

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

   5. metadata-eval 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in a2 around inf 50.2%

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

   1. unpow2 50.2%

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

   2. associate-*r* 50.2%

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

   3. *-commutative 50.2%

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

   4. associate-*l* 50.2%

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

   5. *-commutative 50.2%

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

   6. associate-*r* 50.2%

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

8. Simplified 50.2%

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

9. Final simplification 50.2%

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

Alternative 5: 56.7% accurate, 2.0× speedup

Math

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

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(Float64(a2 * 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[(N[(a2 * a2), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. +-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 50.2%

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

   1. unpow2 50.2%

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

   2. associate-*l/ 50.2%

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

6. Simplified 50.2%

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

   1. associate-*l/ 50.2%

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

8. Applied egg-rr 50.2%

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

   1. div-inv 50.1%

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

   2. *-commutative 50.1%

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

   3. pow1/2 50.1%

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

   4. pow-flip 50.2%

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

   5. metadata-eval 50.2%

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

   6. add-sqr-sqrt 50.0%

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

   7. sqrt-unprod 50.2%

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

   8. pow-prod-up 50.2%

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

   9. metadata-eval 50.2%

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

   10. metadata-eval 50.2%

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

10. Applied egg-rr 50.2%

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

11. Final simplification 50.2%

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

Alternative 6: 66.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return ((a2 * a2) + (a1 * a1)) * pow(0.25, 0.25);
}

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)) * (0.25d0 ** 0.25d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[Power[0.25, 0.25], $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 99.7%

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

   3. times-frac 99.3%

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

   4. pow1/2 99.3%

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

   5. sqrt-pow1 99.3%

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

   6. metadata-eval 99.3%

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

   7. pow1/2 99.3%

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

   8. sqrt-pow1 99.3%

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

   9. metadata-eval 99.3%

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

5. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{0.25}}\right)} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]
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. Taylor expanded in th around 0 67.6%

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

9. Final simplification 67.6%

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

Alternative 7: 66.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $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 67.5%

   \[\leadsto \color{blue}{\frac{{a1}^{2} + {a2}^{2}}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. unpow2 67.5%

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

   2. unpow2 67.5%

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

6. Simplified 67.5%

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

7. Final simplification 67.5%

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

Alternative 8: 39.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-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 67.5%

   \[\leadsto \color{blue}{\frac{{a1}^{2} + {a2}^{2}}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. unpow2 67.5%

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

   2. unpow2 67.5%

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

6. Simplified 67.5%

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

7. Taylor expanded in a1 around 0 36.4%

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

   1. unpow2 36.4%

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

   2. associate-*r/ 36.4%

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

9. Simplified 36.4%

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

    1. clear-num 36.4%

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

    2. associate-/r/ 36.4%

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

    3. pow1/2 36.4%

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

    4. pow-flip 36.4%

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

    5. metadata-eval 36.4%

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

    6. add-sqr-sqrt 36.3%

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

    7. sqrt-unprod 36.4%

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

    8. pow-prod-up 36.4%

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

    9. metadata-eval 36.4%

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

    10. metadata-eval 36.4%

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

11. Applied egg-rr 36.4%

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

12. Final simplification 36.4%

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

Alternative 9: 39.6% 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 th around 0 67.5%

   \[\leadsto \color{blue}{\frac{{a1}^{2} + {a2}^{2}}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. unpow2 67.5%

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

   2. unpow2 67.5%

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

6. Simplified 67.5%

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

7. Taylor expanded in a1 around 0 36.4%

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

   1. unpow2 36.4%

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

   2. associate-*r/ 36.4%

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

9. Simplified 36.4%

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

    1. clear-num 36.4%

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

    2. associate-/r/ 36.4%

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

    3. pow1/2 36.4%

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

    4. pow-flip 36.4%

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

    5. metadata-eval 36.4%

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

    6. add-sqr-sqrt 36.3%

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

    7. sqrt-unprod 36.4%

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

    8. pow-prod-up 36.4%

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

    9. metadata-eval 36.4%

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

    10. metadata-eval 36.4%

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

11. Applied egg-rr 36.4%

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

12. Taylor expanded in a2 around 0 36.4%

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

    1. unpow2 36.4%

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

    2. *-commutative 36.4%

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

14. Simplified 36.4%

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

15. Final simplification 36.4%

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

Reproduce

herbie shell --seed 2023293 
(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))))