Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.5%

Time: 10.1s

Alternatives: 13

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: 99.5% 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: 99.5% 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]]

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. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $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. 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. Taylor expanded in th around inf 99.6%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left(\left({a2}^{2} + {a1}^{2}\right) \cdot \cos th\right)} \]
7. Step-by-step derivation

   1. *-commutative 99.6%

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

   2. *-commutative 99.6%

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

   3. unpow2 99.6%

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

   4. unpow2 99.6%

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

   5. +-commutative 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $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. Final simplification 99.6%

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

Alternative 4: 57.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 * N[(N[Cos[th], $MachinePrecision] * N[Sqrt[0.5], $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 60.2%

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

   1. unpow2 60.2%

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

   2. associate-*l* 60.2%

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

6. Simplified 60.2%

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

   1. div-inv 60.2%

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

   2. associate-*r* 60.2%

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

   3. associate-*r* 60.2%

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

   4. div-inv 60.2%

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

   5. associate-*l* 60.2%

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

   6. div-inv 60.2%

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

   7. add-sqr-sqrt 60.2%

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

   8. sqrt-unprod 60.2%

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

   9. frac-times 60.2%

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

   10. metadata-eval 60.2%

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

   11. add-sqr-sqrt 60.2%

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

   12. metadata-eval 60.2%

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

8. Applied egg-rr 60.2%

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

9. Final simplification 60.2%

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

Alternative 5: 57.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   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 60.2%

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

   1. unpow2 60.2%

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

   2. associate-*l* 60.2%

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

6. Simplified 60.2%

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

   1. div-inv 60.2%

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

   2. associate-*l* 60.2%

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

   3. *-commutative 60.2%

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

   4. add-sqr-sqrt 60.2%

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

   5. sqrt-unprod 60.2%

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

   6. frac-times 60.2%

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

   7. metadata-eval 60.2%

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

   8. add-sqr-sqrt 60.2%

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

   9. metadata-eval 60.2%

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

8. Applied egg-rr 60.2%

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

9. Final simplification 60.2%

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

Alternative 6: 57.2% 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 60.2%

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

   1. unpow2 60.2%

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

   2. associate-*l* 60.2%

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

   3. associate-*r/ 60.2%

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

   4. associate-/l* 60.2%

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

6. Simplified 60.2%

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

7. Final simplification 60.2%

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

Alternative 7: 57.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   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 60.2%

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

   1. unpow2 60.2%

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

   2. associate-*l* 60.2%

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

6. Simplified 60.2%

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

   1. div-inv 60.2%

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

   2. *-commutative 60.2%

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

   3. associate-*l* 60.2%

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

   4. *-commutative 60.2%

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

   5. div-inv 60.3%

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

   6. associate-*r* 60.2%

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

   7. clear-num 60.2%

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

   8. div-inv 60.2%

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

   9. *-commutative 60.2%

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

   10. associate-/r/ 60.2%

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

   11. associate-*l* 60.3%

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

   12. *-commutative 60.3%

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

8. Applied egg-rr 60.3%

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

9. Final simplification 60.3%

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

Alternative 8: 45.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a1 \cdot a1 \leq 2 \cdot 10^{-79}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a2 \cdot a2\right) \cdot \left(-0.5 \cdot \left(th \cdot th\right) + 1\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (* a1 a1) 2e-79)
   (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5))
   (/ (* (* a2 a2) (+ (* -0.5 (* th th)) 1.0)) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((a1 * a1) <= 2e-79) {
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	} else {
		tmp = ((a2 * a2) * ((-0.5 * (th * th)) + 1.0)) / sqrt(2.0);
	}
	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-79) then
        tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
    else
        tmp = ((a2 * a2) * (((-0.5d0) * (th * th)) + 1.0d0)) / sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((a1 * a1) <= 2e-79) {
		tmp = ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
	} else {
		tmp = ((a2 * a2) * ((-0.5 * (th * th)) + 1.0)) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (a1 * a1) <= 2e-79:
		tmp = ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)
	else:
		tmp = ((a2 * a2) * ((-0.5 * (th * th)) + 1.0)) / math.sqrt(2.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (Float64(a1 * a1) <= 2e-79)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5));
	else
		tmp = Float64(Float64(Float64(a2 * a2) * Float64(Float64(-0.5 * Float64(th * th)) + 1.0)) / sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((a1 * a1) <= 2e-79)
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	else
		tmp = ((a2 * a2) * ((-0.5 * (th * th)) + 1.0)) / sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[(a1 * a1), $MachinePrecision], 2e-79], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[(N[(a2 * a2), $MachinePrecision] * N[(N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a1 a1) < 2e-79

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.4%

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

      3. pow1/2 99.4%

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

      4. pow-flip 99.5%

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

      5. metadata-eval 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in th around 0 69.8%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left({a2}^{2} + {a1}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 69.8%

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

      2. unpow2 69.8%

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

      3. unpow2 69.8%

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

      4. +-commutative 69.8%

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

   8. Simplified 69.8%

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

   if 2e-79 < (*.f64 a1 a1)

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around 0 41.6%

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

      1. unpow2 41.6%

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

      2. associate-*l* 41.6%

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

   6. Simplified 41.6%

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

   7. Taylor expanded in th around 0 13.6%

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

      1. unpow2 13.6%

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

      2. +-commutative 13.6%

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

      3. unpow2 13.6%

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

      4. associate-*r* 13.6%

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

      5. *-lft-identity 13.6%

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

      6. distribute-rgt-out 37.6%

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

      7. unpow2 37.6%

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

   9. Simplified 37.6%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a1 \leq 2 \cdot 10^{-79}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a2 \cdot a2\right) \cdot \left(-0.5 \cdot \left(th \cdot th\right) + 1\right)}{\sqrt{2}}\\ \end{array} \]

Alternative 9: 63.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;th \leq 1.4 \cdot 10^{+141}:\\ \;\;\;\;\frac{-0.5}{\frac{\sqrt{2}}{th \cdot \left(th \cdot \left(a2 \cdot a2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 1.55)
   (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5))
   (if (<= th 1.4e+141)
     (/ -0.5 (/ (sqrt 2.0) (* th (* th (* a2 a2)))))
     (* a2 (* a2 (sqrt 0.5))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	} else if (th <= 1.4e+141) {
		tmp = -0.5 / (sqrt(2.0) / (th * (th * (a2 * a2))));
	} else {
		tmp = a2 * (a2 * sqrt(0.5));
	}
	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 (th <= 1.55d0) then
        tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
    else if (th <= 1.4d+141) then
        tmp = (-0.5d0) / (sqrt(2.0d0) / (th * (th * (a2 * a2))))
    else
        tmp = a2 * (a2 * sqrt(0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
	} else if (th <= 1.4e+141) {
		tmp = -0.5 / (Math.sqrt(2.0) / (th * (th * (a2 * a2))));
	} else {
		tmp = a2 * (a2 * Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 1.55:
		tmp = ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)
	elif th <= 1.4e+141:
		tmp = -0.5 / (math.sqrt(2.0) / (th * (th * (a2 * a2))))
	else:
		tmp = a2 * (a2 * math.sqrt(0.5))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 1.55)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5));
	elseif (th <= 1.4e+141)
		tmp = Float64(-0.5 / Float64(sqrt(2.0) / Float64(th * Float64(th * Float64(a2 * a2)))));
	else
		tmp = Float64(a2 * Float64(a2 * sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 1.55)
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	elseif (th <= 1.4e+141)
		tmp = -0.5 / (sqrt(2.0) / (th * (th * (a2 * a2))));
	else
		tmp = a2 * (a2 * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 1.55], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 1.4e+141], N[(-0.5 / N[(N[Sqrt[2.0], $MachinePrecision] / N[(th * N[(th * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 1.55000000000000004

   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/ 99.5%

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

      3. pow1/2 99.5%

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

      4. pow-flip 99.6%

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

      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. Taylor expanded in th around 0 78.0%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left({a2}^{2} + {a1}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 78.0%

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

      2. unpow2 78.0%

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

      3. unpow2 78.0%

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

      4. +-commutative 78.0%

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

   8. Simplified 78.0%

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

   if 1.55000000000000004 < th < 1.39999999999999996e141

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around 0 73.8%

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

      1. unpow2 73.8%

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

      2. associate-*l* 73.7%

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

   6. Simplified 73.7%

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

   7. Taylor expanded in th around 0 14.1%

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

      1. unpow2 14.1%

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

      2. +-commutative 14.1%

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

      3. unpow2 14.1%

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

      4. associate-*r* 14.1%

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

      5. *-lft-identity 14.1%

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

      6. distribute-rgt-out 41.0%

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

      7. unpow2 41.0%

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

   9. Simplified 41.0%

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

   10. Taylor expanded in th around inf 41.0%

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

       1. unpow2 41.0%

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

       2. associate-*r/ 41.0%

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

       3. associate-/l* 41.0%

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

       4. *-commutative 41.0%

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

       5. unpow2 41.0%

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

       6. associate-*l* 41.0%

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

   12. Simplified 41.0%

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

   if 1.39999999999999996e141 < th

   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%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around 0 58.3%

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

      1. unpow2 58.3%

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

      2. associate-*l* 58.2%

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

   6. Simplified 58.2%

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

      1. div-inv 58.1%

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

      2. associate-*l* 58.3%

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

      3. *-commutative 58.3%

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

      4. add-sqr-sqrt 58.3%

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

      5. sqrt-unprod 58.3%

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

      6. frac-times 58.3%

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

      7. metadata-eval 58.3%

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

      8. add-sqr-sqrt 58.2%

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

      9. metadata-eval 58.2%

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

   8. Applied egg-rr 58.2%

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

   9. Taylor expanded in th around 0 25.1%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;th \leq 1.4 \cdot 10^{+141}:\\ \;\;\;\;\frac{-0.5}{\frac{\sqrt{2}}{th \cdot \left(th \cdot \left(a2 \cdot a2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 10: 63.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;th \leq 1.4 \cdot 10^{+141}:\\ \;\;\;\;\frac{\left(a2 \cdot a2\right) \cdot \left(th \cdot \left(th \cdot -0.5\right)\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 1.55)
   (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5))
   (if (<= th 1.4e+141)
     (/ (* (* a2 a2) (* th (* th -0.5))) (sqrt 2.0))
     (* a2 (* a2 (sqrt 0.5))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	} else if (th <= 1.4e+141) {
		tmp = ((a2 * a2) * (th * (th * -0.5))) / sqrt(2.0);
	} else {
		tmp = a2 * (a2 * sqrt(0.5));
	}
	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 (th <= 1.55d0) then
        tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
    else if (th <= 1.4d+141) then
        tmp = ((a2 * a2) * (th * (th * (-0.5d0)))) / sqrt(2.0d0)
    else
        tmp = a2 * (a2 * sqrt(0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
	} else if (th <= 1.4e+141) {
		tmp = ((a2 * a2) * (th * (th * -0.5))) / Math.sqrt(2.0);
	} else {
		tmp = a2 * (a2 * Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 1.55:
		tmp = ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)
	elif th <= 1.4e+141:
		tmp = ((a2 * a2) * (th * (th * -0.5))) / math.sqrt(2.0)
	else:
		tmp = a2 * (a2 * math.sqrt(0.5))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 1.55)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5));
	elseif (th <= 1.4e+141)
		tmp = Float64(Float64(Float64(a2 * a2) * Float64(th * Float64(th * -0.5))) / sqrt(2.0));
	else
		tmp = Float64(a2 * Float64(a2 * sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 1.55)
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	elseif (th <= 1.4e+141)
		tmp = ((a2 * a2) * (th * (th * -0.5))) / sqrt(2.0);
	else
		tmp = a2 * (a2 * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 1.55], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 1.4e+141], N[(N[(N[(a2 * a2), $MachinePrecision] * N[(th * N[(th * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 1.55000000000000004

   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/ 99.5%

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

      3. pow1/2 99.5%

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

      4. pow-flip 99.6%

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

      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. Taylor expanded in th around 0 78.0%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left({a2}^{2} + {a1}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 78.0%

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

      2. unpow2 78.0%

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

      3. unpow2 78.0%

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

      4. +-commutative 78.0%

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

   8. Simplified 78.0%

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

   if 1.55000000000000004 < th < 1.39999999999999996e141

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around 0 73.8%

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

      1. unpow2 73.8%

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

      2. associate-*l* 73.7%

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

   6. Simplified 73.7%

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

   7. Taylor expanded in th around 0 14.1%

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

      1. unpow2 14.1%

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

      2. +-commutative 14.1%

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

      3. unpow2 14.1%

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

      4. associate-*r* 14.1%

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

      5. *-lft-identity 14.1%

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

      6. distribute-rgt-out 41.0%

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

      7. unpow2 41.0%

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

   9. Simplified 41.0%

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

   10. Taylor expanded in th around inf 41.0%

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

       1. unpow2 41.0%

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

       2. associate-*r* 41.0%

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

       3. unpow2 41.0%

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

       4. associate-*r* 41.0%

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

   12. Simplified 41.0%

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

   if 1.39999999999999996e141 < th

   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%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around 0 58.3%

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

      1. unpow2 58.3%

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

      2. associate-*l* 58.2%

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

   6. Simplified 58.2%

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

      1. div-inv 58.1%

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

      2. associate-*l* 58.3%

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

      3. *-commutative 58.3%

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

      4. add-sqr-sqrt 58.3%

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

      5. sqrt-unprod 58.3%

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

      6. frac-times 58.3%

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

      7. metadata-eval 58.3%

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

      8. add-sqr-sqrt 58.2%

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

      9. metadata-eval 58.2%

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

   8. Applied egg-rr 58.2%

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

   9. Taylor expanded in th around 0 25.1%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;th \leq 1.4 \cdot 10^{+141}:\\ \;\;\;\;\frac{\left(a2 \cdot a2\right) \cdot \left(th \cdot \left(th \cdot -0.5\right)\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 11: 66.6% 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. 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. Taylor expanded in th around 0 67.4%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left({a2}^{2} + {a1}^{2}\right)} \]
7. Step-by-step derivation

   1. *-commutative 67.4%

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

   2. unpow2 67.4%

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

   3. unpow2 67.4%

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

   4. +-commutative 67.4%

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

8. Simplified 67.4%

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

9. Final simplification 67.4%

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

Alternative 12: 40.0% 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 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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 60.2%

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

   1. unpow2 60.2%

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

   2. associate-*l* 60.2%

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

6. Simplified 60.2%

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

   1. div-inv 60.2%

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

   2. associate-*l* 60.2%

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

   3. *-commutative 60.2%

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

   4. add-sqr-sqrt 60.2%

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

   5. sqrt-unprod 60.2%

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

   6. frac-times 60.2%

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

   7. metadata-eval 60.2%

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

   8. add-sqr-sqrt 60.2%

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

   9. metadata-eval 60.2%

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

8. Applied egg-rr 60.2%

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

9. Taylor expanded in th around 0 41.1%

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

10. Final simplification 41.1%

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

Alternative 13: 40.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 67.4%

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

5. Taylor expanded in a1 around 0 41.1%

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

   1. unpow2 41.1%

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

   2. associate-*r/ 41.1%

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

7. Simplified 41.1%

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

8. Final simplification 41.1%

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

Reproduce

herbie shell --seed 2023255 
(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))))