Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 10.5s

Alternatives: 7

Speedup: 2.0×

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a1, a2, th)
	tmp = (cos(th) * ((a1 * a1) + (a2 * a2))) / sqrt(2.0);
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[2.0], $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 N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\cos th, \left(a1 \cdot a1 + a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(a1 \cdot a1 + a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right)\right), \left(\sqrt{2}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right)\right), \left(\sqrt{2}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right)\right), \left(\sqrt{2}\right)\right) \]

   8. sqrt-lowering-sqrt.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

6. Applied egg-rr 99.7%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 3: 57.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

3. Simplified 99.6%

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

5. Taylor expanded in a1 around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({a2}^{2} \cdot \cos th\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(a2 \cdot a2\right) \cdot \cos th\right), \left(\sqrt{2}\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(a2 \cdot \left(a2 \cdot \cos th\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(a2 \cdot \left(\cos th \cdot a2\right)\right), \left(\sqrt{2}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, \left(\cos th \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, \left(a2 \cdot \cos th\right)\right), \left(\sqrt{2}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \cos th\right)\right), \left(\sqrt{2}\right)\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \mathsf{cos.f64}\left(th\right)\right)\right), \left(\sqrt{2}\right)\right) \]

   9. sqrt-lowering-sqrt.f64 58.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \mathsf{cos.f64}\left(th\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

7. Simplified 58.5%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. associate-*l* N/A

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

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a2}{\sqrt{2}} \cdot a2\right)}\right) \]

   7. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\color{blue}{\frac{a2}{\sqrt{2}}} \cdot a2\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{*.f64}\left(\left(\frac{a2}{\sqrt{2}}\right), \color{blue}{a2}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(a2, \left(\sqrt{2}\right)\right), a2\right)\right) \]

   10. sqrt-lowering-sqrt.f64 58.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(a2, \mathsf{sqrt.f64}\left(2\right)\right), a2\right)\right) \]

9. Applied egg-rr 58.5%

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

10. Final simplification 58.5%

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

Alternative 4: 64.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 4.8 \cdot 10^{+66}:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 3.6 \cdot 10^{+176}:\\ \;\;\;\;\frac{a2 \cdot a2}{0 - \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 4.8e+66)
   (/ (+ (* a1 a1) (* a2 a2)) (sqrt 2.0))
   (if (<= th 3.6e+176)
     (/ (* a2 a2) (- 0.0 (sqrt 2.0)))
     (/ (* a2 a2) (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 4.8e+66) {
		tmp = ((a1 * a1) + (a2 * a2)) / sqrt(2.0);
	} else if (th <= 3.6e+176) {
		tmp = (a2 * a2) / (0.0 - sqrt(2.0));
	} else {
		tmp = (a2 * a2) / 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 (th <= 4.8d+66) then
        tmp = ((a1 * a1) + (a2 * a2)) / sqrt(2.0d0)
    else if (th <= 3.6d+176) then
        tmp = (a2 * a2) / (0.0d0 - sqrt(2.0d0))
    else
        tmp = (a2 * a2) / sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 4.8e+66) {
		tmp = ((a1 * a1) + (a2 * a2)) / Math.sqrt(2.0);
	} else if (th <= 3.6e+176) {
		tmp = (a2 * a2) / (0.0 - Math.sqrt(2.0));
	} else {
		tmp = (a2 * a2) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 4.8e+66:
		tmp = ((a1 * a1) + (a2 * a2)) / math.sqrt(2.0)
	elif th <= 3.6e+176:
		tmp = (a2 * a2) / (0.0 - math.sqrt(2.0))
	else:
		tmp = (a2 * a2) / math.sqrt(2.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 4.8e+66)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) / sqrt(2.0));
	elseif (th <= 3.6e+176)
		tmp = Float64(Float64(a2 * a2) / Float64(0.0 - sqrt(2.0)));
	else
		tmp = Float64(Float64(a2 * a2) / sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 4.8e+66)
		tmp = ((a1 * a1) + (a2 * a2)) / sqrt(2.0);
	elseif (th <= 3.6e+176)
		tmp = (a2 * a2) / (0.0 - sqrt(2.0));
	else
		tmp = (a2 * a2) / sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 4.8e+66], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 3.6e+176], N[(N[(a2 * a2), $MachinePrecision] / N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 4.8000000000000003e66

   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 N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in th around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({a1}^{2} + {a2}^{2}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({a1}^{2}\right), \left({a2}^{2}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left({a2}^{2}\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left({a2}^{2}\right)\right), \left(\sqrt{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 76.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   7. Simplified 76.8%

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

   if 4.8000000000000003e66 < th < 3.59999999999999991e176

   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 N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 99.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   3. Simplified 99.4%

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

   5. Taylor expanded in th around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({a1}^{2} + {a2}^{2}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({a1}^{2}\right), \left({a2}^{2}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left({a2}^{2}\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left({a2}^{2}\right)\right), \left(\sqrt{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 23.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   7. Simplified 23.6%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)}{a1 \cdot a1 - a2 \cdot a2}\right), \mathsf{sqrt.f64}\left(\color{blue}{2}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{a1 \cdot a1 - a2 \cdot a2}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)}}\right), \mathsf{sqrt.f64}\left(\color{blue}{2}\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{a1 \cdot a1 - a2 \cdot a2} \cdot \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{a1 \cdot a1 - a2 \cdot a2}\right), \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{2}\right)\right) \]

      5. difference-of-squares N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\left(a1 + a2\right) \cdot \left(a1 - a2\right)}\right), \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{a1 + a2}}{a1 - a2}\right), \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{a1 + a2}\right), \left(a1 - a2\right)\right), \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(a1 + a2\right)\right), \left(a1 - a2\right)\right), \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \left(a1 - a2\right)\right), \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      11. difference-of-squares N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \left(\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(a1 \cdot a1 - a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \left(a1 \cdot a1 - a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(a1 \cdot a1 - a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(a1 \cdot a1 - a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(a1 \cdot a1 - a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      16. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{\_.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      18. *-lowering-*.f64 4.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   9. Applied egg-rr 4.0%

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

   10. Taylor expanded in a1 around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \color{blue}{\left(-1 \cdot {a2}^{2}\right)}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\mathsf{neg}\left({a2}^{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(0 - {a2}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{\_.f64}\left(0, \left({a2}^{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{\_.f64}\left(0, \left(a2 \cdot a2\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

       5. *-lowering-*.f64 1.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a2, a2\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   12. Simplified 1.9%

       \[\leadsto \frac{\frac{\frac{1}{a1 + a2}}{a1 - a2} \cdot \left(\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \color{blue}{\left(0 - a2 \cdot a2\right)}\right)}{\sqrt{2}} \]

   13. Taylor expanded in a1 around inf

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

       1. associate-*r/ N/A

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

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot {a2}^{2}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left({a2}^{2}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

       4. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(0 - {a2}^{2}\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left({a2}^{2}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right) \]

       8. sqrt-lowering-sqrt.f64 35.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   15. Simplified 35.6%

       \[\leadsto \color{blue}{\frac{0 - a2 \cdot a2}{\sqrt{2}}} \]

   if 3.59999999999999991e176 < th

   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 N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({a1}^{2} + {a2}^{2}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({a1}^{2}\right), \left({a2}^{2}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left({a2}^{2}\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left({a2}^{2}\right)\right), \left(\sqrt{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 48.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   7. Simplified 48.8%

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

   8. Taylor expanded in a1 around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({a2}^{2}\right)}, \mathsf{sqrt.f64}\left(2\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a2 \cdot a2\right), \mathsf{sqrt.f64}\left(\color{blue}{2}\right)\right) \]

      2. *-lowering-*.f64 16.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \mathsf{sqrt.f64}\left(\color{blue}{2}\right)\right) \]

   10. Simplified 16.4%

       \[\leadsto \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 4.8 \cdot 10^{+66}:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 3.6 \cdot 10^{+176}:\\ \;\;\;\;\frac{a2 \cdot a2}{0 - \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{if}\;th \leq 4.8 \cdot 10^{+66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;th \leq 3.6 \cdot 10^{+176}:\\ \;\;\;\;\frac{a2 \cdot a2}{0 - \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (* a2 a2) (sqrt 2.0))))
   (if (<= th 4.8e+66)
     t_1
     (if (<= th 3.6e+176) (/ (* a2 a2) (- 0.0 (sqrt 2.0))) t_1))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a2 * a2) / sqrt(2.0);
	double tmp;
	if (th <= 4.8e+66) {
		tmp = t_1;
	} else if (th <= 3.6e+176) {
		tmp = (a2 * a2) / (0.0 - sqrt(2.0));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (a2 * a2) / sqrt(2.0d0)
    if (th <= 4.8d+66) then
        tmp = t_1
    else if (th <= 3.6d+176) then
        tmp = (a2 * a2) / (0.0d0 - sqrt(2.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a2 * a2) / Math.sqrt(2.0);
	double tmp;
	if (th <= 4.8e+66) {
		tmp = t_1;
	} else if (th <= 3.6e+176) {
		tmp = (a2 * a2) / (0.0 - Math.sqrt(2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a2 * a2) / math.sqrt(2.0)
	tmp = 0
	if th <= 4.8e+66:
		tmp = t_1
	elif th <= 3.6e+176:
		tmp = (a2 * a2) / (0.0 - math.sqrt(2.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a2 * a2) / sqrt(2.0))
	tmp = 0.0
	if (th <= 4.8e+66)
		tmp = t_1;
	elseif (th <= 3.6e+176)
		tmp = Float64(Float64(a2 * a2) / Float64(0.0 - sqrt(2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a2 * a2) / sqrt(2.0);
	tmp = 0.0;
	if (th <= 4.8e+66)
		tmp = t_1;
	elseif (th <= 3.6e+176)
		tmp = (a2 * a2) / (0.0 - sqrt(2.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[th, 4.8e+66], t$95$1, If[LessEqual[th, 3.6e+176], N[(N[(a2 * a2), $MachinePrecision] / N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if th < 4.8000000000000003e66 or 3.59999999999999991e176 < th

   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 N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in th around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({a1}^{2} + {a2}^{2}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({a1}^{2}\right), \left({a2}^{2}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left({a2}^{2}\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left({a2}^{2}\right)\right), \left(\sqrt{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 73.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   7. Simplified 73.2%

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

   8. Taylor expanded in a1 around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({a2}^{2}\right)}, \mathsf{sqrt.f64}\left(2\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a2 \cdot a2\right), \mathsf{sqrt.f64}\left(\color{blue}{2}\right)\right) \]

      2. *-lowering-*.f64 41.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \mathsf{sqrt.f64}\left(\color{blue}{2}\right)\right) \]

   10. Simplified 41.8%

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

   if 4.8000000000000003e66 < th < 3.59999999999999991e176

   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 N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 99.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   3. Simplified 99.4%

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

   5. Taylor expanded in th around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({a1}^{2} + {a2}^{2}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({a1}^{2}\right), \left({a2}^{2}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left({a2}^{2}\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left({a2}^{2}\right)\right), \left(\sqrt{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 23.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   7. Simplified 23.6%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)}{a1 \cdot a1 - a2 \cdot a2}\right), \mathsf{sqrt.f64}\left(\color{blue}{2}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{a1 \cdot a1 - a2 \cdot a2}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)}}\right), \mathsf{sqrt.f64}\left(\color{blue}{2}\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{a1 \cdot a1 - a2 \cdot a2} \cdot \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{a1 \cdot a1 - a2 \cdot a2}\right), \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{2}\right)\right) \]

      5. difference-of-squares N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\left(a1 + a2\right) \cdot \left(a1 - a2\right)}\right), \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{a1 + a2}}{a1 - a2}\right), \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{a1 + a2}\right), \left(a1 - a2\right)\right), \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(a1 + a2\right)\right), \left(a1 - a2\right)\right), \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \left(a1 - a2\right)\right), \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) - \left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      11. difference-of-squares N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \left(\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(a1 \cdot a1 - a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \left(a1 \cdot a1 - a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(a1 \cdot a1 - a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(a1 \cdot a1 - a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(a1 \cdot a1 - a2 \cdot a2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      16. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{\_.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      18. *-lowering-*.f64 4.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   9. Applied egg-rr 4.0%

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

   10. Taylor expanded in a1 around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \color{blue}{\left(-1 \cdot {a2}^{2}\right)}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\mathsf{neg}\left({a2}^{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(0 - {a2}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{\_.f64}\left(0, \left({a2}^{2}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{\_.f64}\left(0, \left(a2 \cdot a2\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

       5. *-lowering-*.f64 1.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(a1, a2\right)\right), \mathsf{\_.f64}\left(a1, a2\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a2, a2\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   12. Simplified 1.9%

       \[\leadsto \frac{\frac{\frac{1}{a1 + a2}}{a1 - a2} \cdot \left(\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \color{blue}{\left(0 - a2 \cdot a2\right)}\right)}{\sqrt{2}} \]

   13. Taylor expanded in a1 around inf

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

       1. associate-*r/ N/A

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

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot {a2}^{2}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left({a2}^{2}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

       4. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(0 - {a2}^{2}\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left({a2}^{2}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right) \]

       8. sqrt-lowering-sqrt.f64 35.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   15. Simplified 35.6%

       \[\leadsto \color{blue}{\frac{0 - a2 \cdot a2}{\sqrt{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 4.8 \cdot 10^{+66}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 3.6 \cdot 10^{+176}:\\ \;\;\;\;\frac{a2 \cdot a2}{0 - \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.0% accurate, 4.0× speedup

Math

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

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(Float64(a2 * 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[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $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 N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

3. Simplified 99.6%

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

5. Taylor expanded in th around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({a1}^{2} + {a2}^{2}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({a1}^{2}\right), \left({a2}^{2}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left({a2}^{2}\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left({a2}^{2}\right)\right), \left(\sqrt{2}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right) \]

   7. sqrt-lowering-sqrt.f64 68.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

7. Simplified 68.9%

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

8. Taylor expanded in a1 around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({a2}^{2}\right)}, \mathsf{sqrt.f64}\left(2\right)\right) \]
9. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(a2 \cdot a2\right), \mathsf{sqrt.f64}\left(\color{blue}{2}\right)\right) \]

   2. *-lowering-*.f64 39.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a2, a2\right), \mathsf{sqrt.f64}\left(\color{blue}{2}\right)\right) \]

10. Simplified 39.6%

    \[\leadsto \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} \]
11. Add Preprocessing

Alternative 7: 39.5% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a1 * N[(a1 / 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 N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos th, \color{blue}{\left(\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\right)}\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \left(\frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\left(a1 \cdot a1 + a2 \cdot a2\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

3. Simplified 99.6%

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

5. Taylor expanded in th around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({a1}^{2} + {a2}^{2}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({a1}^{2}\right), \left({a2}^{2}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a1 \cdot a1\right), \left({a2}^{2}\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left({a2}^{2}\right)\right), \left(\sqrt{2}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \left(a2 \cdot a2\right)\right), \left(\sqrt{2}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \left(\sqrt{2}\right)\right) \]

   7. sqrt-lowering-sqrt.f64 68.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, a2\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

7. Simplified 68.9%

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

8. Taylor expanded in a1 around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({a1}^{2}\right)}, \mathsf{sqrt.f64}\left(2\right)\right) \]
9. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(a1 \cdot a1\right), \mathsf{sqrt.f64}\left(\color{blue}{2}\right)\right) \]

   2. *-lowering-*.f64 40.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{sqrt.f64}\left(\color{blue}{2}\right)\right) \]

10. Simplified 40.7%

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

    1. associate-/l* N/A

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

    2. *-commutative N/A

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

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{a1}{\sqrt{2}}\right), \color{blue}{a1}\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(a1, \left(\sqrt{2}\right)\right), a1\right) \]

    5. sqrt-lowering-sqrt.f64 40.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(a1, \mathsf{sqrt.f64}\left(2\right)\right), a1\right) \]

12. Applied egg-rr 40.7%

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

13. Final simplification 40.7%

    \[\leadsto a1 \cdot \frac{a1}{\sqrt{2}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(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))))