Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 10.8s

Alternatives: 12

Speedup: 2.0×

• 44.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 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. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

   9. sqrt-lowering-sqrt.f64 99.6%

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

3. Simplified 99.6%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing
3. 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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

   8. *-lowering-*.f64 99.6%

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 56.6% accurate, 2.0× speedup

Math

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

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(Float64(cos(th) * Float64(a2 * 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[(N[Cos[th], $MachinePrecision] * N[(a2 * a2), $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. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

   9. sqrt-lowering-sqrt.f64 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a1 around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 63.3%

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

7. Simplified 63.3%

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

Alternative 4: 56.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(a2 * N[(N[Cos[th], $MachinePrecision] * a2), $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. Add Preprocessing

3. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]
4. 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 63.3%

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

5. Simplified 63.3%

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

6. Final simplification 63.3%

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

Alternative 5: 56.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(a2 * N[(N[Cos[th], $MachinePrecision] * 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. Add Preprocessing
3. 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. clear-num N/A

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

   4. associate-/r/ N/A

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

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

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

   6. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{2}^{\frac{1}{2}}}\right), \left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

   7. pow-flip N/A

      \[\leadsto \mathsf{*.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\color{blue}{\cos th} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{\cos th} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

   9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

   14. *-lowering-*.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right)\right)\right) \]

4. Applied egg-rr 99.6%

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

5. Taylor expanded in a1 around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left({a2}^{2} \cdot \cos th\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\color{blue}{{a2}^{2}} \cdot \cos th\right)\right) \]

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

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

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

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \color{blue}{\cos th}\right)\right)\right) \]

   9. cos-lowering-cos.f64 63.3%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \mathsf{cos.f64}\left(th\right)\right)\right)\right) \]

7. Simplified 63.3%

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

8. Final simplification 63.3%

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

Alternative 6: 67.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;a2 \leq 1.6 \cdot 10^{+225}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{t\_1}{\sqrt{2}}}}\\ \mathbf{elif}\;a2 \leq 5 \cdot 10^{+290}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(th \cdot th\right)\right) \cdot \left(t\_1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= a2 1.6e+225)
     (/ 1.0 (/ 1.0 (/ t_1 (sqrt 2.0))))
     (if (<= a2 5e+290)
       (* (+ 1.0 (* -0.5 (* th th))) (* t_1 (sqrt 0.5)))
       (* a2 (* a2 (sqrt 0.5)))))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (a2 <= 1.6e+225) {
		tmp = 1.0 / (1.0 / (t_1 / sqrt(2.0)));
	} else if (a2 <= 5e+290) {
		tmp = (1.0 + (-0.5 * (th * th))) * (t_1 * sqrt(0.5));
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (a2 <= 1.6d+225) then
        tmp = 1.0d0 / (1.0d0 / (t_1 / sqrt(2.0d0)))
    else if (a2 <= 5d+290) then
        tmp = (1.0d0 + ((-0.5d0) * (th * th))) * (t_1 * sqrt(0.5d0))
    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 t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (a2 <= 1.6e+225) {
		tmp = 1.0 / (1.0 / (t_1 / Math.sqrt(2.0)));
	} else if (a2 <= 5e+290) {
		tmp = (1.0 + (-0.5 * (th * th))) * (t_1 * Math.sqrt(0.5));
	} else {
		tmp = a2 * (a2 * Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if a2 <= 1.6e+225:
		tmp = 1.0 / (1.0 / (t_1 / math.sqrt(2.0)))
	elif a2 <= 5e+290:
		tmp = (1.0 + (-0.5 * (th * th))) * (t_1 * math.sqrt(0.5))
	else:
		tmp = a2 * (a2 * math.sqrt(0.5))
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (a2 <= 1.6e+225)
		tmp = Float64(1.0 / Float64(1.0 / Float64(t_1 / sqrt(2.0))));
	elseif (a2 <= 5e+290)
		tmp = Float64(Float64(1.0 + Float64(-0.5 * Float64(th * th))) * Float64(t_1 * sqrt(0.5)));
	else
		tmp = Float64(a2 * Float64(a2 * sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (a2 <= 1.6e+225)
		tmp = 1.0 / (1.0 / (t_1 / sqrt(2.0)));
	elseif (a2 <= 5e+290)
		tmp = (1.0 + (-0.5 * (th * th))) * (t_1 * sqrt(0.5));
	else
		tmp = a2 * (a2 * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a2, 1.6e+225], N[(1.0 / N[(1.0 / N[(t$95$1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a2, 5e+290], N[(N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a2 < 1.59999999999999995e225

   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. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

      9. sqrt-lowering-sqrt.f64 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]
   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 66.8%

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

   7. Simplified 66.8%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 66.8%

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

   9. Applied egg-rr 66.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}} \]
   10. Step-by-step derivation

       1. flip3-+ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{2}}{\frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}}\right)\right) \]

       2. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{2}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}} \cdot \color{blue}{\left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)\right)}\right)\right) \]

       3. cube-unmult N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{2}}{\left(a1 \cdot a1\right) \cdot \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right)\right) + {\left(a2 \cdot a2\right)}^{3}} \cdot \left(\left(a1 \cdot a1\right) \cdot \left(\color{blue}{a1} \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{2}}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot \left(a1 \cdot \left(a1 \cdot a1\right)\right)\right) + {\left(a2 \cdot a2\right)}^{3}} \cdot \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)\right)\right)\right) \]

       5. cube-unmult N/A

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

       6. swap-sqr N/A

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

       7. clear-num N/A

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

       8. associate-*r* N/A

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

       9. distribute-rgt-out-- N/A

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

   11. Applied egg-rr 66.9%

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

   if 1.59999999999999995e225 < a2 < 4.9999999999999998e290

   1. Initial program 100.0%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Add Preprocessing
   3. 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. clear-num N/A

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

      4. associate-/r/ N/A

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

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

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

      6. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{2}^{\frac{1}{2}}}\right), \left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

      7. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\color{blue}{\cos th} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{\cos th} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

      9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 80.0%

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

   7. Simplified 80.0%

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

   if 4.9999999999999998e290 < a2

   1. Initial program 100.0%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Add Preprocessing
   3. 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. clear-num N/A

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

      4. associate-/r/ N/A

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

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

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

      6. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{2}^{\frac{1}{2}}}\right), \left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

      7. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\color{blue}{\cos th} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{\cos th} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

      9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a1 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left({a2}^{2} \cdot \cos th\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\color{blue}{{a2}^{2}} \cdot \cos th\right)\right) \]

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \color{blue}{\cos th}\right)\right)\right) \]

      9. cos-lowering-cos.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \mathsf{cos.f64}\left(th\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. sqrt-lowering-sqrt.f64 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 1.6 \cdot 10^{+225}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}}}\\ \mathbf{elif}\;a2 \leq 5 \cdot 10^{+290}:\\ \;\;\;\;\left(1 + -0.5 \cdot \left(th \cdot th\right)\right) \cdot \left(\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 3.4 \cdot 10^{+221}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}}}\\ \mathbf{elif}\;a2 \leq 4 \cdot 10^{+290}:\\ \;\;\;\;\frac{\left(a2 \cdot a2\right) \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 3.4e+221)
   (/ 1.0 (/ 1.0 (/ (+ (* a1 a1) (* a2 a2)) (sqrt 2.0))))
   (if (<= a2 4e+290)
     (/ (* (* a2 a2) (+ 1.0 (* -0.5 (* th th)))) (sqrt 2.0))
     (* a2 (* a2 (sqrt 0.5))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 3.4e+221) {
		tmp = 1.0 / (1.0 / (((a1 * a1) + (a2 * a2)) / sqrt(2.0)));
	} else if (a2 <= 4e+290) {
		tmp = ((a2 * a2) * (1.0 + (-0.5 * (th * th)))) / 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 (a2 <= 3.4d+221) then
        tmp = 1.0d0 / (1.0d0 / (((a1 * a1) + (a2 * a2)) / sqrt(2.0d0)))
    else if (a2 <= 4d+290) then
        tmp = ((a2 * a2) * (1.0d0 + ((-0.5d0) * (th * th)))) / 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 (a2 <= 3.4e+221) {
		tmp = 1.0 / (1.0 / (((a1 * a1) + (a2 * a2)) / Math.sqrt(2.0)));
	} else if (a2 <= 4e+290) {
		tmp = ((a2 * a2) * (1.0 + (-0.5 * (th * th)))) / Math.sqrt(2.0);
	} else {
		tmp = a2 * (a2 * Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 3.4e+221:
		tmp = 1.0 / (1.0 / (((a1 * a1) + (a2 * a2)) / math.sqrt(2.0)))
	elif a2 <= 4e+290:
		tmp = ((a2 * a2) * (1.0 + (-0.5 * (th * th)))) / math.sqrt(2.0)
	else:
		tmp = a2 * (a2 * math.sqrt(0.5))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 3.4e+221)
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) / sqrt(2.0))));
	elseif (a2 <= 4e+290)
		tmp = Float64(Float64(Float64(a2 * a2) * Float64(1.0 + Float64(-0.5 * Float64(th * th)))) / 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 (a2 <= 3.4e+221)
		tmp = 1.0 / (1.0 / (((a1 * a1) + (a2 * a2)) / sqrt(2.0)));
	elseif (a2 <= 4e+290)
		tmp = ((a2 * a2) * (1.0 + (-0.5 * (th * th)))) / sqrt(2.0);
	else
		tmp = a2 * (a2 * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 3.4e+221], N[(1.0 / N[(1.0 / N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a2, 4e+290], N[(N[(N[(a2 * a2), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $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 a2 < 3.3999999999999998e221

   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. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

      9. sqrt-lowering-sqrt.f64 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]
   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 66.6%

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

   7. Simplified 66.6%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 66.6%

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

   9. Applied egg-rr 66.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}} \]
   10. Step-by-step derivation

       1. flip3-+ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{2}}{\frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}}\right)\right) \]

       2. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{2}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}} \cdot \color{blue}{\left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)\right)}\right)\right) \]

       3. cube-unmult N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{2}}{\left(a1 \cdot a1\right) \cdot \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right)\right) + {\left(a2 \cdot a2\right)}^{3}} \cdot \left(\left(a1 \cdot a1\right) \cdot \left(\color{blue}{a1} \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{2}}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot \left(a1 \cdot \left(a1 \cdot a1\right)\right)\right) + {\left(a2 \cdot a2\right)}^{3}} \cdot \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)\right)\right)\right) \]

       5. cube-unmult N/A

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

       6. swap-sqr N/A

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

       7. clear-num N/A

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

       8. associate-*r* N/A

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

       9. distribute-rgt-out-- N/A

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

   11. Applied egg-rr 66.6%

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

   if 3.3999999999999998e221 < a2 < 4.00000000000000025e290

   1. Initial program 100.0%

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

   3. Taylor expanded in a1 around 0

      \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]
   4. 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 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-lft-out N/A

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 76.5%

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

   8. Simplified 76.5%

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

   if 4.00000000000000025e290 < a2

   1. Initial program 100.0%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Add Preprocessing
   3. 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. clear-num N/A

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

      4. associate-/r/ N/A

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

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

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

      6. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{2}^{\frac{1}{2}}}\right), \left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

      7. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\color{blue}{\cos th} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{\cos th} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

      9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a1 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left({a2}^{2} \cdot \cos th\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\color{blue}{{a2}^{2}} \cdot \cos th\right)\right) \]

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \color{blue}{\cos th}\right)\right)\right) \]

      9. cos-lowering-cos.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \mathsf{cos.f64}\left(th\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. sqrt-lowering-sqrt.f64 100.0%

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

   10. Simplified 100.0%

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

Alternative 8: 67.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}}} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (/ 1.0 (/ 1.0 (/ (+ (* a1 a1) (* a2 a2)) (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	return 1.0 / (1.0 / (((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 = 1.0d0 / (1.0d0 / (((a1 * a1) + (a2 * a2)) / sqrt(2.0d0)))
end function

Java

public static double code(double a1, double a2, double th) {
	return 1.0 / (1.0 / (((a1 * a1) + (a2 * a2)) / Math.sqrt(2.0)));
}

Python

def code(a1, a2, th):
	return 1.0 / (1.0 / (((a1 * a1) + (a2 * a2)) / math.sqrt(2.0)))

Julia

function code(a1, a2, th)
	return Float64(1.0 / Float64(1.0 / Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) / sqrt(2.0))))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = 1.0 / (1.0 / (((a1 * a1) + (a2 * a2)) / sqrt(2.0)));
end

Wolfram

code[a1_, a2_, th_] := N[(1.0 / N[(1.0 / N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / 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. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

   9. sqrt-lowering-sqrt.f64 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]
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 67.2%

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

7. Simplified 67.2%

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

   1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

   7. *-lowering-*.f64 67.2%

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

9. Applied egg-rr 67.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}} \]
10. Step-by-step derivation

    1. flip3-+ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{2}}{\frac{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}}{\color{blue}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)}}}\right)\right) \]

    2. associate-/r/ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{2}}{{\left(a1 \cdot a1\right)}^{3} + {\left(a2 \cdot a2\right)}^{3}} \cdot \color{blue}{\left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)\right)}\right)\right) \]

    3. cube-unmult N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{2}}{\left(a1 \cdot a1\right) \cdot \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right)\right) + {\left(a2 \cdot a2\right)}^{3}} \cdot \left(\left(a1 \cdot a1\right) \cdot \left(\color{blue}{a1} \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)\right)\right)\right) \]

    4. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sqrt{2}}{\left(a1 \cdot a1\right) \cdot \left(a1 \cdot \left(a1 \cdot \left(a1 \cdot a1\right)\right)\right) + {\left(a2 \cdot a2\right)}^{3}} \cdot \left(\left(a1 \cdot a1\right) \cdot \left(a1 \cdot a1\right) + \left(\left(a2 \cdot a2\right) \cdot \left(a2 \cdot a2\right) - \left(a1 \cdot a1\right) \cdot \left(a2 \cdot a2\right)\right)\right)\right)\right) \]

    5. cube-unmult N/A

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

    6. swap-sqr N/A

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

    7. clear-num N/A

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

    8. associate-*r* N/A

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

    9. distribute-rgt-out-- N/A

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

11. Applied egg-rr 67.2%

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}}}} \]
12. Add Preprocessing

Alternative 9: 67.2% 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. Add Preprocessing
3. 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. clear-num N/A

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

   4. associate-/r/ N/A

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

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

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

   6. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{2}^{\frac{1}{2}}}\right), \left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

   7. pow-flip N/A

      \[\leadsto \mathsf{*.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\color{blue}{\cos th} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{\cos th} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

   9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

   14. *-lowering-*.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right)\right)\right) \]

4. Applied egg-rr 99.6%

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

5. Taylor expanded in th around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. unpow2 N/A

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

   7. *-lowering-*.f64 67.2%

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

7. Simplified 67.2%

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

8. Final simplification 67.2%

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

Alternative 10: 39.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing
3. 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. clear-num N/A

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

   4. associate-/r/ N/A

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

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

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

   6. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{2}^{\frac{1}{2}}}\right), \left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

   7. pow-flip N/A

      \[\leadsto \mathsf{*.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\color{blue}{\cos th} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{\cos th} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

   9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

   14. *-lowering-*.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right)\right)\right) \]

4. Applied egg-rr 99.6%

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

5. Taylor expanded in a1 around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left({a2}^{2} \cdot \cos th\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\color{blue}{{a2}^{2}} \cdot \cos th\right)\right) \]

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

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

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

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \color{blue}{\cos th}\right)\right)\right) \]

   9. cos-lowering-cos.f64 63.3%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \mathsf{cos.f64}\left(th\right)\right)\right)\right) \]

7. Simplified 63.3%

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

8. Taylor expanded in th around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \color{blue}{\left({a2}^{2}\right)}\right) \]
9. Step-by-step derivation

   1. unpow2 N/A

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

   2. *-lowering-*.f64 46.1%

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

10. Simplified 46.1%

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

11. Final simplification 46.1%

    \[\leadsto \left(a2 \cdot a2\right) \cdot \sqrt{0.5} \]
12. Add Preprocessing

Alternative 11: 39.7% 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. Add Preprocessing
3. 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. clear-num N/A

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

   4. associate-/r/ N/A

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

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

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

   6. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{2}^{\frac{1}{2}}}\right), \left(\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

   7. pow-flip N/A

      \[\leadsto \mathsf{*.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(\color{blue}{\cos th} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{\cos th} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right) \]

   9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

   14. *-lowering-*.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(th\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a1, a1\right), \mathsf{*.f64}\left(a2, \color{blue}{a2}\right)\right)\right)\right) \]

4. Applied egg-rr 99.6%

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

5. Taylor expanded in a1 around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left({a2}^{2} \cdot \cos th\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\color{blue}{{a2}^{2}} \cdot \cos th\right)\right) \]

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

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

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

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \color{blue}{\cos th}\right)\right)\right) \]

   9. cos-lowering-cos.f64 63.3%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(a2, \mathsf{*.f64}\left(a2, \mathsf{cos.f64}\left(th\right)\right)\right)\right) \]

7. Simplified 63.3%

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

8. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. sqrt-lowering-sqrt.f64 46.1%

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

10. Simplified 46.1%

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

Alternative 12: 40.6% 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. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

   9. sqrt-lowering-sqrt.f64 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]
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 67.2%

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

7. Simplified 67.2%

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

8. Taylor expanded in a1 around inf

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

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

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

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

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

   5. sqrt-lowering-sqrt.f64 35.8%

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

10. Simplified 35.8%

    \[\leadsto \color{blue}{a1 \cdot \frac{a1}{\sqrt{2}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(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))))