Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 10.3s

Alternatives: 10

Speedup: 2.0×

• 43.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. cos-lowering-cos.f64 99.7%

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

4. Applied egg-rr 99.7%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. 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. associate-/l* N/A

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

   4. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. cos-lowering-cos.f64 99.7%

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 57.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 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 61.0%

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

7. Simplified 61.0%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   9. cos-lowering-cos.f64 61.0%

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

9. Applied egg-rr 61.0%

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

Alternative 4: 57.2% accurate, 2.0× speedup

Math

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

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(cos(th) * Float64(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[Cos[th], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out N/A

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

   2. associate-*l/ N/A

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

   3. /-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 61.0%

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

7. Simplified 61.0%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

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

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

   7. cos-lowering-cos.f64 61.1%

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

9. Applied egg-rr 61.1%

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

10. Final simplification 61.1%

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

Alternative 5: 57.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 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 61.0%

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

7. Simplified 61.0%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   9. cos-lowering-cos.f64 61.0%

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

9. Applied egg-rr 61.0%

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

    1. associate-/r/ N/A

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

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

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

    3. associate-/l* N/A

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

    4. clear-num N/A

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

    5. un-div-inv N/A

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

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

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

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

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

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

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

    9. cos-lowering-cos.f64 61.0%

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

11. Applied egg-rr 61.0%

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

12. Final simplification 61.0%

    \[\leadsto \cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}} \]
13. Add Preprocessing

Alternative 6: 57.2% accurate, 2.0× speedup

Math

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

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 * cos(th)) * Float64(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[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 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 61.0%

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

7. Simplified 61.0%

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

   1. associate-*r* N/A

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

   2. associate-/l* N/A

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

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

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

   4. *-commutative N/A

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

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

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

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

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

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

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

   8. sqrt-lowering-sqrt.f64 61.0%

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

9. Applied egg-rr 61.0%

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

Alternative 7: 66.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 2.85 \cdot 10^{+296}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}} \cdot \left(1 + th \cdot \left(th \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 2.85e+296)
   (/ 1.0 (/ 1.0 (/ (+ (* a1 a1) (* a2 a2)) (sqrt 2.0))))
   (* (/ (* a2 a2) (sqrt 2.0)) (+ 1.0 (* th (* th -0.5))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 2.85e+296) {
		tmp = 1.0 / (1.0 / (((a1 * a1) + (a2 * a2)) / sqrt(2.0)));
	} else {
		tmp = ((a2 * a2) / sqrt(2.0)) * (1.0 + (th * (th * -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 <= 2.85d+296) then
        tmp = 1.0d0 / (1.0d0 / (((a1 * a1) + (a2 * a2)) / sqrt(2.0d0)))
    else
        tmp = ((a2 * a2) / sqrt(2.0d0)) * (1.0d0 + (th * (th * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 2.85e+296) {
		tmp = 1.0 / (1.0 / (((a1 * a1) + (a2 * a2)) / Math.sqrt(2.0)));
	} else {
		tmp = ((a2 * a2) / Math.sqrt(2.0)) * (1.0 + (th * (th * -0.5)));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 2.85e+296:
		tmp = 1.0 / (1.0 / (((a1 * a1) + (a2 * a2)) / math.sqrt(2.0)))
	else:
		tmp = ((a2 * a2) / math.sqrt(2.0)) * (1.0 + (th * (th * -0.5)))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 2.85e+296)
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) / sqrt(2.0))));
	else
		tmp = Float64(Float64(Float64(a2 * a2) / sqrt(2.0)) * Float64(1.0 + Float64(th * Float64(th * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 2.85e+296)
		tmp = 1.0 / (1.0 / (((a1 * a1) + (a2 * a2)) / sqrt(2.0)));
	else
		tmp = ((a2 * a2) / sqrt(2.0)) * (1.0 + (th * (th * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 2.85e+296], 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], N[(N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(th * N[(th * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 2.84999999999999999e296

   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 66.1%

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

   7. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   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 66.1%

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

   9. Applied egg-rr 66.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}} \]
   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. associate-*r* N/A

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

       7. associate-*r* N/A

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

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

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

       9. *-commutative N/A

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

   11. Applied egg-rr 66.1%

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

   if 2.84999999999999999e296 < 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. 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 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]
   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 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \frac{{th}^{2} \cdot {a2}^{2}}{\sqrt{2}} \cdot \frac{-1}{2} + \frac{{a2}^{2}}{\sqrt{2}} \]

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

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

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

   10. Simplified 100.0%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 2.85 \cdot 10^{+296}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}} \cdot \left(1 + th \cdot \left(th \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 10^{+296}:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}} \cdot \left(1 + th \cdot \left(th \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 1e+296)
   (/ (+ (* a1 a1) (* a2 a2)) (sqrt 2.0))
   (* (/ (* a2 a2) (sqrt 2.0)) (+ 1.0 (* th (* th -0.5))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 1e+296) {
		tmp = ((a1 * a1) + (a2 * a2)) / sqrt(2.0);
	} else {
		tmp = ((a2 * a2) / sqrt(2.0)) * (1.0 + (th * (th * -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 <= 1d+296) then
        tmp = ((a1 * a1) + (a2 * a2)) / sqrt(2.0d0)
    else
        tmp = ((a2 * a2) / sqrt(2.0d0)) * (1.0d0 + (th * (th * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 1e+296) {
		tmp = ((a1 * a1) + (a2 * a2)) / Math.sqrt(2.0);
	} else {
		tmp = ((a2 * a2) / Math.sqrt(2.0)) * (1.0 + (th * (th * -0.5)));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 1e+296:
		tmp = ((a1 * a1) + (a2 * a2)) / math.sqrt(2.0)
	else:
		tmp = ((a2 * a2) / math.sqrt(2.0)) * (1.0 + (th * (th * -0.5)))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 1e+296)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) / sqrt(2.0));
	else
		tmp = Float64(Float64(Float64(a2 * a2) / sqrt(2.0)) * Float64(1.0 + Float64(th * Float64(th * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 1e+296)
		tmp = ((a1 * a1) + (a2 * a2)) / sqrt(2.0);
	else
		tmp = ((a2 * a2) / sqrt(2.0)) * (1.0 + (th * (th * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 1e+296], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(th * N[(th * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 9.99999999999999981e295

   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 66.1%

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

   7. Simplified 66.1%

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

   if 9.99999999999999981e295 < 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. 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 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \]
   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 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \frac{{th}^{2} \cdot {a2}^{2}}{\sqrt{2}} \cdot \frac{-1}{2} + \frac{{a2}^{2}}{\sqrt{2}} \]

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

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

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

   10. Simplified 100.0%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 10^{+296}:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}} \cdot \left(1 + th \cdot \left(th \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1), $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 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 65.6%

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

7. Simplified 65.6%

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

Alternative 10: 39.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 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 61.0%

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

7. Simplified 61.0%

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

   1. associate-*r* N/A

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

   2. associate-/l* N/A

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

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

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

   4. *-commutative N/A

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

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

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

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

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

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

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

   8. sqrt-lowering-sqrt.f64 61.0%

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

9. Applied egg-rr 61.0%

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

10. Taylor expanded in th around 0

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

12. Simplified 40.1%

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

Reproduce

herbie shell --seed 2024288 
(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))))