Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 12.0s

Alternatives: 13

Speedup: 2.0×

• 43.0% 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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. sqrt-lowering-sqrt.f64 99.3%

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

3. Simplified 99.3%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. associate-*r* N/A

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

   8. fma-define N/A

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

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

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

   16. sqrt-lowering-sqrt.f64 99.5%

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

6. Applied egg-rr 99.5%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos th \cdot \left({2}^{-0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (cos th) (* (pow 2.0 -0.5) (+ (* a1 a1) (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	return cos(th) * (pow(2.0, -0.5) * ((a1 * a1) + (a2 * a2)));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = cos(th) * ((2.0d0 ** (-0.5d0)) * ((a1 * a1) + (a2 * a2)))
end function

Java

public static double code(double a1, double a2, double th) {
	return Math.cos(th) * (Math.pow(2.0, -0.5) * ((a1 * a1) + (a2 * a2)));
}

Python

def code(a1, a2, th):
	return math.cos(th) * (math.pow(2.0, -0.5) * ((a1 * a1) + (a2 * a2)))

Julia

function code(a1, a2, th)
	return Float64(cos(th) * Float64((2.0 ^ -0.5) * Float64(Float64(a1 * a1) + Float64(a2 * a2))))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = cos(th) * ((2.0 ^ -0.5) * ((a1 * a1) + (a2 * a2)));
end

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[Power[2.0, -0.5], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. sqrt-lowering-sqrt.f64 99.3%

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

3. Simplified 99.3%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

   4. pow1/2 N/A

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

   5. pow-flip N/A

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

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

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

   7. metadata-eval N/A

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

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

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

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

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

   10. *-lowering-*.f64 99.3%

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

6. Applied egg-rr 99.3%

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

Alternative 3: 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 99.3%

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

   1. distribute-lft-out N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. sqrt-lowering-sqrt.f64 99.3%

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

3. Simplified 99.3%

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

   1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   8. sqrt-lowering-sqrt.f64 99.3%

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

6. Applied egg-rr 99.3%

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

Alternative 4: 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 99.3%

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

   1. distribute-lft-out N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. sqrt-lowering-sqrt.f64 99.3%

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

3. Simplified 99.3%

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

Alternative 5: 57.2% accurate, 2.0× speedup

Math

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

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

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. sqrt-lowering-sqrt.f64 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in a1 around 0

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

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

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

   2. unpow2 N/A

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

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

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

   4. sqrt-lowering-sqrt.f64 56.1%

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

7. Simplified 56.1%

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. *-commutative N/A

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

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

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

   7. *-commutative N/A

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

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

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

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

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

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

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

   11. sqrt-lowering-sqrt.f64 56.5%

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

9. Applied egg-rr 56.5%

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

Alternative 6: 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 99.3%

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

   1. distribute-lft-out N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. sqrt-lowering-sqrt.f64 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in a1 around 0

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

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

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

   2. unpow2 N/A

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

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

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

   4. sqrt-lowering-sqrt.f64 56.1%

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

7. Simplified 56.1%

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

   1. associate-*l/ N/A

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

   2. *-commutative N/A

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

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

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

   4. associate-/r/ N/A

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

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

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

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

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

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

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

   8. cos-lowering-cos.f64 56.5%

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

9. Applied egg-rr 56.5%

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

10. Final simplification 56.5%

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

Alternative 7: 57.3% 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 99.3%

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

   1. distribute-lft-out N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. sqrt-lowering-sqrt.f64 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in a1 around 0

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

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

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

   2. unpow2 N/A

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

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

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

   4. sqrt-lowering-sqrt.f64 56.1%

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

7. Simplified 56.1%

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

Alternative 8: 41.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 2.7 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{a2}{\sqrt{2}}}{\frac{1}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \left(th \cdot \left(th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 2.7e+98)
   (/ (/ a2 (sqrt 2.0)) (/ 1.0 a2))
   (/ (* -0.5 (* th (* th (+ (* a1 a1) (* a2 a2))))) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 2.7e+98) {
		tmp = (a2 / sqrt(2.0)) / (1.0 / a2);
	} else {
		tmp = (-0.5 * (th * (th * ((a1 * a1) + (a2 * a2))))) / sqrt(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 2.7d+98) then
        tmp = (a2 / sqrt(2.0d0)) / (1.0d0 / a2)
    else
        tmp = ((-0.5d0) * (th * (th * ((a1 * a1) + (a2 * a2))))) / sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 2.7e+98) {
		tmp = (a2 / Math.sqrt(2.0)) / (1.0 / a2);
	} else {
		tmp = (-0.5 * (th * (th * ((a1 * a1) + (a2 * a2))))) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 2.7e+98:
		tmp = (a2 / math.sqrt(2.0)) / (1.0 / a2)
	else:
		tmp = (-0.5 * (th * (th * ((a1 * a1) + (a2 * a2))))) / math.sqrt(2.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 2.7e+98)
		tmp = Float64(Float64(a2 / sqrt(2.0)) / Float64(1.0 / a2));
	else
		tmp = Float64(Float64(-0.5 * Float64(th * Float64(th * Float64(Float64(a1 * a1) + Float64(a2 * a2))))) / sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 2.7e+98)
		tmp = (a2 / sqrt(2.0)) / (1.0 / a2);
	else
		tmp = (-0.5 * (th * (th * ((a1 * a1) + (a2 * a2))))) / sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 2.7e+98], N[(N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 / a2), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(th * N[(th * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 2.7e98

   1. Initial program 99.2%

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

      1. distribute-lft-out N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in th around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sqrt-lowering-sqrt.f64 75.5%

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

   7. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
   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 75.1%

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

   9. Applied egg-rr 75.1%

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

   10. Taylor expanded in a1 around 0

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

       1. unpow2 N/A

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

       2. associate-/r* N/A

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

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

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

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

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

       5. sqrt-lowering-sqrt.f64 42.0%

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

   12. Simplified 42.0%

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

       1. clear-num N/A

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

       2. div-inv N/A

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

       3. associate-/r* N/A

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

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

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

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

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

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

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

       7. /-lowering-/.f64 42.2%

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

   14. Applied egg-rr 42.2%

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

   if 2.7e98 < th

   1. Initial program 99.7%

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

      1. distribute-lft-out N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in th around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 47.6%

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

   9. Simplified 47.6%

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

   10. Taylor expanded in th around inf

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

       1. associate-*r/ N/A

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

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

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

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

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. +-commutative N/A

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

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

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

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

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

       14. sqrt-lowering-sqrt.f64 49.8%

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

   12. Simplified 49.8%

       \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(th \cdot \left(th \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\right)\right)}{\sqrt{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 2.7 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{a2}{\sqrt{2}}}{\frac{1}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \left(th \cdot \left(th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 9 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{a2}{\sqrt{2}}}{\frac{1}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 9e+159)
   (/ (/ a2 (sqrt 2.0)) (/ 1.0 a2))
   (* (/ (* a2 a2) (sqrt 2.0)) (+ 1.0 (* -0.5 (* th th))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 9e+159) {
		tmp = (a2 / sqrt(2.0)) / (1.0 / a2);
	} else {
		tmp = ((a2 * a2) / sqrt(2.0)) * (1.0 + (-0.5 * (th * th)));
	}
	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 <= 9d+159) then
        tmp = (a2 / sqrt(2.0d0)) / (1.0d0 / a2)
    else
        tmp = ((a2 * a2) / sqrt(2.0d0)) * (1.0d0 + ((-0.5d0) * (th * th)))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 9e+159) {
		tmp = (a2 / Math.sqrt(2.0)) / (1.0 / a2);
	} else {
		tmp = ((a2 * a2) / Math.sqrt(2.0)) * (1.0 + (-0.5 * (th * th)));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 9e+159:
		tmp = (a2 / math.sqrt(2.0)) / (1.0 / a2)
	else:
		tmp = ((a2 * a2) / math.sqrt(2.0)) * (1.0 + (-0.5 * (th * th)))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 9e+159)
		tmp = Float64(Float64(a2 / sqrt(2.0)) / Float64(1.0 / a2));
	else
		tmp = Float64(Float64(Float64(a2 * a2) / sqrt(2.0)) * Float64(1.0 + Float64(-0.5 * Float64(th * th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 9e+159)
		tmp = (a2 / sqrt(2.0)) / (1.0 / a2);
	else
		tmp = ((a2 * a2) / sqrt(2.0)) * (1.0 + (-0.5 * (th * th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 9e+159], N[(N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 / a2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 9.00000000000000053e159

   1. Initial program 99.2%

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

      1. distribute-lft-out N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in th around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sqrt-lowering-sqrt.f64 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 66.9%

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

   9. Applied egg-rr 66.9%

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

   10. Taylor expanded in a1 around 0

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

       1. unpow2 N/A

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

       2. associate-/r* N/A

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

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

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

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

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

       5. sqrt-lowering-sqrt.f64 34.8%

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

   12. Simplified 34.8%

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

       1. clear-num N/A

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

       2. div-inv N/A

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

       3. associate-/r* N/A

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

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

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

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

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

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

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

       7. /-lowering-/.f64 35.0%

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

   14. Applied egg-rr 35.0%

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

   if 9.00000000000000053e159 < 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. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a1 around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 100.0%

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

   7. Simplified 100.0%

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

   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

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 80.8%

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

   10. Simplified 80.8%

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

Alternative 10: 40.6% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 / a2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. sqrt-lowering-sqrt.f64 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in th around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

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

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

   7. sqrt-lowering-sqrt.f64 67.0%

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

7. Simplified 67.0%

   \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
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. Taylor expanded in a1 around 0

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

    1. unpow2 N/A

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

    2. associate-/r* N/A

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

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

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

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

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

    5. sqrt-lowering-sqrt.f64 37.9%

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

12. Simplified 37.9%

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

    1. clear-num N/A

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

    2. div-inv N/A

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

    3. associate-/r* N/A

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

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

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

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

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

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

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

    7. /-lowering-/.f64 38.1%

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

14. Applied egg-rr 38.1%

    \[\leadsto \color{blue}{\frac{\frac{a2}{\sqrt{2}}}{\frac{1}{a2}}} \]
15. Add Preprocessing

Alternative 11: 40.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. sqrt-lowering-sqrt.f64 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in a1 around 0

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

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

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

   2. unpow2 N/A

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

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

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

   4. sqrt-lowering-sqrt.f64 56.1%

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

7. Simplified 56.1%

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. *-commutative N/A

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

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

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

   7. *-commutative N/A

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

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

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

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

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

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

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

   11. sqrt-lowering-sqrt.f64 56.5%

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

9. Applied egg-rr 56.5%

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

10. Taylor expanded in th around 0

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

12. Simplified 38.1%

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

Alternative 12: 40.6% 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 99.3%

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

   1. distribute-lft-out N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. sqrt-lowering-sqrt.f64 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in th around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

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

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

   7. sqrt-lowering-sqrt.f64 67.0%

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

7. Simplified 67.0%

   \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
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. Taylor expanded in a1 around 0

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

    1. unpow2 N/A

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

    2. associate-/r* N/A

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

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

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

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

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

    5. sqrt-lowering-sqrt.f64 37.9%

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

12. Simplified 37.9%

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

    1. clear-num N/A

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

    2. associate-/r/ N/A

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

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

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

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

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

    5. sqrt-lowering-sqrt.f64 38.1%

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

14. Applied egg-rr 38.1%

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

15. Final simplification 38.1%

    \[\leadsto a2 \cdot \frac{a2}{\sqrt{2}} \]
16. Add Preprocessing

Alternative 13: 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 99.3%

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

   1. distribute-lft-out N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. sqrt-lowering-sqrt.f64 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in th around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

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

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

   7. sqrt-lowering-sqrt.f64 67.0%

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

7. Simplified 67.0%

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

8. Taylor expanded in a1 around inf

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

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

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

   2. unpow2 N/A

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

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

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

   4. sqrt-lowering-sqrt.f64 43.0%

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

10. Simplified 43.0%

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

    1. associate-*l/ N/A

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

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

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

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

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

    4. sqrt-lowering-sqrt.f64 43.0%

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

12. Applied egg-rr 43.0%

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

13. Final simplification 43.0%

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

Reproduce

herbie shell --seed 2024164 
(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))))