Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 6.9s

Alternatives: 9

Speedup: 1.0×

• 44.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \left(a2\_m \cdot a2\_m\right) \cdot \left({4}^{-0.25} \cdot \cos th\right) + \left(a1\_m \cdot a1\_m\right) \cdot \frac{\cos th}{\sqrt{2}} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (+
  (* (* a2_m a2_m) (* (pow 4.0 -0.25) (cos th)))
  (* (* a1_m a1_m) (/ (cos th) (sqrt 2.0)))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return ((a2_m * a2_m) * (pow(4.0, -0.25) * cos(th))) + ((a1_m * a1_m) * (cos(th) / sqrt(2.0)));
}

Fortran

a2_m = abs(a2)
a1_m = abs(a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = ((a2_m * a2_m) * ((4.0d0 ** (-0.25d0)) * cos(th))) + ((a1_m * a1_m) * (cos(th) / sqrt(2.0d0)))
end function

Java

a2_m = Math.abs(a2);
a1_m = Math.abs(a1);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return ((a2_m * a2_m) * (Math.pow(4.0, -0.25) * Math.cos(th))) + ((a1_m * a1_m) * (Math.cos(th) / Math.sqrt(2.0)));
}

Python

a2_m = math.fabs(a2)
a1_m = math.fabs(a1)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return ((a2_m * a2_m) * (math.pow(4.0, -0.25) * math.cos(th))) + ((a1_m * a1_m) * (math.cos(th) / math.sqrt(2.0)))

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(Float64(Float64(a2_m * a2_m) * Float64((4.0 ^ -0.25) * cos(th))) + Float64(Float64(a1_m * a1_m) * Float64(cos(th) / sqrt(2.0))))
end

MATLAB

a2_m = abs(a2);
a1_m = abs(a1);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = ((a2_m * a2_m) * ((4.0 ^ -0.25) * cos(th))) + ((a1_m * a1_m) * (cos(th) / sqrt(2.0)));
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * N[(N[Power[4.0, -0.25], $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a1$95$m * a1$95$m), $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. pow1/2 N/A

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

   7. pow-flip N/A

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

   8. sqr-pow N/A

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

   9. pow-prod-down N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

       \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left({\color{blue}{4}}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)} \cdot \cos th\right) \cdot \left(a2 \cdot a2\right) \]

   12. metadata-eval N/A

       \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left({4}^{\left(\frac{\color{blue}{\frac{-1}{2}}}{2}\right)} \cdot \cos th\right) \cdot \left(a2 \cdot a2\right) \]

   13. metadata-eval 99.5

       \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left({4}^{\color{blue}{-0.25}} \cdot \cos th\right) \cdot \left(a2 \cdot a2\right) \]

4. Applied rewrites 99.5%

   \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \color{blue}{\left({4}^{-0.25} \cdot \cos th\right)} \cdot \left(a2 \cdot a2\right) \]

5. Final simplification 99.5%

   \[\leadsto \left(a2 \cdot a2\right) \cdot \left({4}^{-0.25} \cdot \cos th\right) + \left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} \]
6. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \left(\frac{a2\_m}{\sqrt{2}} \cdot \cos th\right) \cdot a2\_m + \left(a1\_m \cdot a1\_m\right) \cdot \frac{\cos th}{\sqrt{2}} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (+
  (* (* (/ a2_m (sqrt 2.0)) (cos th)) a2_m)
  (* (* a1_m a1_m) (/ (cos th) (sqrt 2.0)))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return (((a2_m / sqrt(2.0)) * cos(th)) * a2_m) + ((a1_m * a1_m) * (cos(th) / sqrt(2.0)));
}

Fortran

a2_m = abs(a2)
a1_m = abs(a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = (((a2_m / sqrt(2.0d0)) * cos(th)) * a2_m) + ((a1_m * a1_m) * (cos(th) / sqrt(2.0d0)))
end function

Java

a2_m = Math.abs(a2);
a1_m = Math.abs(a1);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return (((a2_m / Math.sqrt(2.0)) * Math.cos(th)) * a2_m) + ((a1_m * a1_m) * (Math.cos(th) / Math.sqrt(2.0)));
}

Python

a2_m = math.fabs(a2)
a1_m = math.fabs(a1)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return (((a2_m / math.sqrt(2.0)) * math.cos(th)) * a2_m) + ((a1_m * a1_m) * (math.cos(th) / math.sqrt(2.0)))

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(Float64(Float64(Float64(a2_m / sqrt(2.0)) * cos(th)) * a2_m) + Float64(Float64(a1_m * a1_m) * Float64(cos(th) / sqrt(2.0))))
end

MATLAB

a2_m = abs(a2);
a1_m = abs(a1);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = (((a2_m / sqrt(2.0)) * cos(th)) * a2_m) + ((a1_m * a1_m) * (cos(th) / sqrt(2.0)));
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(N[(N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * a2$95$m), $MachinePrecision] + N[(N[(a1$95$m * a1$95$m), $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. frac-2neg N/A

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

   11. metadata-eval N/A

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

   12. associate-*l/ N/A

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

   13. neg-mul-1 N/A

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

   14. frac-2neg-rev N/A

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

   15. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \left(\frac{a2\_m}{\sqrt{2}} \cdot a2\_m\right) \cdot \cos th + \left(a1\_m \cdot a1\_m\right) \cdot \frac{\cos th}{\sqrt{2}} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (+
  (* (* (/ a2_m (sqrt 2.0)) a2_m) (cos th))
  (* (* a1_m a1_m) (/ (cos th) (sqrt 2.0)))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return (((a2_m / sqrt(2.0)) * a2_m) * cos(th)) + ((a1_m * a1_m) * (cos(th) / sqrt(2.0)));
}

Fortran

a2_m = abs(a2)
a1_m = abs(a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = (((a2_m / sqrt(2.0d0)) * a2_m) * cos(th)) + ((a1_m * a1_m) * (cos(th) / sqrt(2.0d0)))
end function

Java

a2_m = Math.abs(a2);
a1_m = Math.abs(a1);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return (((a2_m / Math.sqrt(2.0)) * a2_m) * Math.cos(th)) + ((a1_m * a1_m) * (Math.cos(th) / Math.sqrt(2.0)));
}

Python

a2_m = math.fabs(a2)
a1_m = math.fabs(a1)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return (((a2_m / math.sqrt(2.0)) * a2_m) * math.cos(th)) + ((a1_m * a1_m) * (math.cos(th) / math.sqrt(2.0)))

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(Float64(Float64(Float64(a2_m / sqrt(2.0)) * a2_m) * cos(th)) + Float64(Float64(a1_m * a1_m) * Float64(cos(th) / sqrt(2.0))))
end

MATLAB

a2_m = abs(a2);
a1_m = abs(a1);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = (((a2_m / sqrt(2.0)) * a2_m) * cos(th)) + ((a1_m * a1_m) * (cos(th) / sqrt(2.0)));
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(N[(N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a2$95$m), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] + N[(N[(a1$95$m * a1$95$m), $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. div-inv N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. associate-*l/ N/A

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

   15. neg-mul-1 N/A

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

   16. frac-2neg-rev N/A

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

   17. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 78.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{a1\_m}{\sqrt{2}} \cdot a1\_m\\ \mathbf{if}\;\cos th \leq -0.005:\\ \;\;\;\;\left(-a2\_m\right) \cdot \frac{a2\_m}{\sqrt{2}} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{a2\_m \cdot a2\_m}{2}} \cdot a2\_m + t\_1\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (let* ((t_1 (* (/ a1_m (sqrt 2.0)) a1_m)))
   (if (<= (cos th) -0.005)
     (+ (* (- a2_m) (/ a2_m (sqrt 2.0))) t_1)
     (+ (* (sqrt (/ (* a2_m a2_m) 2.0)) a2_m) t_1))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	double t_1 = (a1_m / sqrt(2.0)) * a1_m;
	double tmp;
	if (cos(th) <= -0.005) {
		tmp = (-a2_m * (a2_m / sqrt(2.0))) + t_1;
	} else {
		tmp = (sqrt(((a2_m * a2_m) / 2.0)) * a2_m) + t_1;
	}
	return tmp;
}

Fortran

a2_m = abs(a2)
a1_m = abs(a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a1_m / sqrt(2.0d0)) * a1_m
    if (cos(th) <= (-0.005d0)) then
        tmp = (-a2_m * (a2_m / sqrt(2.0d0))) + t_1
    else
        tmp = (sqrt(((a2_m * a2_m) / 2.0d0)) * a2_m) + t_1
    end if
    code = tmp
end function

Java

a2_m = Math.abs(a2);
a1_m = Math.abs(a1);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	double t_1 = (a1_m / Math.sqrt(2.0)) * a1_m;
	double tmp;
	if (Math.cos(th) <= -0.005) {
		tmp = (-a2_m * (a2_m / Math.sqrt(2.0))) + t_1;
	} else {
		tmp = (Math.sqrt(((a2_m * a2_m) / 2.0)) * a2_m) + t_1;
	}
	return tmp;
}

Python

a2_m = math.fabs(a2)
a1_m = math.fabs(a1)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	t_1 = (a1_m / math.sqrt(2.0)) * a1_m
	tmp = 0
	if math.cos(th) <= -0.005:
		tmp = (-a2_m * (a2_m / math.sqrt(2.0))) + t_1
	else:
		tmp = (math.sqrt(((a2_m * a2_m) / 2.0)) * a2_m) + t_1
	return tmp

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	t_1 = Float64(Float64(a1_m / sqrt(2.0)) * a1_m)
	tmp = 0.0
	if (cos(th) <= -0.005)
		tmp = Float64(Float64(Float64(-a2_m) * Float64(a2_m / sqrt(2.0))) + t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(a2_m * a2_m) / 2.0)) * a2_m) + t_1);
	end
	return tmp
end

MATLAB

a2_m = abs(a2);
a1_m = abs(a1);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp_2 = code(a1_m, a2_m, th)
	t_1 = (a1_m / sqrt(2.0)) * a1_m;
	tmp = 0.0;
	if (cos(th) <= -0.005)
		tmp = (-a2_m * (a2_m / sqrt(2.0))) + t_1;
	else
		tmp = (sqrt(((a2_m * a2_m) / 2.0)) * a2_m) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := Block[{t$95$1 = N[(N[(a1$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a1$95$m), $MachinePrecision]}, If[LessEqual[N[Cos[th], $MachinePrecision], -0.005], N[(N[((-a2$95$m) * N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(a2$95$m * a2$95$m), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * a2$95$m), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -0.0050000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 55.6

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 8.7

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

   8. Applied rewrites 8.7%

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

   10. Applied rewrites 31.0%

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

   if -0.0050000000000000001 < (cos.f64 th)

   1. Initial program 99.6%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 91.6

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 83.3

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

   8. Applied rewrites 83.3%

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

   10. Applied rewrites 58.4%

       \[\leadsto \frac{a1}{\sqrt{2}} \cdot a1 + a2 \cdot \sqrt{\frac{a2 \cdot a2}{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -0.005:\\ \;\;\;\;\left(-a2\right) \cdot \frac{a2}{\sqrt{2}} + \frac{a1}{\sqrt{2}} \cdot a1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{a2 \cdot a2}{2}} \cdot a2 + \frac{a1}{\sqrt{2}} \cdot a1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \sqrt{\frac{a2\_m \cdot a2\_m}{2}} \cdot \left(a2\_m \cdot \cos th\right) + \frac{a1\_m}{\sqrt{2}} \cdot a1\_m \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (+
  (* (sqrt (/ (* a2_m a2_m) 2.0)) (* a2_m (cos th)))
  (* (/ a1_m (sqrt 2.0)) a1_m)))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return (sqrt(((a2_m * a2_m) / 2.0)) * (a2_m * cos(th))) + ((a1_m / sqrt(2.0)) * a1_m);
}

Fortran

a2_m = abs(a2)
a1_m = abs(a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = (sqrt(((a2_m * a2_m) / 2.0d0)) * (a2_m * cos(th))) + ((a1_m / sqrt(2.0d0)) * a1_m)
end function

Java

a2_m = Math.abs(a2);
a1_m = Math.abs(a1);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return (Math.sqrt(((a2_m * a2_m) / 2.0)) * (a2_m * Math.cos(th))) + ((a1_m / Math.sqrt(2.0)) * a1_m);
}

Python

a2_m = math.fabs(a2)
a1_m = math.fabs(a1)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return (math.sqrt(((a2_m * a2_m) / 2.0)) * (a2_m * math.cos(th))) + ((a1_m / math.sqrt(2.0)) * a1_m)

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(Float64(sqrt(Float64(Float64(a2_m * a2_m) / 2.0)) * Float64(a2_m * cos(th))) + Float64(Float64(a1_m / sqrt(2.0)) * a1_m))
end

MATLAB

a2_m = abs(a2);
a1_m = abs(a1);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = (sqrt(((a2_m * a2_m) / 2.0)) * (a2_m * cos(th))) + ((a1_m / sqrt(2.0)) * a1_m);
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(N[Sqrt[N[(N[(a2$95$m * a2$95$m), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(a2$95$m * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a1$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a1$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 82.5

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

5. Applied rewrites 82.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. /-rgt-identity N/A

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

   4. associate-/r/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-2neg N/A

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

   8. associate-/r/ N/A

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

   9. sqr-neg-rev N/A

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

   10. times-frac N/A

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

   11. frac-2neg N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. frac-2neg N/A

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

   16. metadata-eval N/A

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

   17. remove-double-neg N/A

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

   18. lower-/.f64 N/A

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

   19. lower-/.f64 82.5

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

7. Applied rewrites 82.5%

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

   1. lift-/.f64 N/A

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

   2. unpow1 N/A

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

   3. metadata-eval N/A

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

   4. sqrt-pow1 N/A

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

   5. pow2 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. sqrt-undiv N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 56.5

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

9. Applied rewrites 56.5%

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

10. Taylor expanded in a2 around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower-cos.f64 56.5

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

12. Applied rewrites 56.5%

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

13. Final simplification 56.5%

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

Alternative 6: 78.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{a2\_m}{\sqrt{2}}\\ t_2 := \frac{a1\_m}{\sqrt{2}} \cdot a1\_m\\ \mathbf{if}\;\cos th \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-a2\_m\right) \cdot t\_1 + t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot a2\_m + t\_2\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (let* ((t_1 (/ a2_m (sqrt 2.0))) (t_2 (* (/ a1_m (sqrt 2.0)) a1_m)))
   (if (<= (cos th) -5e-310) (+ (* (- a2_m) t_1) t_2) (+ (* t_1 a2_m) t_2))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	double t_1 = a2_m / sqrt(2.0);
	double t_2 = (a1_m / sqrt(2.0)) * a1_m;
	double tmp;
	if (cos(th) <= -5e-310) {
		tmp = (-a2_m * t_1) + t_2;
	} else {
		tmp = (t_1 * a2_m) + t_2;
	}
	return tmp;
}

Fortran

a2_m = abs(a2)
a1_m = abs(a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a2_m / sqrt(2.0d0)
    t_2 = (a1_m / sqrt(2.0d0)) * a1_m
    if (cos(th) <= (-5d-310)) then
        tmp = (-a2_m * t_1) + t_2
    else
        tmp = (t_1 * a2_m) + t_2
    end if
    code = tmp
end function

Java

a2_m = Math.abs(a2);
a1_m = Math.abs(a1);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	double t_1 = a2_m / Math.sqrt(2.0);
	double t_2 = (a1_m / Math.sqrt(2.0)) * a1_m;
	double tmp;
	if (Math.cos(th) <= -5e-310) {
		tmp = (-a2_m * t_1) + t_2;
	} else {
		tmp = (t_1 * a2_m) + t_2;
	}
	return tmp;
}

Python

a2_m = math.fabs(a2)
a1_m = math.fabs(a1)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	t_1 = a2_m / math.sqrt(2.0)
	t_2 = (a1_m / math.sqrt(2.0)) * a1_m
	tmp = 0
	if math.cos(th) <= -5e-310:
		tmp = (-a2_m * t_1) + t_2
	else:
		tmp = (t_1 * a2_m) + t_2
	return tmp

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	t_1 = Float64(a2_m / sqrt(2.0))
	t_2 = Float64(Float64(a1_m / sqrt(2.0)) * a1_m)
	tmp = 0.0
	if (cos(th) <= -5e-310)
		tmp = Float64(Float64(Float64(-a2_m) * t_1) + t_2);
	else
		tmp = Float64(Float64(t_1 * a2_m) + t_2);
	end
	return tmp
end

MATLAB

a2_m = abs(a2);
a1_m = abs(a1);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp_2 = code(a1_m, a2_m, th)
	t_1 = a2_m / sqrt(2.0);
	t_2 = (a1_m / sqrt(2.0)) * a1_m;
	tmp = 0.0;
	if (cos(th) <= -5e-310)
		tmp = (-a2_m * t_1) + t_2;
	else
		tmp = (t_1 * a2_m) + t_2;
	end
	tmp_2 = tmp;
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := Block[{t$95$1 = N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a1$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a1$95$m), $MachinePrecision]}, If[LessEqual[N[Cos[th], $MachinePrecision], -5e-310], N[(N[((-a2$95$m) * t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[(t$95$1 * a2$95$m), $MachinePrecision] + t$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -4.999999999999985e-310

   1. Initial program 99.6%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 55.6

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 8.7

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

   8. Applied rewrites 8.7%

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

   10. Applied rewrites 31.0%

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

   if -4.999999999999985e-310 < (cos.f64 th)

   1. Initial program 99.6%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 91.6

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 83.3

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

   8. Applied rewrites 83.3%

      \[\leadsto \frac{a1}{\sqrt{2}} \cdot a1 + \color{blue}{a2 \cdot \frac{a2}{\sqrt{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-a2\right) \cdot \frac{a2}{\sqrt{2}} + \frac{a1}{\sqrt{2}} \cdot a1\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot a2 + \frac{a1}{\sqrt{2}} \cdot a1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \left(\frac{a2\_m}{\sqrt{2}} \cdot \cos th\right) \cdot a2\_m + \frac{a1\_m}{\sqrt{2}} \cdot a1\_m \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (+ (* (* (/ a2_m (sqrt 2.0)) (cos th)) a2_m) (* (/ a1_m (sqrt 2.0)) a1_m)))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return (((a2_m / sqrt(2.0)) * cos(th)) * a2_m) + ((a1_m / sqrt(2.0)) * a1_m);
}

Fortran

a2_m = abs(a2)
a1_m = abs(a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = (((a2_m / sqrt(2.0d0)) * cos(th)) * a2_m) + ((a1_m / sqrt(2.0d0)) * a1_m)
end function

Java

a2_m = Math.abs(a2);
a1_m = Math.abs(a1);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return (((a2_m / Math.sqrt(2.0)) * Math.cos(th)) * a2_m) + ((a1_m / Math.sqrt(2.0)) * a1_m);
}

Python

a2_m = math.fabs(a2)
a1_m = math.fabs(a1)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return (((a2_m / math.sqrt(2.0)) * math.cos(th)) * a2_m) + ((a1_m / math.sqrt(2.0)) * a1_m)

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(Float64(Float64(Float64(a2_m / sqrt(2.0)) * cos(th)) * a2_m) + Float64(Float64(a1_m / sqrt(2.0)) * a1_m))
end

MATLAB

a2_m = abs(a2);
a1_m = abs(a1);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = (((a2_m / sqrt(2.0)) * cos(th)) * a2_m) + ((a1_m / sqrt(2.0)) * a1_m);
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(N[(N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * a2$95$m), $MachinePrecision] + N[(N[(a1$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a1$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 82.5

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

5. Applied rewrites 82.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. frac-2neg N/A

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

   7. associate-*l/ N/A

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

   8. neg-mul-1 N/A

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

   9. times-frac N/A

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

   10. metadata-eval N/A

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

   11. frac-2neg N/A

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

   12. /-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 82.5

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

7. Applied rewrites 82.5%

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

8. Final simplification 82.5%

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

Alternative 8: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \left(\frac{a2\_m}{\sqrt{2}} \cdot a2\_m\right) \cdot \cos th + \frac{a1\_m}{\sqrt{2}} \cdot a1\_m \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (+ (* (* (/ a2_m (sqrt 2.0)) a2_m) (cos th)) (* (/ a1_m (sqrt 2.0)) a1_m)))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return (((a2_m / sqrt(2.0)) * a2_m) * cos(th)) + ((a1_m / sqrt(2.0)) * a1_m);
}

Fortran

a2_m = abs(a2)
a1_m = abs(a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = (((a2_m / sqrt(2.0d0)) * a2_m) * cos(th)) + ((a1_m / sqrt(2.0d0)) * a1_m)
end function

Java

a2_m = Math.abs(a2);
a1_m = Math.abs(a1);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return (((a2_m / Math.sqrt(2.0)) * a2_m) * Math.cos(th)) + ((a1_m / Math.sqrt(2.0)) * a1_m);
}

Python

a2_m = math.fabs(a2)
a1_m = math.fabs(a1)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return (((a2_m / math.sqrt(2.0)) * a2_m) * math.cos(th)) + ((a1_m / math.sqrt(2.0)) * a1_m)

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(Float64(Float64(Float64(a2_m / sqrt(2.0)) * a2_m) * cos(th)) + Float64(Float64(a1_m / sqrt(2.0)) * a1_m))
end

MATLAB

a2_m = abs(a2);
a1_m = abs(a1);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = (((a2_m / sqrt(2.0)) * a2_m) * cos(th)) + ((a1_m / sqrt(2.0)) * a1_m);
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(N[(N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a2$95$m), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] + N[(N[(a1$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a1$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 82.5

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

5. Applied rewrites 82.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. lift-/.f64 N/A

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

   8. associate-/r/ N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 82.5

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

7. Applied rewrites 82.5%

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

8. Final simplification 82.5%

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

Alternative 9: 66.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \frac{a2\_m}{\sqrt{2}} \cdot a2\_m + \frac{a1\_m}{\sqrt{2}} \cdot a1\_m \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (+ (* (/ a2_m (sqrt 2.0)) a2_m) (* (/ a1_m (sqrt 2.0)) a1_m)))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return ((a2_m / sqrt(2.0)) * a2_m) + ((a1_m / sqrt(2.0)) * a1_m);
}

Fortran

a2_m = abs(a2)
a1_m = abs(a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = ((a2_m / sqrt(2.0d0)) * a2_m) + ((a1_m / sqrt(2.0d0)) * a1_m)
end function

Java

a2_m = Math.abs(a2);
a1_m = Math.abs(a1);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return ((a2_m / Math.sqrt(2.0)) * a2_m) + ((a1_m / Math.sqrt(2.0)) * a1_m);
}

Python

a2_m = math.fabs(a2)
a1_m = math.fabs(a1)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return ((a2_m / math.sqrt(2.0)) * a2_m) + ((a1_m / math.sqrt(2.0)) * a1_m)

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(Float64(Float64(a2_m / sqrt(2.0)) * a2_m) + Float64(Float64(a1_m / sqrt(2.0)) * a1_m))
end

MATLAB

a2_m = abs(a2);
a1_m = abs(a1);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = ((a2_m / sqrt(2.0)) * a2_m) + ((a1_m / sqrt(2.0)) * a1_m);
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a2$95$m), $MachinePrecision] + N[(N[(a1$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a1$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 82.5

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

5. Applied rewrites 82.5%

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

6. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 64.4

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

8. Applied rewrites 64.4%

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

9. Final simplification 64.4%

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

Reproduce

herbie shell --seed 2024312 
(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))))