Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.4%

Time: 10.2s

Alternatives: 16

Speedup: 2.0×

• 44.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% 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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[N[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

   1. fma-def 99.6%

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

   2. clear-num 99.6%

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

   3. un-div-inv 99.6%

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

   4. add-sqr-sqrt 99.6%

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

   5. pow2 99.6%

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

   6. hypot-def 99.7%

      \[\leadsto \frac{\cos th}{\frac{\sqrt{2}}{{\color{blue}{\left(\mathsf{hypot}\left(a1, a2\right)\right)}}^{2}}} \]

5. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\cos th}{\frac{\sqrt{2}}{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}}} \]

6. Final simplification 99.7%

   \[\leadsto \frac{\cos th}{\frac{\sqrt{2}}{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}} \]

Alternative 2: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 71.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.7)
   (* (cos th) (* a2 a2))
   (* (sqrt 0.5) (+ (* a2 a2) (* a1 a1)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = cos(th) * (a2 * a2);
	} else {
		tmp = sqrt(0.5) * ((a2 * a2) + (a1 * a1));
	}
	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 (cos(th) <= 0.7d0) then
        tmp = cos(th) * (a2 * a2)
    else
        tmp = sqrt(0.5d0) * ((a2 * a2) + (a1 * a1))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.7) {
		tmp = Math.cos(th) * (a2 * a2);
	} else {
		tmp = Math.sqrt(0.5) * ((a2 * a2) + (a1 * a1));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.7:
		tmp = math.cos(th) * (a2 * a2)
	else:
		tmp = math.sqrt(0.5) * ((a2 * a2) + (a1 * a1))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.7)
		tmp = Float64(cos(th) * Float64(a2 * a2));
	else
		tmp = Float64(sqrt(0.5) * Float64(Float64(a2 * a2) + Float64(a1 * a1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.7)
		tmp = cos(th) * (a2 * a2);
	else
		tmp = sqrt(0.5) * ((a2 * a2) + (a1 * a1));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.7], N[(N[Cos[th], $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69999999999999996

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. fma-def 99.6%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.6%

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

      4. add-sqr-sqrt 99.6%

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

      5. pow2 99.6%

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

      6. hypot-def 99.6%

         \[\leadsto \frac{\cos th}{\frac{\sqrt{2}}{{\color{blue}{\left(\mathsf{hypot}\left(a1, a2\right)\right)}}^{2}}} \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\cos th}{\frac{\sqrt{2}}{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}}} \]

   6. Taylor expanded in a1 around 0 58.2%

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

      1. unpow2 58.2%

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

   8. Simplified 58.2%

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

   10. Applied egg-rr 41.9%

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

       1. *-commutative 41.9%

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

       2. *-commutative 41.9%

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

       3. associate-*r* 41.9%

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

   12. Simplified 41.9%

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

   if 0.69999999999999996 < (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. Step-by-step derivation

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

      3. pow1/2 99.6%

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

      4. pow-flip 99.7%

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

      5. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in th around 0 90.9%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \end{array} \]

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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) * sqrt(0.5d0)) * ((a2 * a2) + (a1 * a1))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.6%

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

   3. pow1/2 99.6%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around inf 99.6%

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

7. Final simplification 99.6%

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

Alternative 5: 47.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.7) (* (cos th) (* a2 a2)) (* a2 (/ a2 (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = cos(th) * (a2 * a2);
	} else {
		tmp = a2 * (a2 / sqrt(2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= 0.7d0) then
        tmp = cos(th) * (a2 * a2)
    else
        tmp = a2 * (a2 / sqrt(2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.7) {
		tmp = Math.cos(th) * (a2 * a2);
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.7:
		tmp = math.cos(th) * (a2 * a2)
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.7)
		tmp = Float64(cos(th) * Float64(a2 * a2));
	else
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.7)
		tmp = cos(th) * (a2 * a2);
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.7], N[(N[Cos[th], $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69999999999999996

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. fma-def 99.6%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.6%

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

      4. add-sqr-sqrt 99.6%

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

      5. pow2 99.6%

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

      6. hypot-def 99.6%

         \[\leadsto \frac{\cos th}{\frac{\sqrt{2}}{{\color{blue}{\left(\mathsf{hypot}\left(a1, a2\right)\right)}}^{2}}} \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\cos th}{\frac{\sqrt{2}}{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}}} \]

   6. Taylor expanded in a1 around 0 58.2%

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

      1. unpow2 58.2%

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

   8. Simplified 58.2%

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

   10. Applied egg-rr 41.9%

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

       1. *-commutative 41.9%

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

       2. *-commutative 41.9%

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

       3. associate-*r* 41.9%

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

   12. Simplified 41.9%

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

   if 0.69999999999999996 < (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. Step-by-step derivation

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around 0 54.2%

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

      1. unpow2 54.2%

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

      2. associate-*l* 54.2%

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

      3. associate-*r/ 54.3%

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

      4. associate-/l* 54.3%

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

   6. Simplified 54.3%

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

   7. Taylor expanded in th around 0 50.2%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 6: 47.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.7) (* (cos th) (* a2 a2)) (/ a2 (/ (sqrt 2.0) a2))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = cos(th) * (a2 * a2);
	} else {
		tmp = a2 / (sqrt(2.0) / a2);
	}
	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 (cos(th) <= 0.7d0) then
        tmp = cos(th) * (a2 * a2)
    else
        tmp = a2 / (sqrt(2.0d0) / a2)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.7) {
		tmp = Math.cos(th) * (a2 * a2);
	} else {
		tmp = a2 / (Math.sqrt(2.0) / a2);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.7:
		tmp = math.cos(th) * (a2 * a2)
	else:
		tmp = a2 / (math.sqrt(2.0) / a2)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.7)
		tmp = Float64(cos(th) * Float64(a2 * a2));
	else
		tmp = Float64(a2 / Float64(sqrt(2.0) / a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.7)
		tmp = cos(th) * (a2 * a2);
	else
		tmp = a2 / (sqrt(2.0) / a2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.7], N[(N[Cos[th], $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision], N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69999999999999996

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. fma-def 99.6%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.6%

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

      4. add-sqr-sqrt 99.6%

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

      5. pow2 99.6%

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

      6. hypot-def 99.6%

         \[\leadsto \frac{\cos th}{\frac{\sqrt{2}}{{\color{blue}{\left(\mathsf{hypot}\left(a1, a2\right)\right)}}^{2}}} \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\cos th}{\frac{\sqrt{2}}{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}}} \]

   6. Taylor expanded in a1 around 0 58.2%

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

      1. unpow2 58.2%

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

   8. Simplified 58.2%

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

   10. Applied egg-rr 41.9%

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

       1. *-commutative 41.9%

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

       2. *-commutative 41.9%

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

       3. associate-*r* 41.9%

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

   12. Simplified 41.9%

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

   if 0.69999999999999996 < (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. Step-by-step derivation

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around 0 54.2%

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

      1. unpow2 54.2%

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

      2. associate-*l* 54.2%

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

      3. associate-*r/ 54.3%

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

      4. associate-/l* 54.3%

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

   6. Simplified 54.3%

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

   7. Taylor expanded in th around 0 50.2%

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

   8. Taylor expanded in a2 around 0 50.2%

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

      1. unpow2 50.2%

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

      2. associate-/l* 50.2%

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

   10. Simplified 50.2%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 7: 47.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.7) (* (cos th) (* a2 a2)) (/ (* a2 a2) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = cos(th) * (a2 * a2);
	} else {
		tmp = (a2 * a2) / sqrt(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= 0.7d0) then
        tmp = cos(th) * (a2 * a2)
    else
        tmp = (a2 * a2) / sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.7) {
		tmp = Math.cos(th) * (a2 * a2);
	} else {
		tmp = (a2 * a2) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.7:
		tmp = math.cos(th) * (a2 * a2)
	else:
		tmp = (a2 * a2) / math.sqrt(2.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.7)
		tmp = Float64(cos(th) * Float64(a2 * a2));
	else
		tmp = Float64(Float64(a2 * a2) / sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.7)
		tmp = cos(th) * (a2 * a2);
	else
		tmp = (a2 * a2) / sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.7], N[(N[Cos[th], $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69999999999999996

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. fma-def 99.6%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.6%

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

      4. add-sqr-sqrt 99.6%

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

      5. pow2 99.6%

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

      6. hypot-def 99.6%

         \[\leadsto \frac{\cos th}{\frac{\sqrt{2}}{{\color{blue}{\left(\mathsf{hypot}\left(a1, a2\right)\right)}}^{2}}} \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\cos th}{\frac{\sqrt{2}}{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}}} \]

   6. Taylor expanded in a1 around 0 58.2%

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

      1. unpow2 58.2%

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

   8. Simplified 58.2%

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

   10. Applied egg-rr 41.9%

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

       1. *-commutative 41.9%

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

       2. *-commutative 41.9%

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

       3. associate-*r* 41.9%

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

   12. Simplified 41.9%

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

   if 0.69999999999999996 < (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. Step-by-step derivation

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 90.8%

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

   5. Taylor expanded in a1 around 0 50.2%

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

      1. unpow2 50.2%

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

   7. Simplified 50.2%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]

Alternative 8: 57.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 55.6%

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

   1. unpow2 55.6%

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

   2. associate-*l* 55.6%

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

   3. associate-*r/ 55.6%

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

   4. associate-/l* 55.6%

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

6. Simplified 55.6%

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

7. Final simplification 55.6%

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

Alternative 9: 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 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 55.6%

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

   1. unpow2 55.6%

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

6. Simplified 55.6%

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

7. Final simplification 55.6%

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

Alternative 10: 57.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

   1. fma-def 99.6%

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

   2. clear-num 99.6%

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

   3. un-div-inv 99.6%

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

   4. add-sqr-sqrt 99.6%

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

   5. pow2 99.6%

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

   6. hypot-def 99.7%

      \[\leadsto \frac{\cos th}{\frac{\sqrt{2}}{{\color{blue}{\left(\mathsf{hypot}\left(a1, a2\right)\right)}}^{2}}} \]

5. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\cos th}{\frac{\sqrt{2}}{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}}} \]

6. Taylor expanded in a1 around 0 55.6%

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

   1. unpow2 55.6%

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

8. Simplified 55.6%

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

9. Final simplification 55.6%

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

Alternative 11: 57.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 55.6%

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

   1. unpow2 55.6%

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

   2. *-commutative 55.6%

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

6. Simplified 55.6%

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

7. Final simplification 55.6%

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

Alternative 12: 39.6% 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 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 55.6%

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

   1. unpow2 55.6%

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

   2. associate-*l* 55.6%

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

   3. associate-*r/ 55.6%

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

   4. associate-/l* 55.6%

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

6. Simplified 55.6%

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

7. Taylor expanded in th around 0 39.8%

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

8. Final simplification 39.8%

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

Alternative 13: 29.9% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 4.4 \cdot 10^{+236}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(--0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(0.5 \cdot \left(-a2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 4.4e+236) (* a2 (* a2 (- -0.5))) (* a2 (* 0.5 (- a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 4.4e+236) {
		tmp = a2 * (a2 * -(-0.5));
	} else {
		tmp = a2 * (0.5 * -a2);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 4.4d+236) then
        tmp = a2 * (a2 * -(-0.5d0))
    else
        tmp = a2 * (0.5d0 * -a2)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 4.4e+236) {
		tmp = a2 * (a2 * -(-0.5));
	} else {
		tmp = a2 * (0.5 * -a2);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 4.4e+236:
		tmp = a2 * (a2 * -(-0.5))
	else:
		tmp = a2 * (0.5 * -a2)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 4.4e+236)
		tmp = Float64(a2 * Float64(a2 * Float64(-(-0.5))));
	else
		tmp = Float64(a2 * Float64(0.5 * Float64(-a2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 4.4e+236)
		tmp = a2 * (a2 * -(-0.5));
	else
		tmp = a2 * (0.5 * -a2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 4.4e+236], N[(a2 * N[(a2 * (--0.5)), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(0.5 * (-a2)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 4.39999999999999956e236

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around 0 55.8%

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

      1. unpow2 55.8%

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

      2. associate-*l* 55.8%

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

      3. associate-*r/ 55.8%

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

      4. associate-/l* 55.8%

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

   6. Simplified 55.8%

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

   7. Taylor expanded in th around 0 41.1%

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

      1. frac-2neg 41.1%

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

      2. div-inv 41.0%

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

   9. Applied egg-rr 41.0%

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

   11. Applied egg-rr 31.3%

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

   if 4.39999999999999956e236 < th

   1. Initial program 99.4%

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

      1. distribute-lft-out 99.4%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 51.9%

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

      1. unpow2 51.9%

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

      2. associate-*l* 51.8%

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

      3. associate-*r/ 51.8%

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

      4. associate-/l* 51.9%

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

   6. Simplified 51.9%

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

   7. Taylor expanded in th around 0 18.0%

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

      1. frac-2neg 18.0%

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

      2. div-inv 18.0%

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

   9. Applied egg-rr 18.0%

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

   11. Applied egg-rr 33.7%

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

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 4.4 \cdot 10^{+236}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(--0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(0.5 \cdot \left(-a2\right)\right)\\ \end{array} \]

Alternative 14: 29.9% accurate, 69.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a1, a2, th):
	return a2 * (a2 * -(-0.5))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 55.6%

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

   1. unpow2 55.6%

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

   2. associate-*l* 55.6%

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

   3. associate-*r/ 55.6%

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

   4. associate-/l* 55.6%

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

6. Simplified 55.6%

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

7. Taylor expanded in th around 0 39.8%

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

   1. frac-2neg 39.8%

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

   2. div-inv 39.8%

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

9. Applied egg-rr 39.8%

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

11. Applied egg-rr 30.6%

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

12. Final simplification 30.6%

    \[\leadsto a2 \cdot \left(a2 \cdot \left(--0.5\right)\right) \]

Alternative 15: 29.8% accurate, 138.3× speedup

Math

\[\begin{array}{l} \\ a2 \cdot a2 \end{array} \]

FPCore

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

C

double code(double a1, double a2, double th) {
	return 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 = a2 * a2
end function

Java

public static double code(double a1, double a2, double th) {
	return a2 * a2;
}

Python

def code(a1, a2, th):
	return a2 * a2

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 55.6%

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

   1. unpow2 55.6%

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

   2. associate-*l* 55.6%

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

   3. associate-*r/ 55.6%

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

   4. associate-/l* 55.6%

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

6. Simplified 55.6%

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

7. Taylor expanded in th around 0 39.8%

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

   1. frac-2neg 39.8%

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

   2. div-inv 39.8%

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

9. Applied egg-rr 39.8%

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

11. Applied egg-rr 30.4%

    \[\leadsto a2 \cdot \left(\left(-a2\right) \cdot \color{blue}{-1}\right) \]

12. Final simplification 30.4%

    \[\leadsto a2 \cdot a2 \]

Alternative 16: 3.4% accurate, 415.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 1.0)

C

double code(double a1, double a2, double th) {
	return 1.0;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = 1.0d0
end function

Java

public static double code(double a1, double a2, double th) {
	return 1.0;
}

Python

def code(a1, a2, th):
	return 1.0

Julia

function code(a1, a2, th)
	return 1.0
end

MATLAB

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

Wolfram

code[a1_, a2_, th_] := 1.0

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 55.6%

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

   1. unpow2 55.6%

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

   2. *-commutative 55.6%

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

6. Simplified 55.6%

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

   1. div-inv 55.5%

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

   2. *-commutative 55.5%

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

   3. associate-*r* 55.5%

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

   4. add-sqr-sqrt 55.5%

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

   5. sqrt-unprod 55.5%

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

   6. frac-times 55.5%

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

   7. metadata-eval 55.5%

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

   8. add-sqr-sqrt 55.6%

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

   9. metadata-eval 55.6%

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

8. Applied egg-rr 55.6%

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

9. Taylor expanded in th around 0 39.8%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot {a2}^{2}} \]
10. Step-by-step derivation

    1. unpow2 39.8%

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

11. Simplified 39.8%

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

13. Applied egg-rr 3.4%

    \[\leadsto \color{blue}{\frac{a2}{a2}} \]
14. Step-by-step derivation

    1. *-inverses 3.4%

       \[\leadsto \color{blue}{1} \]

15. Simplified 3.4%

    \[\leadsto \color{blue}{1} \]

16. Final simplification 3.4%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023254 
(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))))