Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 11.5s

Alternatives: 14

Speedup: 2.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[Power[N[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 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/ 99.5%

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

   4. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. fma-def 99.5%

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

   2. div-inv 99.5%

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

   3. add-sqr-sqrt 99.5%

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

   4. pow2 99.5%

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

   5. hypot-def 99.5%

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

   6. pow1/2 99.5%

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

   7. pow-flip 99.6%

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

   8. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

   \[\leadsto \cos th \cdot \left({\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2} \cdot {2}^{-0.5}\right) \]

Alternative 2: 78.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.9:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.9)
   (* a2 (/ a2 (/ (sqrt 2.0) (cos th))))
   (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.9) {
		tmp = a2 * (a2 / (sqrt(2.0) / cos(th)));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= 0.9d0) then
        tmp = a2 * (a2 / (sqrt(2.0d0) / cos(th)))
    else
        tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.9) {
		tmp = a2 * (a2 / (Math.sqrt(2.0) / Math.cos(th)));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.9:
		tmp = a2 * (a2 / (math.sqrt(2.0) / math.cos(th)))
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.9)
		tmp = Float64(a2 * Float64(a2 / Float64(sqrt(2.0) / cos(th))));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.9)
		tmp = a2 * (a2 / (sqrt(2.0) / cos(th)));
	else
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.9], N[(a2 * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.900000000000000022

   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/ 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a1 around 0 58.2%

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

      1. unpow2 58.2%

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

      2. associate-*l* 58.2%

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

      3. associate-*r/ 58.1%

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

      4. associate-/l* 58.2%

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

   6. Simplified 58.2%

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

   if 0.900000000000000022 < (cos.f64 th)

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

      3. pow1/2 99.5%

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

      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 94.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.9:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 3: 68.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 4.5 \cdot 10^{-109}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 4 \cdot 10^{-76} \lor \neg \left(a2 \leq 3.6 \cdot 10^{-35}\right):\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 4.5e-109)
   (* (cos th) (* (* a1 a1) (sqrt 0.5)))
   (if (or (<= a2 4e-76) (not (<= a2 3.6e-35)))
     (* a2 (/ a2 (/ (sqrt 2.0) (cos th))))
     (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 4.5e-109) {
		tmp = cos(th) * ((a1 * a1) * sqrt(0.5));
	} else if ((a2 <= 4e-76) || !(a2 <= 3.6e-35)) {
		tmp = a2 * (a2 / (sqrt(2.0) / cos(th)));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (a2 <= 4.5d-109) then
        tmp = cos(th) * ((a1 * a1) * sqrt(0.5d0))
    else if ((a2 <= 4d-76) .or. (.not. (a2 <= 3.6d-35))) then
        tmp = a2 * (a2 / (sqrt(2.0d0) / cos(th)))
    else
        tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 4.5e-109) {
		tmp = Math.cos(th) * ((a1 * a1) * Math.sqrt(0.5));
	} else if ((a2 <= 4e-76) || !(a2 <= 3.6e-35)) {
		tmp = a2 * (a2 / (Math.sqrt(2.0) / Math.cos(th)));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 4.5e-109:
		tmp = math.cos(th) * ((a1 * a1) * math.sqrt(0.5))
	elif (a2 <= 4e-76) or not (a2 <= 3.6e-35):
		tmp = a2 * (a2 / (math.sqrt(2.0) / math.cos(th)))
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 4.5e-109)
		tmp = Float64(cos(th) * Float64(Float64(a1 * a1) * sqrt(0.5)));
	elseif ((a2 <= 4e-76) || !(a2 <= 3.6e-35))
		tmp = Float64(a2 * Float64(a2 / Float64(sqrt(2.0) / cos(th))));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 4.5e-109)
		tmp = cos(th) * ((a1 * a1) * sqrt(0.5));
	elseif ((a2 <= 4e-76) || ~((a2 <= 3.6e-35)))
		tmp = a2 * (a2 / (sqrt(2.0) / cos(th)));
	else
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 4.5e-109], N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[a2, 4e-76], N[Not[LessEqual[a2, 3.6e-35]], $MachinePrecision]], N[(a2 * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a2 < 4.5000000000000001e-109

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

      1. fma-def 99.5%

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

      2. div-inv 99.5%

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

      3. add-sqr-sqrt 99.5%

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

      4. pow2 99.5%

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

      5. hypot-def 99.5%

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

      6. pow1/2 99.5%

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

      7. pow-flip 99.6%

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

      8. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in a1 around inf 71.3%

      \[\leadsto \cos th \cdot \color{blue}{\left(\sqrt{0.5} \cdot {a1}^{2}\right)} \]
   7. Step-by-step derivation

      1. unpow2 71.3%

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

   8. Simplified 71.3%

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

   if 4.5000000000000001e-109 < a2 < 3.99999999999999971e-76 or 3.60000000000000019e-35 < a2

   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/ 99.6%

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

      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 73.1%

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

      1. unpow2 73.1%

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

      2. associate-*l* 73.1%

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

      3. associate-*r/ 73.0%

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

      4. associate-/l* 73.1%

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

   6. Simplified 73.1%

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

   if 3.99999999999999971e-76 < a2 < 3.60000000000000019e-35

   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%

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

   3. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 99.4%

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

      3. pow1/2 99.4%

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

      4. pow-flip 99.8%

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

      5. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in th around 0 62.2%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 4.5 \cdot 10^{-109}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 4 \cdot 10^{-76} \lor \neg \left(a2 \leq 3.6 \cdot 10^{-35}\right):\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 4: 68.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 3.25 \cdot 10^{-109}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 1.5 \cdot 10^{-76} \lor \neg \left(a2 \leq 2.7 \cdot 10^{-32}\right):\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 3.25e-109)
   (* (cos th) (* (* a1 a1) (sqrt 0.5)))
   (if (or (<= a2 1.5e-76) (not (<= a2 2.7e-32)))
     (* (cos th) (/ a2 (/ (sqrt 2.0) a2)))
     (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 3.25e-109) {
		tmp = cos(th) * ((a1 * a1) * sqrt(0.5));
	} else if ((a2 <= 1.5e-76) || !(a2 <= 2.7e-32)) {
		tmp = cos(th) * (a2 / (sqrt(2.0) / a2));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (a2 <= 3.25d-109) then
        tmp = cos(th) * ((a1 * a1) * sqrt(0.5d0))
    else if ((a2 <= 1.5d-76) .or. (.not. (a2 <= 2.7d-32))) then
        tmp = cos(th) * (a2 / (sqrt(2.0d0) / a2))
    else
        tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 3.25e-109) {
		tmp = Math.cos(th) * ((a1 * a1) * Math.sqrt(0.5));
	} else if ((a2 <= 1.5e-76) || !(a2 <= 2.7e-32)) {
		tmp = Math.cos(th) * (a2 / (Math.sqrt(2.0) / a2));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 3.25e-109:
		tmp = math.cos(th) * ((a1 * a1) * math.sqrt(0.5))
	elif (a2 <= 1.5e-76) or not (a2 <= 2.7e-32):
		tmp = math.cos(th) * (a2 / (math.sqrt(2.0) / a2))
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 3.25e-109)
		tmp = Float64(cos(th) * Float64(Float64(a1 * a1) * sqrt(0.5)));
	elseif ((a2 <= 1.5e-76) || !(a2 <= 2.7e-32))
		tmp = Float64(cos(th) * Float64(a2 / Float64(sqrt(2.0) / a2)));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 3.25e-109)
		tmp = cos(th) * ((a1 * a1) * sqrt(0.5));
	elseif ((a2 <= 1.5e-76) || ~((a2 <= 2.7e-32)))
		tmp = cos(th) * (a2 / (sqrt(2.0) / a2));
	else
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 3.25e-109], N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[a2, 1.5e-76], N[Not[LessEqual[a2, 2.7e-32]], $MachinePrecision]], N[(N[Cos[th], $MachinePrecision] * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a2 < 3.2499999999999998e-109

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

      1. fma-def 99.5%

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

      2. div-inv 99.5%

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

      3. add-sqr-sqrt 99.5%

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

      4. pow2 99.5%

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

      5. hypot-def 99.5%

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

      6. pow1/2 99.5%

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

      7. pow-flip 99.6%

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

      8. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in a1 around inf 71.3%

      \[\leadsto \cos th \cdot \color{blue}{\left(\sqrt{0.5} \cdot {a1}^{2}\right)} \]
   7. Step-by-step derivation

      1. unpow2 71.3%

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

   8. Simplified 71.3%

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

   if 3.2499999999999998e-109 < a2 < 1.50000000000000012e-76 or 2.69999999999999981e-32 < a2

   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/ 99.6%

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

      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 73.1%

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

      1. unpow2 73.1%

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

      2. associate-/l* 73.1%

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

   6. Simplified 73.1%

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

   if 1.50000000000000012e-76 < a2 < 2.69999999999999981e-32

   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%

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

   3. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 99.4%

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

      3. pow1/2 99.4%

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

      4. pow-flip 99.8%

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

      5. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in th around 0 62.2%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 3.25 \cdot 10^{-109}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 1.5 \cdot 10^{-76} \lor \neg \left(a2 \leq 2.7 \cdot 10^{-32}\right):\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 5: 68.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 1.05 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{elif}\;a2 \leq 1.65 \cdot 10^{-76} \lor \neg \left(a2 \leq 5.3 \cdot 10^{-33}\right):\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 1.05e-109)
   (* (sqrt 0.5) (* a1 (* (cos th) a1)))
   (if (or (<= a2 1.65e-76) (not (<= a2 5.3e-33)))
     (* (cos th) (/ a2 (/ (sqrt 2.0) a2)))
     (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 1.05e-109) {
		tmp = sqrt(0.5) * (a1 * (cos(th) * a1));
	} else if ((a2 <= 1.65e-76) || !(a2 <= 5.3e-33)) {
		tmp = cos(th) * (a2 / (sqrt(2.0) / a2));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (a2 <= 1.05d-109) then
        tmp = sqrt(0.5d0) * (a1 * (cos(th) * a1))
    else if ((a2 <= 1.65d-76) .or. (.not. (a2 <= 5.3d-33))) then
        tmp = cos(th) * (a2 / (sqrt(2.0d0) / a2))
    else
        tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 1.05e-109) {
		tmp = Math.sqrt(0.5) * (a1 * (Math.cos(th) * a1));
	} else if ((a2 <= 1.65e-76) || !(a2 <= 5.3e-33)) {
		tmp = Math.cos(th) * (a2 / (Math.sqrt(2.0) / a2));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 1.05e-109:
		tmp = math.sqrt(0.5) * (a1 * (math.cos(th) * a1))
	elif (a2 <= 1.65e-76) or not (a2 <= 5.3e-33):
		tmp = math.cos(th) * (a2 / (math.sqrt(2.0) / a2))
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 1.05e-109)
		tmp = Float64(sqrt(0.5) * Float64(a1 * Float64(cos(th) * a1)));
	elseif ((a2 <= 1.65e-76) || !(a2 <= 5.3e-33))
		tmp = Float64(cos(th) * Float64(a2 / Float64(sqrt(2.0) / a2)));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 1.05e-109)
		tmp = sqrt(0.5) * (a1 * (cos(th) * a1));
	elseif ((a2 <= 1.65e-76) || ~((a2 <= 5.3e-33)))
		tmp = cos(th) * (a2 / (sqrt(2.0) / a2));
	else
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 1.05e-109], N[(N[Sqrt[0.5], $MachinePrecision] * N[(a1 * N[(N[Cos[th], $MachinePrecision] * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[a2, 1.65e-76], N[Not[LessEqual[a2, 5.3e-33]], $MachinePrecision]], N[(N[Cos[th], $MachinePrecision] * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a2 < 1.04999999999999998e-109

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

      1. fma-def 99.5%

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

      2. div-inv 99.5%

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

      3. add-sqr-sqrt 99.5%

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

      4. pow2 99.5%

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

      5. hypot-def 99.5%

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

      6. pow1/2 99.5%

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

      7. pow-flip 99.6%

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

      8. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in a1 around inf 71.4%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left({a1}^{2} \cdot \cos th\right)} \]
   7. Step-by-step derivation

      1. unpow2 71.4%

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

      2. associate-*r* 71.4%

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

   8. Simplified 71.4%

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

   if 1.04999999999999998e-109 < a2 < 1.64999999999999992e-76 or 5.29999999999999968e-33 < a2

   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/ 99.6%

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

      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 73.1%

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

      1. unpow2 73.1%

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

      2. associate-/l* 73.1%

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

   6. Simplified 73.1%

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

   if 1.64999999999999992e-76 < a2 < 5.29999999999999968e-33

   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%

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

   3. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 99.4%

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

      3. pow1/2 99.4%

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

      4. pow-flip 99.8%

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

      5. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in th around 0 62.2%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 1.05 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{elif}\;a2 \leq 1.65 \cdot 10^{-76} \lor \neg \left(a2 \leq 5.3 \cdot 10^{-33}\right):\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 6: 68.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 3.75 \cdot 10^{-109}:\\ \;\;\;\;\frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1}}\\ \mathbf{elif}\;a2 \leq 2.2 \cdot 10^{-76} \lor \neg \left(a2 \leq 2.2 \cdot 10^{-32}\right):\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 3.75e-109)
   (/ (cos th) (/ (sqrt 2.0) (* a1 a1)))
   (if (or (<= a2 2.2e-76) (not (<= a2 2.2e-32)))
     (* (cos th) (/ a2 (/ (sqrt 2.0) a2)))
     (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 3.75e-109) {
		tmp = cos(th) / (sqrt(2.0) / (a1 * a1));
	} else if ((a2 <= 2.2e-76) || !(a2 <= 2.2e-32)) {
		tmp = cos(th) * (a2 / (sqrt(2.0) / a2));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (a2 <= 3.75d-109) then
        tmp = cos(th) / (sqrt(2.0d0) / (a1 * a1))
    else if ((a2 <= 2.2d-76) .or. (.not. (a2 <= 2.2d-32))) then
        tmp = cos(th) * (a2 / (sqrt(2.0d0) / a2))
    else
        tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 3.75e-109) {
		tmp = Math.cos(th) / (Math.sqrt(2.0) / (a1 * a1));
	} else if ((a2 <= 2.2e-76) || !(a2 <= 2.2e-32)) {
		tmp = Math.cos(th) * (a2 / (Math.sqrt(2.0) / a2));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 3.75e-109:
		tmp = math.cos(th) / (math.sqrt(2.0) / (a1 * a1))
	elif (a2 <= 2.2e-76) or not (a2 <= 2.2e-32):
		tmp = math.cos(th) * (a2 / (math.sqrt(2.0) / a2))
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 3.75e-109)
		tmp = Float64(cos(th) / Float64(sqrt(2.0) / Float64(a1 * a1)));
	elseif ((a2 <= 2.2e-76) || !(a2 <= 2.2e-32))
		tmp = Float64(cos(th) * Float64(a2 / Float64(sqrt(2.0) / a2)));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 3.75e-109)
		tmp = cos(th) / (sqrt(2.0) / (a1 * a1));
	elseif ((a2 <= 2.2e-76) || ~((a2 <= 2.2e-32)))
		tmp = cos(th) * (a2 / (sqrt(2.0) / a2));
	else
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 3.75e-109], N[(N[Cos[th], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[a2, 2.2e-76], N[Not[LessEqual[a2, 2.2e-32]], $MachinePrecision]], N[(N[Cos[th], $MachinePrecision] * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a2 < 3.74999999999999991e-109

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a1 around inf 71.3%

      \[\leadsto \cos th \cdot \color{blue}{\frac{{a1}^{2}}{\sqrt{2}}} \]
   5. Step-by-step derivation

      1. unpow2 71.3%

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

      2. associate-/l* 71.3%

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

   6. Simplified 71.3%

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

   7. Taylor expanded in th around inf 71.3%

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

      1. unpow2 71.3%

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

      2. *-commutative 71.3%

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

      3. associate-/l* 71.3%

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

   9. Simplified 71.3%

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

   if 3.74999999999999991e-109 < a2 < 2.19999999999999999e-76 or 2.2e-32 < a2

   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/ 99.6%

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

      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 73.1%

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

      1. unpow2 73.1%

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

      2. associate-/l* 73.1%

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

   6. Simplified 73.1%

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

   if 2.19999999999999999e-76 < a2 < 2.2e-32

   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%

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

   3. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 99.4%

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

      3. pow1/2 99.4%

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

      4. pow-flip 99.8%

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

      5. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in th around 0 62.2%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 3.75 \cdot 10^{-109}:\\ \;\;\;\;\frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1}}\\ \mathbf{elif}\;a2 \leq 2.2 \cdot 10^{-76} \lor \neg \left(a2 \leq 2.2 \cdot 10^{-32}\right):\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 7: 68.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sqrt{2}}{a2}\\ \mathbf{if}\;a2 \leq 4.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1}}\\ \mathbf{elif}\;a2 \leq 1.5 \cdot 10^{-76}:\\ \;\;\;\;\cos th \cdot \frac{a2}{t_1}\\ \mathbf{elif}\;a2 \leq 9 \cdot 10^{-33}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos th \cdot a2}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (sqrt 2.0) a2)))
   (if (<= a2 4.5e-109)
     (/ (cos th) (/ (sqrt 2.0) (* a1 a1)))
     (if (<= a2 1.5e-76)
       (* (cos th) (/ a2 t_1))
       (if (<= a2 9e-33)
         (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5))
         (/ (* (cos th) a2) t_1))))))

C

double code(double a1, double a2, double th) {
	double t_1 = sqrt(2.0) / a2;
	double tmp;
	if (a2 <= 4.5e-109) {
		tmp = cos(th) / (sqrt(2.0) / (a1 * a1));
	} else if (a2 <= 1.5e-76) {
		tmp = cos(th) * (a2 / t_1);
	} else if (a2 <= 9e-33) {
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	} else {
		tmp = (cos(th) * a2) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(2.0d0) / a2
    if (a2 <= 4.5d-109) then
        tmp = cos(th) / (sqrt(2.0d0) / (a1 * a1))
    else if (a2 <= 1.5d-76) then
        tmp = cos(th) * (a2 / t_1)
    else if (a2 <= 9d-33) then
        tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
    else
        tmp = (cos(th) * a2) / t_1
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.sqrt(2.0) / a2;
	double tmp;
	if (a2 <= 4.5e-109) {
		tmp = Math.cos(th) / (Math.sqrt(2.0) / (a1 * a1));
	} else if (a2 <= 1.5e-76) {
		tmp = Math.cos(th) * (a2 / t_1);
	} else if (a2 <= 9e-33) {
		tmp = ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
	} else {
		tmp = (Math.cos(th) * a2) / t_1;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = math.sqrt(2.0) / a2
	tmp = 0
	if a2 <= 4.5e-109:
		tmp = math.cos(th) / (math.sqrt(2.0) / (a1 * a1))
	elif a2 <= 1.5e-76:
		tmp = math.cos(th) * (a2 / t_1)
	elif a2 <= 9e-33:
		tmp = ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)
	else:
		tmp = (math.cos(th) * a2) / t_1
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(sqrt(2.0) / a2)
	tmp = 0.0
	if (a2 <= 4.5e-109)
		tmp = Float64(cos(th) / Float64(sqrt(2.0) / Float64(a1 * a1)));
	elseif (a2 <= 1.5e-76)
		tmp = Float64(cos(th) * Float64(a2 / t_1));
	elseif (a2 <= 9e-33)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5));
	else
		tmp = Float64(Float64(cos(th) * a2) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = sqrt(2.0) / a2;
	tmp = 0.0;
	if (a2 <= 4.5e-109)
		tmp = cos(th) / (sqrt(2.0) / (a1 * a1));
	elseif (a2 <= 1.5e-76)
		tmp = cos(th) * (a2 / t_1);
	elseif (a2 <= 9e-33)
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	else
		tmp = (cos(th) * a2) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]}, If[LessEqual[a2, 4.5e-109], N[(N[Cos[th], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a2, 1.5e-76], N[(N[Cos[th], $MachinePrecision] * N[(a2 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[a2, 9e-33], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if a2 < 4.5000000000000001e-109

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a1 around inf 71.3%

      \[\leadsto \cos th \cdot \color{blue}{\frac{{a1}^{2}}{\sqrt{2}}} \]
   5. Step-by-step derivation

      1. unpow2 71.3%

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

      2. associate-/l* 71.3%

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

   6. Simplified 71.3%

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

   7. Taylor expanded in th around inf 71.3%

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

      1. unpow2 71.3%

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

      2. *-commutative 71.3%

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

      3. associate-/l* 71.3%

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

   9. Simplified 71.3%

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

   if 4.5000000000000001e-109 < a2 < 1.50000000000000012e-76

   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/ 99.8%

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

      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 27.2%

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

      1. unpow2 27.2%

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

      2. associate-/l* 27.2%

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

   6. Simplified 27.2%

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

   if 1.50000000000000012e-76 < a2 < 8.99999999999999982e-33

   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%

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

   3. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 99.4%

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

      3. pow1/2 99.4%

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

      4. pow-flip 99.8%

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

      5. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in th around 0 62.2%

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

   if 8.99999999999999982e-33 < a2

   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/ 99.6%

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

      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 78.0%

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

      1. unpow2 78.0%

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

      2. associate-/l* 78.1%

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

   6. Simplified 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-*l/ 78.0%

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

   8. Applied egg-rr 78.0%

      \[\leadsto \color{blue}{\frac{a2 \cdot \cos th}{\frac{\sqrt{2}}{a2}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 4.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1}}\\ \mathbf{elif}\;a2 \leq 1.5 \cdot 10^{-76}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{elif}\;a2 \leq 9 \cdot 10^{-33}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos th \cdot a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 8: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 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/ 99.5%

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

   3. pow1/2 99.5%

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

   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. Final simplification 99.6%

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

Alternative 9: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 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. Final simplification 99.6%

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

Alternative 10: 67.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a1 \leq -1 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot \left(a1 + -0.5 \cdot \left(a1 \cdot \left(th \cdot th\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a1 -1e+156)
   (* (sqrt 0.5) (* a1 (+ a1 (* -0.5 (* a1 (* th th))))))
   (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= -1e+156) {
		tmp = sqrt(0.5) * (a1 * (a1 + (-0.5 * (a1 * (th * th)))));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (a1 <= (-1d+156)) then
        tmp = sqrt(0.5d0) * (a1 * (a1 + ((-0.5d0) * (a1 * (th * th)))))
    else
        tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= -1e+156) {
		tmp = Math.sqrt(0.5) * (a1 * (a1 + (-0.5 * (a1 * (th * th)))));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a1 <= -1e+156:
		tmp = math.sqrt(0.5) * (a1 * (a1 + (-0.5 * (a1 * (th * th)))))
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a1 <= -1e+156)
		tmp = Float64(sqrt(0.5) * Float64(a1 * Float64(a1 + Float64(-0.5 * Float64(a1 * Float64(th * th))))));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a1 <= -1e+156)
		tmp = sqrt(0.5) * (a1 * (a1 + (-0.5 * (a1 * (th * th)))));
	else
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a1, -1e+156], N[(N[Sqrt[0.5], $MachinePrecision] * N[(a1 * N[(a1 + N[(-0.5 * N[(a1 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a1 < -9.9999999999999998e155

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-def 100.0%

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

      2. div-inv 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. pow2 100.0%

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

      5. hypot-def 100.0%

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

      6. pow1/2 100.0%

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

      7. pow-flip 100.0%

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

      8. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in a1 around inf 100.0%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left({a1}^{2} \cdot \cos th\right)} \]
   7. Step-by-step derivation

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in th around 0 90.0%

      \[\leadsto \sqrt{0.5} \cdot \left(a1 \cdot \color{blue}{\left(a1 + -0.5 \cdot \left(a1 \cdot {th}^{2}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. *-commutative 90.0%

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

       2. unpow2 90.0%

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

   11. Simplified 90.0%

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

   if -9.9999999999999998e155 < a1

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

      3. pow1/2 99.5%

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

      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 0 63.5%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \leq -1 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot \left(a1 + -0.5 \cdot \left(a1 \cdot \left(th \cdot th\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 11: 46.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 3.5 \cdot 10^{-66}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;a2 \leq 1.9 \cdot 10^{+164} \lor \neg \left(a2 \leq 1.45 \cdot 10^{+173}\right):\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \left(-a2\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 3.5e-66)
   (* (* a1 a1) (sqrt 0.5))
   (if (or (<= a2 1.9e+164) (not (<= a2 1.45e+173)))
     (* (* a2 a2) (sqrt 0.5))
     (/ (* a2 (- a2)) (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 3.5e-66) {
		tmp = (a1 * a1) * sqrt(0.5);
	} else if ((a2 <= 1.9e+164) || !(a2 <= 1.45e+173)) {
		tmp = (a2 * a2) * sqrt(0.5);
	} 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 (a2 <= 3.5d-66) then
        tmp = (a1 * a1) * sqrt(0.5d0)
    else if ((a2 <= 1.9d+164) .or. (.not. (a2 <= 1.45d+173))) then
        tmp = (a2 * a2) * sqrt(0.5d0)
    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 (a2 <= 3.5e-66) {
		tmp = (a1 * a1) * Math.sqrt(0.5);
	} else if ((a2 <= 1.9e+164) || !(a2 <= 1.45e+173)) {
		tmp = (a2 * a2) * Math.sqrt(0.5);
	} else {
		tmp = (a2 * -a2) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 3.5e-66:
		tmp = (a1 * a1) * math.sqrt(0.5)
	elif (a2 <= 1.9e+164) or not (a2 <= 1.45e+173):
		tmp = (a2 * a2) * math.sqrt(0.5)
	else:
		tmp = (a2 * -a2) / math.sqrt(2.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 3.5e-66)
		tmp = Float64(Float64(a1 * a1) * sqrt(0.5));
	elseif ((a2 <= 1.9e+164) || !(a2 <= 1.45e+173))
		tmp = Float64(Float64(a2 * a2) * sqrt(0.5));
	else
		tmp = Float64(Float64(a2 * Float64(-a2)) / sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 3.5e-66)
		tmp = (a1 * a1) * sqrt(0.5);
	elseif ((a2 <= 1.9e+164) || ~((a2 <= 1.45e+173)))
		tmp = (a2 * a2) * sqrt(0.5);
	else
		tmp = (a2 * -a2) / sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 3.5e-66], N[(N[(a1 * a1), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[a2, 1.9e+164], N[Not[LessEqual[a2, 1.45e+173]], $MachinePrecision]], N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[(a2 * (-a2)), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a2 < 3.5e-66

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 64.3%

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

      1. unpow2 64.3%

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

      2. unpow2 64.3%

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

      3. +-commutative 64.3%

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

   6. Simplified 64.3%

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

   7. Taylor expanded in a1 around inf 49.1%

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

      1. unpow2 49.1%

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

   9. Simplified 49.1%

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

       1. div-inv 49.1%

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

       2. add-sqr-sqrt 49.1%

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

       3. sqrt-unprod 49.1%

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

       4. frac-times 49.1%

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

       5. metadata-eval 49.1%

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

       6. add-sqr-sqrt 49.2%

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

       7. metadata-eval 49.2%

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

   11. Applied egg-rr 49.2%

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

   if 3.5e-66 < a2 < 1.90000000000000011e164 or 1.45000000000000003e173 < a2

   1. Initial program 99.5%

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

      1. distribute-lft-out 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/ 99.5%

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

      3. pow1/2 99.5%

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

      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 0 65.1%

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

   7. Taylor expanded in a1 around 0 47.6%

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

      1. unpow2 47.6%

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

   9. Simplified 47.6%

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

   if 1.90000000000000011e164 < a2 < 1.45000000000000003e173

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in th around 0 0.0%

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

      1. unpow2 0.0%

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

      2. unpow2 0.0%

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

      3. +-commutative 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in a1 around 0 0.0%

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

      1. unpow2 0.0%

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

   9. Simplified 0.0%

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

   11. Applied egg-rr 100.0%

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

       1. distribute-neg-frac 100.0%

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

       2. distribute-rgt-neg-out 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 3.5 \cdot 10^{-66}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;a2 \leq 1.9 \cdot 10^{+164} \lor \neg \left(a2 \leq 1.45 \cdot 10^{+173}\right):\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \left(-a2\right)}{\sqrt{2}}\\ \end{array} \]

Alternative 12: 66.8% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 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/ 99.5%

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

   3. pow1/2 99.5%

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

   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 0 63.9%

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

7. Final simplification 63.9%

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

Alternative 13: 47.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 4.4 \cdot 10^{-66}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 4.4e-66) (* (* a1 a1) (sqrt 0.5)) (* (* a2 a2) (sqrt 0.5))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 4.4e-66) {
		tmp = (a1 * a1) * sqrt(0.5);
	} else {
		tmp = (a2 * a2) * sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (a2 <= 4.4d-66) then
        tmp = (a1 * a1) * sqrt(0.5d0)
    else
        tmp = (a2 * a2) * sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 4.4e-66) {
		tmp = (a1 * a1) * Math.sqrt(0.5);
	} else {
		tmp = (a2 * a2) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 4.4e-66:
		tmp = (a1 * a1) * math.sqrt(0.5)
	else:
		tmp = (a2 * a2) * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 4.4e-66)
		tmp = Float64(Float64(a1 * a1) * sqrt(0.5));
	else
		tmp = Float64(Float64(a2 * a2) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 4.4e-66)
		tmp = (a1 * a1) * sqrt(0.5);
	else
		tmp = (a2 * a2) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 4.4e-66], N[(N[(a1 * a1), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 4.4000000000000002e-66

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 64.3%

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

      1. unpow2 64.3%

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

      2. unpow2 64.3%

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

      3. +-commutative 64.3%

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

   6. Simplified 64.3%

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

   7. Taylor expanded in a1 around inf 49.1%

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

      1. unpow2 49.1%

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

   9. Simplified 49.1%

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

       1. div-inv 49.1%

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

       2. add-sqr-sqrt 49.1%

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

       3. sqrt-unprod 49.1%

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

       4. frac-times 49.1%

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

       5. metadata-eval 49.1%

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

       6. add-sqr-sqrt 49.2%

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

       7. metadata-eval 49.2%

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

   11. Applied egg-rr 49.2%

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

   if 4.4000000000000002e-66 < a2

   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/ 99.5%

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

      3. pow1/2 99.5%

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

      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 0 62.7%

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

   7. Taylor expanded in a1 around 0 45.9%

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

      1. unpow2 45.9%

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

   9. Simplified 45.9%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 4.4 \cdot 10^{-66}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 14: 39.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 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/ 99.5%

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

   3. pow1/2 99.5%

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

   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 0 63.9%

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

7. Taylor expanded in a1 around 0 35.0%

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

   1. unpow2 35.0%

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

9. Simplified 35.0%

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

10. Final simplification 35.0%

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

Reproduce

herbie shell --seed 2023171 
(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))))