Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 10.9s

Alternatives: 14

Speedup: 2.0×

• 43.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.3× speedup

Math

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

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (* (cos th) (/ (fma a1 a1 (* a2 a2)) (sqrt 2.0))))

C

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

Julia

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

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.7%

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

   4. fma-def 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 78.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.115:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.115)
   (* a2 (/ a2 (/ (sqrt 2.0) (cos th))))
   (* (sqrt 0.5) (+ (* a2 a2) (* a1 a1)))))

C

assert(a1 < a2);
double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.115) {
		tmp = a2 * (a2 / (sqrt(2.0) / cos(th)));
	} else {
		tmp = sqrt(0.5) * ((a2 * a2) + (a1 * a1));
	}
	return tmp;
}

Fortran

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
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) <= 0.115d0) then
        tmp = a2 * (a2 / (sqrt(2.0d0) / cos(th)))
    else
        tmp = sqrt(0.5d0) * ((a2 * a2) + (a1 * a1))
    end if
    code = tmp
end function

Java

assert a1 < a2;
public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.115) {
		tmp = a2 * (a2 / (Math.sqrt(2.0) / Math.cos(th)));
	} else {
		tmp = Math.sqrt(0.5) * ((a2 * a2) + (a1 * a1));
	}
	return tmp;
}

Python

[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.115:
		tmp = a2 * (a2 / (math.sqrt(2.0) / math.cos(th)))
	else:
		tmp = math.sqrt(0.5) * ((a2 * a2) + (a1 * a1))
	return tmp

Julia

a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.115)
		tmp = Float64(a2 * Float64(a2 / Float64(sqrt(2.0) / cos(th))));
	else
		tmp = Float64(sqrt(0.5) * Float64(Float64(a2 * a2) + Float64(a1 * a1)));
	end
	return tmp
end

MATLAB

a1, a2 = num2cell(sort([a1, a2])){:}
function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.115)
		tmp = a2 * (a2 / (sqrt(2.0) / cos(th)));
	else
		tmp = sqrt(0.5) * ((a2 * a2) + (a1 * a1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.115], N[(a2 * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.115000000000000005

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around 0 61.2%

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

      1. unpow2 61.2%

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

      2. associate-*l* 61.3%

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

      3. associate-*r/ 61.2%

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

      4. associate-/l* 61.2%

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

   6. Simplified 61.2%

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

   if 0.115000000000000005 < (cos.f64 th)

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.5%

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

      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 83.7%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left({a2}^{2} + {a1}^{2}\right)} \]
   7. Step-by-step derivation

      1. unpow2 83.7%

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

      2. unpow2 83.7%

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

   8. Simplified 83.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.115:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (* (* (cos th) (pow 2.0 -0.5)) (+ (* a2 a2) (* a1 a1))))

C

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

Fortran

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
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))) * ((a2 * a2) + (a1 * a1))
end function

Java

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

Python

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

Julia

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

MATLAB

a1, a2 = num2cell(sort([a1, a2])){:}
function tmp = code(a1, a2, th)
	tmp = (cos(th) * (2.0 ^ -0.5)) * ((a2 * a2) + (a1 * a1));
end

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.5%

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

   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%

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

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (* (sqrt 0.5) (* (cos th) (+ (* a2 a2) (* a1 a1)))))

C

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

Fortran

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = sqrt(0.5d0) * (cos(th) * ((a2 * a2) + (a1 * a1)))
end function

Java

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

Python

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

Julia

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

MATLAB

a1, a2 = num2cell(sort([a1, a2])){:}
function tmp = code(a1, a2, th)
	tmp = sqrt(0.5) * (cos(th) * ((a2 * a2) + (a1 * a1)));
end

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

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 inf 99.6%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left(\left({a2}^{2} + {a1}^{2}\right) \cdot \cos th\right)} \]
7. Step-by-step derivation

   1. unpow2 99.6%

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

   2. unpow2 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 5: 87.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ \begin{array}{l} \mathbf{if}\;a2 \leq 3.3 \cdot 10^{-127}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \end{array} \]

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 3.3e-127)
   (* (cos th) (/ a1 (/ (sqrt 2.0) a1)))
   (* a2 (/ a2 (/ (sqrt 2.0) (cos th))))))

C

assert(a1 < a2);
double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 3.3e-127) {
		tmp = cos(th) * (a1 / (sqrt(2.0) / a1));
	} else {
		tmp = a2 * (a2 / (sqrt(2.0) / cos(th)));
	}
	return tmp;
}

Fortran

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
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 <= 3.3d-127) then
        tmp = cos(th) * (a1 / (sqrt(2.0d0) / a1))
    else
        tmp = a2 * (a2 / (sqrt(2.0d0) / cos(th)))
    end if
    code = tmp
end function

Java

assert a1 < a2;
public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 3.3e-127) {
		tmp = Math.cos(th) * (a1 / (Math.sqrt(2.0) / a1));
	} else {
		tmp = a2 * (a2 / (Math.sqrt(2.0) / Math.cos(th)));
	}
	return tmp;
}

Python

[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	tmp = 0
	if a2 <= 3.3e-127:
		tmp = math.cos(th) * (a1 / (math.sqrt(2.0) / a1))
	else:
		tmp = a2 * (a2 / (math.sqrt(2.0) / math.cos(th)))
	return tmp

Julia

a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 3.3e-127)
		tmp = Float64(cos(th) * Float64(a1 / Float64(sqrt(2.0) / a1)));
	else
		tmp = Float64(a2 * Float64(a2 / Float64(sqrt(2.0) / cos(th))));
	end
	return tmp
end

MATLAB

a1, a2 = num2cell(sort([a1, a2])){:}
function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 3.3e-127)
		tmp = cos(th) * (a1 / (sqrt(2.0) / a1));
	else
		tmp = a2 * (a2 / (sqrt(2.0) / cos(th)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := If[LessEqual[a2, 3.3e-127], N[(N[Cos[th], $MachinePrecision] * N[(a1 / N[(N[Sqrt[2.0], $MachinePrecision] / a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 3.29999999999999981e-127

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around inf 64.6%

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

      1. unpow2 64.6%

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

      2. associate-/l* 64.6%

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

   6. Simplified 64.6%

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

   if 3.29999999999999981e-127 < 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/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 75.9%

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

      1. unpow2 75.9%

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

      2. associate-*l* 75.9%

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

      3. associate-*r/ 75.8%

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

      4. associate-/l* 75.9%

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

   6. Simplified 75.9%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 3.3 \cdot 10^{-127}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \]

Alternative 6: 87.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ \begin{array}{l} \mathbf{if}\;a2 \leq 6 \cdot 10^{-130}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \end{array} \]

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 6e-130)
   (* (cos th) (/ (* a1 a1) (sqrt 2.0)))
   (* a2 (/ a2 (/ (sqrt 2.0) (cos th))))))

C

assert(a1 < a2);
double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 6e-130) {
		tmp = cos(th) * ((a1 * a1) / sqrt(2.0));
	} else {
		tmp = a2 * (a2 / (sqrt(2.0) / cos(th)));
	}
	return tmp;
}

Fortran

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
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 <= 6d-130) then
        tmp = cos(th) * ((a1 * a1) / sqrt(2.0d0))
    else
        tmp = a2 * (a2 / (sqrt(2.0d0) / cos(th)))
    end if
    code = tmp
end function

Java

assert a1 < a2;
public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 6e-130) {
		tmp = Math.cos(th) * ((a1 * a1) / Math.sqrt(2.0));
	} else {
		tmp = a2 * (a2 / (Math.sqrt(2.0) / Math.cos(th)));
	}
	return tmp;
}

Python

[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	tmp = 0
	if a2 <= 6e-130:
		tmp = math.cos(th) * ((a1 * a1) / math.sqrt(2.0))
	else:
		tmp = a2 * (a2 / (math.sqrt(2.0) / math.cos(th)))
	return tmp

Julia

a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 6e-130)
		tmp = Float64(cos(th) * Float64(Float64(a1 * a1) / sqrt(2.0)));
	else
		tmp = Float64(a2 * Float64(a2 / Float64(sqrt(2.0) / cos(th))));
	end
	return tmp
end

MATLAB

a1, a2 = num2cell(sort([a1, a2])){:}
function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 6e-130)
		tmp = cos(th) * ((a1 * a1) / sqrt(2.0));
	else
		tmp = a2 * (a2 / (sqrt(2.0) / cos(th)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := If[LessEqual[a2, 6e-130], N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 5.99999999999999972e-130

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around inf 64.4%

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

      1. unpow2 64.4%

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

   6. Simplified 64.4%

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

   if 5.99999999999999972e-130 < 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/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 75.0%

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

      1. unpow2 75.0%

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

      2. associate-*l* 75.0%

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

      3. associate-*r/ 75.0%

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

      4. associate-/l* 75.0%

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

   6. Simplified 75.0%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 6 \cdot 10^{-130}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \]

Alternative 7: 87.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ \begin{array}{l} \mathbf{if}\;a2 \leq 3.3 \cdot 10^{-127}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 3.3e-127)
   (* (cos th) (/ (* a1 a1) (sqrt 2.0)))
   (* (cos th) (/ (* a2 a2) (sqrt 2.0)))))

C

assert(a1 < a2);
double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 3.3e-127) {
		tmp = cos(th) * ((a1 * a1) / sqrt(2.0));
	} else {
		tmp = cos(th) * ((a2 * a2) / sqrt(2.0));
	}
	return tmp;
}

Fortran

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
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 <= 3.3d-127) then
        tmp = cos(th) * ((a1 * a1) / sqrt(2.0d0))
    else
        tmp = cos(th) * ((a2 * a2) / sqrt(2.0d0))
    end if
    code = tmp
end function

Java

assert a1 < a2;
public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 3.3e-127) {
		tmp = Math.cos(th) * ((a1 * a1) / Math.sqrt(2.0));
	} else {
		tmp = Math.cos(th) * ((a2 * a2) / Math.sqrt(2.0));
	}
	return tmp;
}

Python

[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	tmp = 0
	if a2 <= 3.3e-127:
		tmp = math.cos(th) * ((a1 * a1) / math.sqrt(2.0))
	else:
		tmp = math.cos(th) * ((a2 * a2) / math.sqrt(2.0))
	return tmp

Julia

a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 3.3e-127)
		tmp = Float64(cos(th) * Float64(Float64(a1 * a1) / sqrt(2.0)));
	else
		tmp = Float64(cos(th) * Float64(Float64(a2 * a2) / sqrt(2.0)));
	end
	return tmp
end

MATLAB

a1, a2 = num2cell(sort([a1, a2])){:}
function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 3.3e-127)
		tmp = cos(th) * ((a1 * a1) / sqrt(2.0));
	else
		tmp = cos(th) * ((a2 * a2) / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := If[LessEqual[a2, 3.3e-127], N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 3.29999999999999981e-127

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around inf 64.6%

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

      1. unpow2 64.6%

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

   6. Simplified 64.6%

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

   if 3.29999999999999981e-127 < 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/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 75.9%

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

      1. unpow2 75.9%

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

   6. Simplified 75.9%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 3.3 \cdot 10^{-127}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]

Alternative 8: 87.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ \begin{array}{l} \mathbf{if}\;a2 \leq 3.3 \cdot 10^{-127}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(\cos th \cdot \sqrt{0.5}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 3.3e-127)
   (* (cos th) (/ (* a1 a1) (sqrt 2.0)))
   (* a2 (* a2 (* (cos th) (sqrt 0.5))))))

C

assert(a1 < a2);
double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 3.3e-127) {
		tmp = cos(th) * ((a1 * a1) / sqrt(2.0));
	} else {
		tmp = a2 * (a2 * (cos(th) * sqrt(0.5)));
	}
	return tmp;
}

Fortran

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
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 <= 3.3d-127) then
        tmp = cos(th) * ((a1 * a1) / sqrt(2.0d0))
    else
        tmp = a2 * (a2 * (cos(th) * sqrt(0.5d0)))
    end if
    code = tmp
end function

Java

assert a1 < a2;
public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 3.3e-127) {
		tmp = Math.cos(th) * ((a1 * a1) / Math.sqrt(2.0));
	} else {
		tmp = a2 * (a2 * (Math.cos(th) * Math.sqrt(0.5)));
	}
	return tmp;
}

Python

[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	tmp = 0
	if a2 <= 3.3e-127:
		tmp = math.cos(th) * ((a1 * a1) / math.sqrt(2.0))
	else:
		tmp = a2 * (a2 * (math.cos(th) * math.sqrt(0.5)))
	return tmp

Julia

a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 3.3e-127)
		tmp = Float64(cos(th) * Float64(Float64(a1 * a1) / sqrt(2.0)));
	else
		tmp = Float64(a2 * Float64(a2 * Float64(cos(th) * sqrt(0.5))));
	end
	return tmp
end

MATLAB

a1, a2 = num2cell(sort([a1, a2])){:}
function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 3.3e-127)
		tmp = cos(th) * ((a1 * a1) / sqrt(2.0));
	else
		tmp = a2 * (a2 * (cos(th) * sqrt(0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := If[LessEqual[a2, 3.3e-127], N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * N[(N[Cos[th], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 3.29999999999999981e-127

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around inf 64.6%

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

      1. unpow2 64.6%

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

   6. Simplified 64.6%

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

   if 3.29999999999999981e-127 < 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/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 75.9%

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

      1. unpow2 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-/l* 75.9%

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

   6. Simplified 75.9%

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

      1. associate-/r/ 75.9%

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

      2. div-inv 75.8%

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

      3. *-commutative 75.8%

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

      4. pow1/2 75.8%

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

      5. pow-flip 75.9%

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

      6. metadata-eval 75.9%

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

      7. associate-*r* 75.9%

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

      8. *-commutative 75.9%

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

      9. metadata-eval 75.9%

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

      10. pow-flip 75.8%

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

      11. pow1/2 75.8%

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

      12. add-sqr-sqrt 75.8%

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

      13. sqrt-unprod 75.8%

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

      14. frac-times 75.8%

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

      15. metadata-eval 75.8%

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

      16. add-sqr-sqrt 75.9%

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

      17. metadata-eval 75.9%

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

   8. Applied egg-rr 75.9%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 3.3 \cdot 10^{-127}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(\cos th \cdot \sqrt{0.5}\right)\right)\\ \end{array} \]

Alternative 9: 58.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ \begin{array}{l} \mathbf{if}\;a2 \leq 8.8 \cdot 10^{-91}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}\\ \end{array} \end{array} \]

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 8.8e-91)
   (* a1 (* a1 (sqrt 0.5)))
   (/ 1.0 (/ (sqrt 2.0) (* a2 a2)))))

C

assert(a1 < a2);
double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 8.8e-91) {
		tmp = a1 * (a1 * sqrt(0.5));
	} else {
		tmp = 1.0 / (sqrt(2.0) / (a2 * a2));
	}
	return tmp;
}

Fortran

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
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 <= 8.8d-91) then
        tmp = a1 * (a1 * sqrt(0.5d0))
    else
        tmp = 1.0d0 / (sqrt(2.0d0) / (a2 * a2))
    end if
    code = tmp
end function

Java

assert a1 < a2;
public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 8.8e-91) {
		tmp = a1 * (a1 * Math.sqrt(0.5));
	} else {
		tmp = 1.0 / (Math.sqrt(2.0) / (a2 * a2));
	}
	return tmp;
}

Python

[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	tmp = 0
	if a2 <= 8.8e-91:
		tmp = a1 * (a1 * math.sqrt(0.5))
	else:
		tmp = 1.0 / (math.sqrt(2.0) / (a2 * a2))
	return tmp

Julia

a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 8.8e-91)
		tmp = Float64(a1 * Float64(a1 * sqrt(0.5)));
	else
		tmp = Float64(1.0 / Float64(sqrt(2.0) / Float64(a2 * a2)));
	end
	return tmp
end

MATLAB

a1, a2 = num2cell(sort([a1, a2])){:}
function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 8.8e-91)
		tmp = a1 * (a1 * sqrt(0.5));
	else
		tmp = 1.0 / (sqrt(2.0) / (a2 * a2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := If[LessEqual[a2, 8.8e-91], N[(a1 * N[(a1 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 8.8000000000000003e-91

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 66.8%

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

   5. Taylor expanded in a1 around inf 45.3%

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

      1. unpow2 45.3%

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

      2. associate-*r/ 45.2%

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

   7. Simplified 45.2%

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

      1. clear-num 45.2%

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

      2. associate-/r/ 45.2%

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

      3. add-sqr-sqrt 45.2%

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

      4. sqrt-unprod 45.2%

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

      5. frac-times 45.2%

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

      6. metadata-eval 45.2%

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

      7. add-sqr-sqrt 45.3%

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

      8. metadata-eval 45.3%

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

   9. Applied egg-rr 45.3%

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

   if 8.8000000000000003e-91 < 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. Taylor expanded in th around 0 61.6%

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

   5. Taylor expanded in a1 around 0 47.4%

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

      1. unpow2 47.4%

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

   7. Simplified 47.4%

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

      1. associate-*l/ 47.5%

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

      2. associate-/l* 47.5%

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

   9. Applied egg-rr 47.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 8.8 \cdot 10^{-91}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}\\ \end{array} \]

Alternative 10: 66.3% accurate, 3.8× speedup

Math

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

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th) :precision binary64 (* (sqrt 0.5) (+ (* a2 a2) (* a1 a1))))

C

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

Fortran

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = sqrt(0.5d0) * ((a2 * a2) + (a1 * a1))
end function

Java

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

Python

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

Julia

a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	return Float64(sqrt(0.5) * Float64(Float64(a2 * a2) + Float64(a1 * a1)))
end

MATLAB

a1, a2 = num2cell(sort([a1, a2])){:}
function tmp = code(a1, a2, th)
	tmp = sqrt(0.5) * ((a2 * a2) + (a1 * a1));
end

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.5%

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

   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 65.3%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left({a2}^{2} + {a1}^{2}\right)} \]
7. Step-by-step derivation

   1. unpow2 65.3%

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

   2. unpow2 65.3%

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

8. Simplified 65.3%

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

9. Final simplification 65.3%

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

Alternative 11: 58.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ \begin{array}{l} \mathbf{if}\;a2 \leq 8.5 \cdot 10^{-92}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 8.5e-92) (* a1 (* a1 (sqrt 0.5))) (* a2 (/ a2 (sqrt 2.0)))))

C

assert(a1 < a2);
double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 8.5e-92) {
		tmp = a1 * (a1 * sqrt(0.5));
	} else {
		tmp = a2 * (a2 / sqrt(2.0));
	}
	return tmp;
}

Fortran

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
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 <= 8.5d-92) then
        tmp = a1 * (a1 * sqrt(0.5d0))
    else
        tmp = a2 * (a2 / sqrt(2.0d0))
    end if
    code = tmp
end function

Java

assert a1 < a2;
public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 8.5e-92) {
		tmp = a1 * (a1 * Math.sqrt(0.5));
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	tmp = 0
	if a2 <= 8.5e-92:
		tmp = a1 * (a1 * math.sqrt(0.5))
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 8.5e-92)
		tmp = Float64(a1 * Float64(a1 * sqrt(0.5)));
	else
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	end
	return tmp
end

MATLAB

a1, a2 = num2cell(sort([a1, a2])){:}
function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 8.5e-92)
		tmp = a1 * (a1 * sqrt(0.5));
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := If[LessEqual[a2, 8.5e-92], N[(a1 * N[(a1 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 8.50000000000000067e-92

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 66.8%

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

   5. Taylor expanded in a1 around inf 45.3%

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

      1. unpow2 45.3%

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

      2. associate-*r/ 45.2%

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

   7. Simplified 45.2%

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

      1. clear-num 45.2%

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

      2. associate-/r/ 45.2%

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

      3. add-sqr-sqrt 45.2%

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

      4. sqrt-unprod 45.2%

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

      5. frac-times 45.2%

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

      6. metadata-eval 45.2%

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

      7. add-sqr-sqrt 45.3%

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

      8. metadata-eval 45.3%

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

   9. Applied egg-rr 45.3%

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

   if 8.50000000000000067e-92 < 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. Taylor expanded in th around 0 61.6%

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

   5. Taylor expanded in a1 around 0 47.4%

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

      1. unpow2 47.4%

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

   7. Simplified 47.4%

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

   8. Taylor expanded in a2 around 0 47.5%

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

      1. unpow2 47.5%

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

      2. associate-*l/ 47.4%

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

      3. *-commutative 47.4%

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

   10. Simplified 47.4%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 8.5 \cdot 10^{-92}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 12: 58.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ \begin{array}{l} \mathbf{if}\;a2 \leq 1.1 \cdot 10^{-90}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 1.1e-90) (* a1 (* a1 (sqrt 0.5))) (* a2 (* a2 (sqrt 0.5)))))

C

assert(a1 < a2);
double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 1.1e-90) {
		tmp = a1 * (a1 * sqrt(0.5));
	} else {
		tmp = a2 * (a2 * sqrt(0.5));
	}
	return tmp;
}

Fortran

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
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 <= 1.1d-90) then
        tmp = a1 * (a1 * sqrt(0.5d0))
    else
        tmp = a2 * (a2 * sqrt(0.5d0))
    end if
    code = tmp
end function

Java

assert a1 < a2;
public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 1.1e-90) {
		tmp = a1 * (a1 * Math.sqrt(0.5));
	} else {
		tmp = a2 * (a2 * Math.sqrt(0.5));
	}
	return tmp;
}

Python

[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	tmp = 0
	if a2 <= 1.1e-90:
		tmp = a1 * (a1 * math.sqrt(0.5))
	else:
		tmp = a2 * (a2 * math.sqrt(0.5))
	return tmp

Julia

a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 1.1e-90)
		tmp = Float64(a1 * Float64(a1 * sqrt(0.5)));
	else
		tmp = Float64(a2 * Float64(a2 * sqrt(0.5)));
	end
	return tmp
end

MATLAB

a1, a2 = num2cell(sort([a1, a2])){:}
function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 1.1e-90)
		tmp = a1 * (a1 * sqrt(0.5));
	else
		tmp = a2 * (a2 * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := If[LessEqual[a2, 1.1e-90], N[(a1 * N[(a1 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 1.09999999999999993e-90

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 66.8%

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

   5. Taylor expanded in a1 around inf 45.3%

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

      1. unpow2 45.3%

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

      2. associate-*r/ 45.2%

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

   7. Simplified 45.2%

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

      1. clear-num 45.2%

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

      2. associate-/r/ 45.2%

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

      3. add-sqr-sqrt 45.2%

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

      4. sqrt-unprod 45.2%

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

      5. frac-times 45.2%

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

      6. metadata-eval 45.2%

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

      7. add-sqr-sqrt 45.3%

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

      8. metadata-eval 45.3%

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

   9. Applied egg-rr 45.3%

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

   if 1.09999999999999993e-90 < 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/ 99.6%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around 0 78.2%

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

      1. unpow2 78.2%

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

      2. *-commutative 78.2%

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

      3. associate-/l* 78.2%

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

   6. Simplified 78.2%

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

      1. associate-/r/ 78.1%

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

      2. div-inv 78.1%

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

      3. *-commutative 78.1%

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

      4. pow1/2 78.1%

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

      5. pow-flip 78.1%

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

      6. metadata-eval 78.1%

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

      7. associate-*r* 78.2%

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

      8. *-commutative 78.2%

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

      9. metadata-eval 78.2%

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

      10. pow-flip 78.1%

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

      11. pow1/2 78.1%

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

      12. add-sqr-sqrt 78.1%

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

      13. sqrt-unprod 78.1%

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

      14. frac-times 78.1%

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

      15. metadata-eval 78.1%

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

      16. add-sqr-sqrt 78.2%

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

      17. metadata-eval 78.2%

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

   8. Applied egg-rr 78.2%

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

   9. Taylor expanded in th around 0 47.4%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 1.1 \cdot 10^{-90}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 13: 58.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ \begin{array}{l} \mathbf{if}\;a2 \leq 1.7 \cdot 10^{-90}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \end{array} \]

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 1.7e-90) (* a1 (* a1 (sqrt 0.5))) (/ a2 (/ (sqrt 2.0) a2))))

C

assert(a1 < a2);
double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 1.7e-90) {
		tmp = a1 * (a1 * sqrt(0.5));
	} else {
		tmp = a2 / (sqrt(2.0) / a2);
	}
	return tmp;
}

Fortran

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
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 <= 1.7d-90) then
        tmp = a1 * (a1 * sqrt(0.5d0))
    else
        tmp = a2 / (sqrt(2.0d0) / a2)
    end if
    code = tmp
end function

Java

assert a1 < a2;
public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 1.7e-90) {
		tmp = a1 * (a1 * Math.sqrt(0.5));
	} else {
		tmp = a2 / (Math.sqrt(2.0) / a2);
	}
	return tmp;
}

Python

[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	tmp = 0
	if a2 <= 1.7e-90:
		tmp = a1 * (a1 * math.sqrt(0.5))
	else:
		tmp = a2 / (math.sqrt(2.0) / a2)
	return tmp

Julia

a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 1.7e-90)
		tmp = Float64(a1 * Float64(a1 * sqrt(0.5)));
	else
		tmp = Float64(a2 / Float64(sqrt(2.0) / a2));
	end
	return tmp
end

MATLAB

a1, a2 = num2cell(sort([a1, a2])){:}
function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 1.7e-90)
		tmp = a1 * (a1 * sqrt(0.5));
	else
		tmp = a2 / (sqrt(2.0) / a2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := If[LessEqual[a2, 1.7e-90], N[(a1 * N[(a1 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 1.69999999999999997e-90

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 66.8%

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

   5. Taylor expanded in a1 around inf 45.3%

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

      1. unpow2 45.3%

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

      2. associate-*r/ 45.2%

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

   7. Simplified 45.2%

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

      1. clear-num 45.2%

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

      2. associate-/r/ 45.2%

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

      3. add-sqr-sqrt 45.2%

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

      4. sqrt-unprod 45.2%

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

      5. frac-times 45.2%

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

      6. metadata-eval 45.2%

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

      7. add-sqr-sqrt 45.3%

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

      8. metadata-eval 45.3%

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

   9. Applied egg-rr 45.3%

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

   if 1.69999999999999997e-90 < 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. Taylor expanded in th around 0 61.6%

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

   5. Taylor expanded in a1 around 0 47.4%

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

      1. unpow2 47.4%

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

   7. Simplified 47.4%

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

      1. associate-*l/ 47.5%

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

      2. *-un-lft-identity 47.5%

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

      3. associate-/l* 47.5%

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

   9. Applied egg-rr 47.5%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 1.7 \cdot 10^{-90}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 14: 40.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ a1 \cdot \left(a1 \cdot \sqrt{0.5}\right) \end{array} \]

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th) :precision binary64 (* a1 (* a1 (sqrt 0.5))))

C

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

Fortran

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a1 * (a1 * sqrt(0.5d0))
end function

Java

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

Python

[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	return a1 * (a1 * math.sqrt(0.5))

Julia

a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	return Float64(a1 * Float64(a1 * sqrt(0.5)))
end

MATLAB

a1, a2 = num2cell(sort([a1, a2])){:}
function tmp = code(a1, a2, th)
	tmp = a1 * (a1 * sqrt(0.5));
end

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(a1 * N[(a1 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 65.3%

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

5. Taylor expanded in a1 around inf 39.0%

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

   1. unpow2 39.0%

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

   2. associate-*r/ 39.0%

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

7. Simplified 39.0%

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

   1. clear-num 39.0%

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

   2. associate-/r/ 39.0%

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

   3. add-sqr-sqrt 39.0%

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

   4. sqrt-unprod 39.0%

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

   5. frac-times 39.0%

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

   6. metadata-eval 39.0%

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

   7. add-sqr-sqrt 39.0%

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

   8. metadata-eval 39.0%

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

9. Applied egg-rr 39.0%

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

10. Final simplification 39.0%

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

Reproduce

herbie shell --seed 2023181 
(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))))