Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.1%

Time: 11.1s

Alternatives: 6

Speedup: N/A×

• 43.9% 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: 99.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a1 should be positive before calling this function
NOTE: a2 should be positive before calling this function
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 = (a2 * sqrt(0.5d0)) * (a2 * cos(th))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: a1 should be positive before calling this function
NOTE: a2 should be positive before calling this function
NOTE: a1 and a2 should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[(a2 * N[Cos[th], $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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a2 around inf 56.0%

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

   1. unpow2 56.0%

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

   2. associate-/l* 56.0%

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

6. Simplified 56.0%

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

   1. associate-/l* 56.0%

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

   2. div-inv 56.0%

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

   3. *-commutative 56.0%

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

   4. inv-pow 56.0%

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

   5. sqrt-pow2 56.0%

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

   6. metadata-eval 56.0%

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

   7. associate-*l* 56.0%

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

   8. *-commutative 56.0%

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

8. Applied egg-rr 56.0%

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

   1. associate-*r* 56.0%

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

   2. *-commutative 56.0%

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

10. Simplified 56.0%

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

11. Taylor expanded in a2 around 0 56.0%

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

12. Final simplification 56.0%

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

Alternative 2: 99.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a1 should be positive before calling this function
NOTE: a2 should be positive before calling this function
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) * (a2 / (sqrt(2.0d0) / a2))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a2 around inf 56.0%

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

   1. unpow2 56.0%

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

   2. *-commutative 56.0%

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

   3. associate-*r/ 56.0%

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

   4. associate-/l* 56.0%

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

6. Simplified 56.0%

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

7. Final simplification 56.0%

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

Alternative 3: 66.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} a1 = |a1|\\ a2 = |a2|\\ [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ \begin{array}{l} \mathbf{if}\;a2 \leq 7 \cdot 10^{+167}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{elif}\;a2 \leq 2.4 \cdot 10^{+259}:\\ \;\;\;\;\left(\sqrt{0.5} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right) \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a1 should be positive before calling this function
NOTE: a2 should be positive before calling this function
NOTE: a1 and a2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 7e+167)
   (/ a2 (/ (sqrt 2.0) a2))
   (if (<= a2 2.4e+259)
     (* (* (sqrt 0.5) (+ 1.0 (* -0.5 (* th th)))) (* a2 a2))
     (* (sqrt 0.5) (* a2 a2)))))

C

a1 = abs(a1);
a2 = abs(a2);
assert(a1 < a2);
double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 7e+167) {
		tmp = a2 / (sqrt(2.0) / a2);
	} else if (a2 <= 2.4e+259) {
		tmp = (sqrt(0.5) * (1.0 + (-0.5 * (th * th)))) * (a2 * a2);
	} else {
		tmp = sqrt(0.5) * (a2 * a2);
	}
	return tmp;
}

Fortran

NOTE: a1 should be positive before calling this function
NOTE: a2 should be positive before calling this function
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 <= 7d+167) then
        tmp = a2 / (sqrt(2.0d0) / a2)
    else if (a2 <= 2.4d+259) then
        tmp = (sqrt(0.5d0) * (1.0d0 + ((-0.5d0) * (th * th)))) * (a2 * a2)
    else
        tmp = sqrt(0.5d0) * (a2 * a2)
    end if
    code = tmp
end function

Java

a1 = Math.abs(a1);
a2 = Math.abs(a2);
assert a1 < a2;
public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 7e+167) {
		tmp = a2 / (Math.sqrt(2.0) / a2);
	} else if (a2 <= 2.4e+259) {
		tmp = (Math.sqrt(0.5) * (1.0 + (-0.5 * (th * th)))) * (a2 * a2);
	} else {
		tmp = Math.sqrt(0.5) * (a2 * a2);
	}
	return tmp;
}

Python

a1 = abs(a1)
a2 = abs(a2)
[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	tmp = 0
	if a2 <= 7e+167:
		tmp = a2 / (math.sqrt(2.0) / a2)
	elif a2 <= 2.4e+259:
		tmp = (math.sqrt(0.5) * (1.0 + (-0.5 * (th * th)))) * (a2 * a2)
	else:
		tmp = math.sqrt(0.5) * (a2 * a2)
	return tmp

Julia

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 7e+167)
		tmp = Float64(a2 / Float64(sqrt(2.0) / a2));
	elseif (a2 <= 2.4e+259)
		tmp = Float64(Float64(sqrt(0.5) * Float64(1.0 + Float64(-0.5 * Float64(th * th)))) * Float64(a2 * a2));
	else
		tmp = Float64(sqrt(0.5) * Float64(a2 * a2));
	end
	return tmp
end

MATLAB

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = num2cell(sort([a1, a2])){:}
function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 7e+167)
		tmp = a2 / (sqrt(2.0) / a2);
	elseif (a2 <= 2.4e+259)
		tmp = (sqrt(0.5) * (1.0 + (-0.5 * (th * th)))) * (a2 * a2);
	else
		tmp = sqrt(0.5) * (a2 * a2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a1 should be positive before calling this function
NOTE: a2 should be positive before calling this function
NOTE: a1 and a2 should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := If[LessEqual[a2, 7e+167], N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision], If[LessEqual[a2, 2.4e+259], N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a2 < 6.99999999999999975e167

   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. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 65.3%

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

   5. Taylor expanded in a2 around inf 37.1%

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

      1. unpow2 37.1%

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

   7. Simplified 37.1%

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

   8. Taylor expanded in a2 around 0 37.1%

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

      1. unpow2 37.1%

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

      2. associate-/l* 37.1%

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

   10. Simplified 37.1%

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

   if 6.99999999999999975e167 < a2 < 2.4e259

   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. +-commutative 100.0%

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

      2. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a2 around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-/l* 100.0%

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

   6. Simplified 100.0%

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

      1. associate-/l* 100.0%

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

      2. div-inv 100.0%

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

      3. *-commutative 100.0%

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

      4. inv-pow 100.0%

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

      5. sqrt-pow2 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-*l* 100.0%

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

      8. *-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in th around 0 92.3%

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

       1. associate-*r* 92.3%

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

       2. unpow2 92.3%

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

   13. Simplified 92.3%

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

   14. Taylor expanded in a2 around 0 92.3%

       \[\leadsto \color{blue}{{a2}^{2} \cdot \left(\sqrt{0.5} \cdot \left(1 + -0.5 \cdot {th}^{2}\right)\right)} \]
   15. Step-by-step derivation

       1. unpow2 92.3%

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

       2. *-commutative 92.3%

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

       3. unpow2 92.3%

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

   16. Simplified 92.3%

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

   if 2.4e259 < a2

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a2 around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-/l* 100.0%

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

   6. Simplified 100.0%

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

      1. associate-/l* 100.0%

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

      2. div-inv 100.0%

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

      3. *-commutative 100.0%

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

      4. inv-pow 100.0%

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

      5. sqrt-pow2 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-*l* 100.0%

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

      8. *-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in th around 0 81.8%

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

       1. unpow2 81.8%

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

       2. *-commutative 81.8%

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

   13. Simplified 81.8%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 7 \cdot 10^{+167}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{elif}\;a2 \leq 2.4 \cdot 10^{+259}:\\ \;\;\;\;\left(\sqrt{0.5} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right) \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\\ \end{array} \]

Alternative 4: 63.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} a1 = |a1|\\ a2 = |a2|\\ [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ \begin{array}{l} \mathbf{if}\;th \leq 1.02 \cdot 10^{+175}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sqrt{0.5} \cdot \left(\left(th \cdot th\right) \cdot \left(a2 \cdot a2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a1 should be positive before calling this function
NOTE: a2 should be positive before calling this function
NOTE: a1 and a2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 1.02e+175)
   (/ a2 (/ (sqrt 2.0) a2))
   (* -0.5 (* (sqrt 0.5) (* (* th th) (* a2 a2))))))

C

a1 = abs(a1);
a2 = abs(a2);
assert(a1 < a2);
double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.02e+175) {
		tmp = a2 / (sqrt(2.0) / a2);
	} else {
		tmp = -0.5 * (sqrt(0.5) * ((th * th) * (a2 * a2)));
	}
	return tmp;
}

Fortran

NOTE: a1 should be positive before calling this function
NOTE: a2 should be positive before calling this function
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 (th <= 1.02d+175) then
        tmp = a2 / (sqrt(2.0d0) / a2)
    else
        tmp = (-0.5d0) * (sqrt(0.5d0) * ((th * th) * (a2 * a2)))
    end if
    code = tmp
end function

Java

a1 = Math.abs(a1);
a2 = Math.abs(a2);
assert a1 < a2;
public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.02e+175) {
		tmp = a2 / (Math.sqrt(2.0) / a2);
	} else {
		tmp = -0.5 * (Math.sqrt(0.5) * ((th * th) * (a2 * a2)));
	}
	return tmp;
}

Python

a1 = abs(a1)
a2 = abs(a2)
[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	tmp = 0
	if th <= 1.02e+175:
		tmp = a2 / (math.sqrt(2.0) / a2)
	else:
		tmp = -0.5 * (math.sqrt(0.5) * ((th * th) * (a2 * a2)))
	return tmp

Julia

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	tmp = 0.0
	if (th <= 1.02e+175)
		tmp = Float64(a2 / Float64(sqrt(2.0) / a2));
	else
		tmp = Float64(-0.5 * Float64(sqrt(0.5) * Float64(Float64(th * th) * Float64(a2 * a2))));
	end
	return tmp
end

MATLAB

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = num2cell(sort([a1, a2])){:}
function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 1.02e+175)
		tmp = a2 / (sqrt(2.0) / a2);
	else
		tmp = -0.5 * (sqrt(0.5) * ((th * th) * (a2 * a2)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a1 should be positive before calling this function
NOTE: a2 should be positive before calling this function
NOTE: a1 and a2 should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := If[LessEqual[th, 1.02e+175], N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(th * th), $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 1.0199999999999999e175

   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. +-commutative 99.6%

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

      2. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 70.6%

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

   5. Taylor expanded in a2 around inf 42.0%

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

      1. unpow2 42.0%

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

   7. Simplified 42.0%

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

   8. Taylor expanded in a2 around 0 42.0%

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

      1. unpow2 42.0%

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

      2. associate-/l* 42.0%

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

   10. Simplified 42.0%

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

   if 1.0199999999999999e175 < th

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in a2 around inf 48.0%

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

      1. unpow2 48.0%

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

      2. associate-/l* 48.1%

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

   6. Simplified 48.1%

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

      1. associate-/l* 48.0%

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

      2. div-inv 48.0%

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

      3. *-commutative 48.0%

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

      4. inv-pow 48.0%

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

      5. sqrt-pow2 48.1%

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

      6. metadata-eval 48.1%

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

      7. associate-*l* 48.1%

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

      8. *-commutative 48.1%

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

   8. Applied egg-rr 48.1%

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

      1. associate-*r* 48.2%

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

      2. *-commutative 48.2%

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

   10. Simplified 48.2%

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

   11. Taylor expanded in th around 0 29.9%

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

       1. associate-*r* 29.9%

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

       2. unpow2 29.9%

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

   13. Simplified 29.9%

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

   14. Taylor expanded in th around inf 26.1%

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

       1. unpow2 26.1%

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

       2. unpow2 26.1%

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

       3. associate-*r* 26.1%

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

       4. *-commutative 26.1%

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

   16. Simplified 26.1%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.02 \cdot 10^{+175}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sqrt{0.5} \cdot \left(\left(th \cdot th\right) \cdot \left(a2 \cdot a2\right)\right)\right)\\ \end{array} \]

Alternative 5: 64.9% accurate, 4.0× speedup

Math

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

FPCore

NOTE: a1 should be positive before calling this function
NOTE: a2 should be positive before calling this function
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)))

C

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

Fortran

NOTE: a1 should be positive before calling this function
NOTE: a2 should be positive before calling this function
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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: a1 should be positive before calling this function
NOTE: a2 should be positive before calling this function
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[(a2 * a2), $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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a2 around inf 56.0%

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

   1. unpow2 56.0%

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

   2. associate-/l* 56.0%

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

6. Simplified 56.0%

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

   1. associate-/l* 56.0%

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

   2. div-inv 56.0%

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

   3. *-commutative 56.0%

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

   4. inv-pow 56.0%

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

   5. sqrt-pow2 56.0%

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

   6. metadata-eval 56.0%

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

   7. associate-*l* 56.0%

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

   8. *-commutative 56.0%

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

8. Applied egg-rr 56.0%

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

   1. associate-*r* 56.0%

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

   2. *-commutative 56.0%

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

10. Simplified 56.0%

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

11. Taylor expanded in th around 0 39.4%

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

    1. unpow2 39.4%

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

    2. *-commutative 39.4%

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

13. Simplified 39.4%

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

14. Final simplification 39.4%

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

Alternative 6: 64.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} a1 = |a1|\\ a2 = |a2|\\ [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ \frac{a2}{\frac{\sqrt{2}}{a2}} \end{array} \]

FPCore

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

C

a1 = abs(a1);
a2 = abs(a2);
assert(a1 < a2);
double code(double a1, double a2, double th) {
	return a2 / (sqrt(2.0) / a2);
}

Fortran

NOTE: a1 should be positive before calling this function
NOTE: a2 should be positive before calling this function
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 = a2 / (sqrt(2.0d0) / a2)
end function

Java

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

Python

a1 = abs(a1)
a2 = abs(a2)
[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	return a2 / (math.sqrt(2.0) / a2)

Julia

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

MATLAB

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

Wolfram

NOTE: a1 should be positive before calling this function
NOTE: a2 should be positive before calling this function
NOTE: a1 and a2 should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 65.1%

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

5. Taylor expanded in a2 around inf 39.4%

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

   1. unpow2 39.4%

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

7. Simplified 39.4%

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

8. Taylor expanded in a2 around 0 39.4%

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

   1. unpow2 39.4%

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

   2. associate-/l* 39.4%

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

10. Simplified 39.4%

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

11. Final simplification 39.4%

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

Reproduce

herbie shell --seed 2023297 
(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))))