Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.1%

Time: 8.9s

Alternatives: 8

Speedup: N/A×

• 43.5% 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

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

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 (* a2 (cos th))) (sqrt 2.0)))

C

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

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 * (a2 * cos(th))) / sqrt(2.0d0)
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 * (a2 * Math.cos(th))) / Math.sqrt(2.0);
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.7%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 58.0%

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

5. Taylor expanded in a2 around 0 58.0%

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

   1. unpow2 58.0%

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

   2. *-commutative 58.0%

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

   3. associate-*r* 58.0%

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

7. Simplified 58.0%

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

8. Final simplification 58.0%

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

Alternative 2: 99.1% accurate, 2.0× speedup

Math

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

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 (* a2 (* (cos th) (sqrt 0.5)))))

C

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

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 * (a2 * (cos(th) * sqrt(0.5d0)))
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 * (a2 * (Math.cos(th) * Math.sqrt(0.5)));
}

Python

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

Julia

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

MATLAB

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = num2cell(sort([a1, a2])){:}
function tmp = code(a1, a2, th)
	tmp = a2 * (a2 * (cos(th) * sqrt(0.5)));
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[(a2 * N[(N[Cos[th], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.7%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 58.0%

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

5. Taylor expanded in a2 around 0 58.0%

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

   1. unpow2 58.0%

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

7. Simplified 58.0%

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

   1. div-inv 57.9%

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

   2. *-commutative 57.9%

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

   3. associate-*l* 57.9%

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

   4. add-sqr-sqrt 57.9%

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

   5. sqrt-unprod 57.9%

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

   6. frac-times 57.9%

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

   7. metadata-eval 57.9%

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

   8. add-sqr-sqrt 57.9%

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

   9. metadata-eval 57.9%

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

9. Applied egg-rr 57.9%

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

    1. associate-*l* 57.9%

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

11. Simplified 57.9%

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

12. Final simplification 57.9%

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

Alternative 3: 99.0% accurate, 2.0× speedup

Math

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

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 a2) (sqrt 2.0))))

C

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

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 * a2) / sqrt(2.0d0))
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 * a2) / Math.sqrt(2.0));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.7%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 57.9%

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

   1. unpow2 57.9%

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

6. Simplified 57.9%

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

7. Final simplification 57.9%

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

Alternative 4: 65.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} a1 = |a1|\\ a2 = |a2|\\ [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ \begin{array}{l} \mathbf{if}\;a1 \cdot a1 \leq 2.4 \cdot 10^{-79}:\\ \;\;\;\;a2 \cdot \sqrt{a2 \cdot \frac{a2}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a2 \cdot a2\right) \cdot \left(-0.5 \cdot \left(th \cdot th\right) + 1\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

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 (<= (* a1 a1) 2.4e-79)
   (* a2 (sqrt (* a2 (/ a2 2.0))))
   (/ (* (* a2 a2) (+ (* -0.5 (* th th)) 1.0)) (sqrt 2.0))))

C

a1 = abs(a1);
a2 = abs(a2);
assert(a1 < a2);
double code(double a1, double a2, double th) {
	double tmp;
	if ((a1 * a1) <= 2.4e-79) {
		tmp = a2 * sqrt((a2 * (a2 / 2.0)));
	} else {
		tmp = ((a2 * a2) * ((-0.5 * (th * th)) + 1.0)) / sqrt(2.0);
	}
	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 ((a1 * a1) <= 2.4d-79) then
        tmp = a2 * sqrt((a2 * (a2 / 2.0d0)))
    else
        tmp = ((a2 * a2) * (((-0.5d0) * (th * th)) + 1.0d0)) / sqrt(2.0d0)
    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 ((a1 * a1) <= 2.4e-79) {
		tmp = a2 * Math.sqrt((a2 * (a2 / 2.0)));
	} else {
		tmp = ((a2 * a2) * ((-0.5 * (th * th)) + 1.0)) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

a1 = abs(a1)
a2 = abs(a2)
[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	tmp = 0
	if (a1 * a1) <= 2.4e-79:
		tmp = a2 * math.sqrt((a2 * (a2 / 2.0)))
	else:
		tmp = ((a2 * a2) * ((-0.5 * (th * th)) + 1.0)) / math.sqrt(2.0)
	return tmp

Julia

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	tmp = 0.0
	if (Float64(a1 * a1) <= 2.4e-79)
		tmp = Float64(a2 * sqrt(Float64(a2 * Float64(a2 / 2.0))));
	else
		tmp = Float64(Float64(Float64(a2 * a2) * Float64(Float64(-0.5 * Float64(th * th)) + 1.0)) / sqrt(2.0));
	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 ((a1 * a1) <= 2.4e-79)
		tmp = a2 * sqrt((a2 * (a2 / 2.0)));
	else
		tmp = ((a2 * a2) * ((-0.5 * (th * th)) + 1.0)) / sqrt(2.0);
	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[N[(a1 * a1), $MachinePrecision], 2.4e-79], N[(a2 * N[Sqrt[N[(a2 * N[(a2 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(a2 * a2), $MachinePrecision] * N[(N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a1 a1) < 2.40000000000000006e-79

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a1 around 0 81.9%

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

      1. unpow2 81.9%

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

      2. associate-*l* 81.9%

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

      3. associate-*r/ 81.9%

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

      4. associate-/l* 81.9%

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

   6. Simplified 81.9%

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

   7. Taylor expanded in th around 0 60.7%

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

      1. add-sqr-sqrt 32.5%

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

      2. sqrt-unprod 39.8%

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

      3. frac-times 39.8%

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

      4. add-sqr-sqrt 39.9%

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

   9. Applied egg-rr 39.9%

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

       1. associate-/l* 39.9%

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

       2. associate-/r/ 39.9%

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

   11. Simplified 39.9%

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

   if 2.40000000000000006e-79 < (*.f64 a1 a1)

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around 0 35.1%

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

   5. Taylor expanded in th around 0 10.9%

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

      1. unpow2 10.9%

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

      2. unpow2 10.9%

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

      3. associate-*r* 10.9%

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

      4. distribute-rgt1-in 30.0%

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

      5. unpow2 30.0%

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

   7. Simplified 30.0%

      \[\leadsto \frac{\color{blue}{\left(-0.5 \cdot \left(th \cdot th\right) + 1\right) \cdot \left(a2 \cdot a2\right)}}{\sqrt{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a1 \leq 2.4 \cdot 10^{-79}:\\ \;\;\;\;a2 \cdot \sqrt{a2 \cdot \frac{a2}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a2 \cdot a2\right) \cdot \left(-0.5 \cdot \left(th \cdot th\right) + 1\right)}{\sqrt{2}}\\ \end{array} \]

Alternative 5: 66.2% accurate, 3.9× speedup

Math

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

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 (* a2 (/ a2 2.0)))))

C

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

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((a2 * (a2 / 2.0d0)))
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((a2 * (a2 / 2.0)));
}

Python

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

Julia

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

MATLAB

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = num2cell(sort([a1, a2])){:}
function tmp = code(a1, a2, th)
	tmp = a2 * sqrt((a2 * (a2 / 2.0)));
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[Sqrt[N[(a2 * N[(a2 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.7%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 58.0%

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

   1. unpow2 58.0%

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

   2. associate-*l* 58.0%

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

   3. associate-*r/ 57.9%

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

   4. associate-/l* 57.9%

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

6. Simplified 57.9%

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

7. Taylor expanded in th around 0 40.9%

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

   1. add-sqr-sqrt 20.0%

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

   2. sqrt-unprod 26.1%

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

   3. frac-times 26.0%

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

   4. add-sqr-sqrt 26.1%

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

9. Applied egg-rr 26.1%

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

    1. associate-/l* 26.1%

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

    2. associate-/r/ 26.1%

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

11. Simplified 26.1%

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

12. Final simplification 26.1%

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

Alternative 6: 66.1% accurate, 4.0× speedup

Math

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

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 (/ a2 (sqrt 2.0))))

C

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

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 * (a2 / sqrt(2.0d0))
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 * (a2 / Math.sqrt(2.0));
}

Python

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

Julia

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

MATLAB

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = num2cell(sort([a1, a2])){:}
function tmp = code(a1, a2, th)
	tmp = a2 * (a2 / sqrt(2.0));
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[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.7%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 58.0%

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

   1. unpow2 58.0%

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

   2. associate-*l* 58.0%

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

   3. associate-*r/ 57.9%

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

   4. associate-/l* 57.9%

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

6. Simplified 57.9%

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

7. Taylor expanded in th around 0 40.9%

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

8. Final simplification 40.9%

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

Alternative 7: 66.1% 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. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.7%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 58.0%

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

5. Taylor expanded in th around 0 40.9%

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

   1. unpow2 40.9%

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

   2. associate-/l* 40.9%

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

7. Simplified 40.9%

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

8. Final simplification 40.9%

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

Alternative 8: 66.1% accurate, 4.0× speedup

Math

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

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 a2) (sqrt 2.0)))

C

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

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 * a2) / sqrt(2.0d0)
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 * a2) / Math.sqrt(2.0);
}

Python

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

Julia

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

MATLAB

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = num2cell(sort([a1, a2])){:}
function tmp = code(a1, a2, th)
	tmp = (a2 * a2) / sqrt(2.0);
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 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.7%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 58.0%

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

5. Taylor expanded in th around 0 40.9%

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

   1. unpow2 40.9%

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

7. Simplified 40.9%

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

8. Final simplification 40.9%

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

Reproduce

herbie shell --seed 2023213 
(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))))