Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.1%

Time: 15.8s

Alternatives: 12

Speedup: N/A×

• 44.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 2.0× speedup

Math

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

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

C

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

Python

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

Julia

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	return Float64(Float64(a2 * cos(th)) * Float64(a2 * 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 * cos(th)) * (a2 * 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[(N[(a2 * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.9%

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

   1. distribute-lft-out 98.9%

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

   2. cos-neg 98.9%

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

   3. associate-*l/ 99.0%

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

   4. cos-neg 99.0%

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

   5. fma-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in a1 around 0 59.2%

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

   1. unpow2 59.2%

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

   2. associate-*l* 59.5%

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

6. Simplified 59.5%

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

   1. div-inv 59.5%

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

   2. pow1/2 59.5%

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

   3. pow-flip 59.5%

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

   4. metadata-eval 59.5%

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

   5. *-commutative 59.5%

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

   6. associate-*l* 59.5%

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

   7. add-sqr-sqrt 59.2%

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

   8. sqrt-unprod 59.5%

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

   9. pow-prod-up 59.5%

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

   10. metadata-eval 59.5%

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

   11. metadata-eval 59.5%

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

8. Applied egg-rr 59.5%

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

9. Final simplification 59.5%

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

Alternative 2: 99.1% accurate, 2.0× speedup

Math

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

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 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(a2 * Float64(a2 * Float64(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[(a2 * N[(a2 * N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.9%

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

   1. +-commutative 98.9%

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

   2. distribute-lft-out 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in a2 around inf 59.2%

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

   1. unpow2 59.2%

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

   2. associate-*r/ 59.1%

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

   3. associate-*r* 59.5%

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

6. Simplified 59.5%

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

7. Final simplification 59.5%

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

Alternative 3: 64.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} a1 = |a1|\\ a2 = |a2|\\ [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ \begin{array}{l} \mathbf{if}\;th \leq 2.35 \cdot 10^{+72}:\\ \;\;\;\;\left(a2 \cdot \sqrt{0.5}\right) \cdot \left(a2 + -0.5 \cdot \left(a2 \cdot \left(th \cdot th\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

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 (<= th 2.35e+72)
   (* (* a2 (sqrt 0.5)) (+ a2 (* -0.5 (* a2 (* th th)))))
   (* a2 (/ a2 (sqrt 2.0)))))

C

a1 = abs(a1);
a2 = abs(a2);
assert(a1 < a2);
double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 2.35e+72) {
		tmp = (a2 * sqrt(0.5)) * (a2 + (-0.5 * (a2 * (th * th))));
	} else {
		tmp = a2 * (a2 / 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 (th <= 2.35d+72) then
        tmp = (a2 * sqrt(0.5d0)) * (a2 + ((-0.5d0) * (a2 * (th * th))))
    else
        tmp = a2 * (a2 / 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 (th <= 2.35e+72) {
		tmp = (a2 * Math.sqrt(0.5)) * (a2 + (-0.5 * (a2 * (th * th))));
	} else {
		tmp = a2 * (a2 / 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 th <= 2.35e+72:
		tmp = (a2 * math.sqrt(0.5)) * (a2 + (-0.5 * (a2 * (th * th))))
	else:
		tmp = a2 * (a2 / 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 (th <= 2.35e+72)
		tmp = Float64(Float64(a2 * sqrt(0.5)) * Float64(a2 + Float64(-0.5 * Float64(a2 * Float64(th * th)))));
	else
		tmp = Float64(a2 * Float64(a2 / 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 (th <= 2.35e+72)
		tmp = (a2 * sqrt(0.5)) * (a2 + (-0.5 * (a2 * (th * th))));
	else
		tmp = a2 * (a2 / 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[th, 2.35e+72], N[(N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[(a2 + N[(-0.5 * N[(a2 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 2.35000000000000017e72

   1. Initial program 99.2%

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

      1. distribute-lft-out 99.2%

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

      2. cos-neg 99.2%

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

      3. associate-*l/ 99.3%

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

      4. cos-neg 99.3%

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

      5. fma-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a1 around 0 58.5%

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

      1. unpow2 58.5%

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

      2. associate-*l* 58.9%

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

   6. Simplified 58.9%

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

      1. div-inv 58.9%

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

      2. pow1/2 58.9%

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

      3. pow-flip 58.9%

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

      4. metadata-eval 58.9%

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

      5. *-commutative 58.9%

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

      6. associate-*l* 58.9%

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

      7. add-sqr-sqrt 58.6%

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

      8. sqrt-unprod 58.9%

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

      9. pow-prod-up 58.9%

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

      10. metadata-eval 58.9%

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

      11. metadata-eval 58.9%

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

   8. Applied egg-rr 58.9%

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

   9. Taylor expanded in th around 0 40.5%

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

       1. unpow2 40.5%

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

   11. Simplified 40.5%

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

   if 2.35000000000000017e72 < th

   1. Initial program 97.9%

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

      1. +-commutative 97.9%

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

      2. distribute-lft-out 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in a2 around inf 61.7%

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

      1. unpow2 61.7%

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

      2. associate-*r/ 61.7%

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

      3. associate-*r* 61.8%

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

   6. Simplified 61.8%

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

   7. Taylor expanded in th around 0 25.2%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 2.35 \cdot 10^{+72}:\\ \;\;\;\;\left(a2 \cdot \sqrt{0.5}\right) \cdot \left(a2 + -0.5 \cdot \left(a2 \cdot \left(th \cdot th\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 4: 66.2% accurate, 4.0× speedup

Math

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

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 (sqrt 0.5))))

C

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

Python

a1 = abs(a1)
a2 = abs(a2)
[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	return a2 * (a2 * 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 * 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 * 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[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.9%

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

   1. distribute-lft-out 98.9%

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

   2. cos-neg 98.9%

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

   3. associate-*l/ 99.0%

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

   4. cos-neg 99.0%

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

   5. fma-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in a1 around 0 59.2%

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

   1. unpow2 59.2%

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

   2. associate-*l* 59.5%

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

6. Simplified 59.5%

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

   1. div-inv 59.5%

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

   2. pow1/2 59.5%

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

   3. pow-flip 59.5%

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

   4. metadata-eval 59.5%

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

   5. *-commutative 59.5%

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

   6. associate-*l* 59.5%

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

   7. add-sqr-sqrt 59.2%

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

   8. sqrt-unprod 59.5%

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

   9. pow-prod-up 59.5%

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

   10. metadata-eval 59.5%

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

   11. metadata-eval 59.5%

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

8. Applied egg-rr 59.5%

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

9. Taylor expanded in th around 0 39.6%

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

    1. unpow2 39.6%

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

    2. associate-*l* 39.6%

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

11. Simplified 39.6%

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

12. Final simplification 39.6%

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

Alternative 5: 25.7% accurate, 41.2× speedup

Math

\[\begin{array}{l} a1 = |a1|\\ a2 = |a2|\\ [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ \begin{array}{l} \mathbf{if}\;a2 \cdot a2 \leq 5 \cdot 10^{-233}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \left(-64\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot th\right)\\ \end{array} \end{array} \]

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 a2) 5e-233) (* (* a1 a1) (- 64.0)) (* a2 (* a2 th))))

C

a1 = abs(a1);
a2 = abs(a2);
assert(a1 < a2);
double code(double a1, double a2, double th) {
	double tmp;
	if ((a2 * a2) <= 5e-233) {
		tmp = (a1 * a1) * -64.0;
	} else {
		tmp = a2 * (a2 * th);
	}
	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 * a2) <= 5d-233) then
        tmp = (a1 * a1) * -64.0d0
    else
        tmp = a2 * (a2 * th)
    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 * a2) <= 5e-233) {
		tmp = (a1 * a1) * -64.0;
	} else {
		tmp = a2 * (a2 * th);
	}
	return tmp;
}

Python

a1 = abs(a1)
a2 = abs(a2)
[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	tmp = 0
	if (a2 * a2) <= 5e-233:
		tmp = (a1 * a1) * -64.0
	else:
		tmp = a2 * (a2 * th)
	return tmp

Julia

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	tmp = 0.0
	if (Float64(a2 * a2) <= 5e-233)
		tmp = Float64(Float64(a1 * a1) * Float64(-64.0));
	else
		tmp = Float64(a2 * Float64(a2 * th));
	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 * a2) <= 5e-233)
		tmp = (a1 * a1) * -64.0;
	else
		tmp = a2 * (a2 * th);
	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[(a2 * a2), $MachinePrecision], 5e-233], N[(N[(a1 * a1), $MachinePrecision] * (-64.0)), $MachinePrecision], N[(a2 * N[(a2 * th), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a2 a2) < 5.00000000000000012e-233

   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 th around 0 66.7%

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

      1. unpow2 66.7%

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

      2. unpow2 66.7%

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

      3. +-commutative 66.7%

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

   6. Simplified 66.7%

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

   7. Taylor expanded in a2 around 0 64.3%

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

      1. unpow2 64.3%

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

   9. Simplified 64.3%

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

       1. frac-2neg 64.3%

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

       2. div-inv 64.2%

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

   11. Applied egg-rr 64.2%

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

   13. Applied egg-rr 25.1%

       \[\leadsto \left(-a1 \cdot a1\right) \cdot \color{blue}{64} \]

   if 5.00000000000000012e-233 < (*.f64 a2 a2)

   1. Initial program 98.7%

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

      1. distribute-lft-out 98.7%

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

      2. cos-neg 98.7%

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

      3. associate-*l/ 98.7%

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

      4. cos-neg 98.7%

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

      5. fma-def 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in a1 around 0 75.7%

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

      1. unpow2 75.7%

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

      2. associate-*l* 76.2%

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

   6. Simplified 76.2%

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

   7. Taylor expanded in th around 0 45.8%

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

      1. unpow2 45.8%

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

   9. Simplified 45.8%

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

   10. Taylor expanded in th around inf 12.7%

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

       1. unpow2 12.7%

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

       2. unpow2 12.7%

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

       3. associate-*l* 12.8%

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

       4. associate-*l* 12.8%

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

       5. *-commutative 12.8%

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

       6. associate-*l* 12.8%

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

       7. *-commutative 12.8%

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

   12. Simplified 12.8%

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

   14. Applied egg-rr 20.9%

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

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \cdot a2 \leq 5 \cdot 10^{-233}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \left(-64\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot th\right)\\ \end{array} \]

Alternative 6: 25.8% accurate, 51.5× speedup

Math

\[\begin{array}{l} a1 = |a1|\\ a2 = |a2|\\ [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ \begin{array}{l} \mathbf{if}\;a2 \leq 2.1 \cdot 10^{+29}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \left(--2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot th\right)\\ \end{array} \end{array} \]

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 2.1e+29) (* (* a1 a1) (- -2.0)) (* a2 (* a2 th))))

C

a1 = abs(a1);
a2 = abs(a2);
assert(a1 < a2);
double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 2.1e+29) {
		tmp = (a1 * a1) * -(-2.0);
	} else {
		tmp = a2 * (a2 * th);
	}
	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 <= 2.1d+29) then
        tmp = (a1 * a1) * -(-2.0d0)
    else
        tmp = a2 * (a2 * th)
    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 <= 2.1e+29) {
		tmp = (a1 * a1) * -(-2.0);
	} else {
		tmp = a2 * (a2 * th);
	}
	return tmp;
}

Python

a1 = abs(a1)
a2 = abs(a2)
[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	tmp = 0
	if a2 <= 2.1e+29:
		tmp = (a1 * a1) * -(-2.0)
	else:
		tmp = a2 * (a2 * th)
	return tmp

Julia

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 2.1e+29)
		tmp = Float64(Float64(a1 * a1) * Float64(-(-2.0)));
	else
		tmp = Float64(a2 * Float64(a2 * th));
	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 <= 2.1e+29)
		tmp = (a1 * a1) * -(-2.0);
	else
		tmp = a2 * (a2 * th);
	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, 2.1e+29], N[(N[(a1 * a1), $MachinePrecision] * (--2.0)), $MachinePrecision], N[(a2 * N[(a2 * th), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 2.1000000000000002e29

   1. Initial program 98.7%

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

      1. +-commutative 98.7%

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

      2. distribute-lft-out 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in th around 0 63.9%

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

      1. unpow2 63.9%

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

      2. unpow2 63.9%

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

      3. +-commutative 63.9%

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

   6. Simplified 63.9%

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

   7. Taylor expanded in a2 around 0 46.0%

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

      1. unpow2 46.0%

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

   9. Simplified 46.0%

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

       1. frac-2neg 46.0%

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

       2. div-inv 45.9%

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

   11. Applied egg-rr 45.9%

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

   13. Applied egg-rr 33.9%

       \[\leadsto \left(-a1 \cdot a1\right) \cdot \color{blue}{-2} \]

   if 2.1000000000000002e29 < a2

   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. cos-neg 99.7%

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

      3. associate-*l/ 99.8%

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

      4. cos-neg 99.8%

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

      5. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a1 around 0 90.4%

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

      1. unpow2 90.4%

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

      2. associate-*l* 90.3%

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

   6. Simplified 90.3%

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

   7. Taylor expanded in th around 0 50.9%

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

      1. unpow2 50.9%

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

   9. Simplified 50.9%

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

   10. Taylor expanded in th around inf 10.9%

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

       1. unpow2 10.9%

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

       2. unpow2 10.9%

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

       3. associate-*l* 11.1%

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

       4. associate-*l* 11.1%

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

       5. *-commutative 11.1%

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

       6. associate-*l* 11.1%

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

       7. *-commutative 11.1%

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

   12. Simplified 11.1%

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

   14. Applied egg-rr 30.0%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 2.1 \cdot 10^{+29}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \left(--2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot th\right)\\ \end{array} \]

Alternative 7: 25.7% accurate, 58.8× speedup

Math

\[\begin{array}{l} a1 = |a1|\\ a2 = |a2|\\ [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ \begin{array}{l} \mathbf{if}\;a2 \leq 5.1 \cdot 10^{-113}:\\ \;\;\;\;\frac{a1 \cdot a1}{-2}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot th\right)\\ \end{array} \end{array} \]

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 5.1e-113) (/ (* a1 a1) -2.0) (* a2 (* a2 th))))

C

a1 = abs(a1);
a2 = abs(a2);
assert(a1 < a2);
double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 5.1e-113) {
		tmp = (a1 * a1) / -2.0;
	} else {
		tmp = a2 * (a2 * th);
	}
	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 <= 5.1d-113) then
        tmp = (a1 * a1) / (-2.0d0)
    else
        tmp = a2 * (a2 * th)
    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 <= 5.1e-113) {
		tmp = (a1 * a1) / -2.0;
	} else {
		tmp = a2 * (a2 * th);
	}
	return tmp;
}

Python

a1 = abs(a1)
a2 = abs(a2)
[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	tmp = 0
	if a2 <= 5.1e-113:
		tmp = (a1 * a1) / -2.0
	else:
		tmp = a2 * (a2 * th)
	return tmp

Julia

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 5.1e-113)
		tmp = Float64(Float64(a1 * a1) / -2.0);
	else
		tmp = Float64(a2 * Float64(a2 * th));
	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 <= 5.1e-113)
		tmp = (a1 * a1) / -2.0;
	else
		tmp = a2 * (a2 * th);
	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, 5.1e-113], N[(N[(a1 * a1), $MachinePrecision] / -2.0), $MachinePrecision], N[(a2 * N[(a2 * th), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 5.09999999999999979e-113

   1. Initial program 98.6%

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

      1. +-commutative 98.6%

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

      2. distribute-lft-out 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in th around 0 66.5%

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

      1. unpow2 66.5%

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

      2. unpow2 66.5%

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

      3. +-commutative 66.5%

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

   6. Simplified 66.5%

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

   7. Taylor expanded in a2 around 0 49.1%

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

      1. unpow2 49.1%

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

   9. Simplified 49.1%

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

       1. frac-2neg 49.1%

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

       2. div-inv 49.1%

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

   11. Applied egg-rr 49.1%

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

   13. Applied egg-rr 16.3%

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

       1. associate-/l* 16.3%

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

   15. Simplified 16.3%

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

   if 5.09999999999999979e-113 < 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. cos-neg 99.6%

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

      3. associate-*l/ 99.6%

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

      4. cos-neg 99.6%

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

      5. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 85.4%

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

      1. unpow2 85.4%

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

      2. associate-*l* 85.4%

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

   6. Simplified 85.4%

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

   7. Taylor expanded in th around 0 43.7%

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

      1. unpow2 43.7%

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

   9. Simplified 43.7%

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

   10. Taylor expanded in th around inf 8.9%

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

       1. unpow2 8.9%

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

       2. unpow2 8.9%

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

       3. associate-*l* 9.1%

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

       4. associate-*l* 9.1%

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

       5. *-commutative 9.1%

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

       6. associate-*l* 9.1%

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

       7. *-commutative 9.1%

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

   12. Simplified 9.1%

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

   14. Applied egg-rr 23.2%

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

4. Final simplification 18.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 5.1 \cdot 10^{-113}:\\ \;\;\;\;\frac{a1 \cdot a1}{-2}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot th\right)\\ \end{array} \]

Alternative 8: 23.9% accurate, 83.0× speedup

Math

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

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 (* a2 th)))

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

   1. distribute-lft-out 98.9%

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

   2. cos-neg 98.9%

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

   3. associate-*l/ 99.0%

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

   4. cos-neg 99.0%

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

   5. fma-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in a1 around 0 59.2%

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

   1. unpow2 59.2%

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

   2. associate-*l* 59.5%

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

6. Simplified 59.5%

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

7. Taylor expanded in th around 0 35.8%

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

   1. unpow2 35.8%

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

9. Simplified 35.8%

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

10. Taylor expanded in th around inf 12.8%

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

    1. unpow2 12.8%

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

    2. unpow2 12.8%

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

    3. associate-*l* 12.5%

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

    4. associate-*l* 12.5%

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

    5. *-commutative 12.5%

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

    6. associate-*l* 12.5%

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

    7. *-commutative 12.5%

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

12. Simplified 12.5%

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

14. Applied egg-rr 19.0%

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

15. Final simplification 19.0%

    \[\leadsto a2 \cdot \left(a2 \cdot th\right) \]

Alternative 9: 13.5% accurate, 138.3× speedup

Math

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

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 th))

C

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

Python

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

Julia

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

MATLAB

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = num2cell(sort([a1, a2])){:}
function tmp = code(a1, a2, th)
	tmp = a2 * 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[(a2 * th), $MachinePrecision]

Derivation

1. Initial program 98.9%

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

   1. distribute-lft-out 98.9%

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

   2. cos-neg 98.9%

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

   3. associate-*l/ 99.0%

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

   4. cos-neg 99.0%

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

   5. fma-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in a1 around 0 59.2%

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

   1. unpow2 59.2%

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

   2. associate-*l* 59.5%

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

6. Simplified 59.5%

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

7. Taylor expanded in th around 0 35.8%

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

   1. unpow2 35.8%

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

9. Simplified 35.8%

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

10. Taylor expanded in th around inf 12.8%

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

    1. unpow2 12.8%

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

    2. unpow2 12.8%

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

    3. associate-*l* 12.5%

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

    4. associate-*l* 12.5%

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

    5. *-commutative 12.5%

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

    6. associate-*l* 12.5%

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

    7. *-commutative 12.5%

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

12. Simplified 12.5%

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

14. Applied egg-rr 11.3%

    \[\leadsto \color{blue}{th \cdot a2} \]

15. Final simplification 11.3%

    \[\leadsto a2 \cdot th \]

Alternative 10: 3.5% accurate, 415.0× speedup

Math

\[\begin{array}{l} a1 = |a1|\\ a2 = |a2|\\ [a1, a2] = \mathsf{sort}([a1, a2])\\ \\ 1 \end{array} \]

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 1.0)

C

a1 = abs(a1);
a2 = abs(a2);
assert(a1 < a2);
double code(double a1, double a2, double th) {
	return 1.0;
}

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 = 1.0d0
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 1.0;
}

Python

a1 = abs(a1)
a2 = abs(a2)
[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	return 1.0

Julia

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	return 1.0
end

MATLAB

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = num2cell(sort([a1, a2])){:}
function tmp = code(a1, a2, th)
	tmp = 1.0;
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_] := 1.0

Derivation

1. Initial program 98.9%

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

   1. distribute-lft-out 98.9%

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

   2. cos-neg 98.9%

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

   3. associate-*l/ 99.0%

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

   4. cos-neg 99.0%

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

   5. fma-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in a1 around 0 59.2%

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

   1. unpow2 59.2%

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

   2. associate-*l* 59.5%

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

6. Simplified 59.5%

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

   1. div-inv 59.5%

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

   2. pow1/2 59.5%

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

   3. pow-flip 59.5%

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

   4. metadata-eval 59.5%

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

   5. *-commutative 59.5%

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

   6. associate-*l* 59.5%

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

   7. add-sqr-sqrt 59.2%

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

   8. sqrt-unprod 59.5%

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

   9. pow-prod-up 59.5%

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

   10. metadata-eval 59.5%

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

   11. metadata-eval 59.5%

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

8. Applied egg-rr 59.5%

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

9. Taylor expanded in th around 0 39.6%

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

    1. unpow2 39.6%

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

    2. associate-*l* 39.6%

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

11. Simplified 39.6%

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

13. Applied egg-rr 3.3%

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

    1. *-inverses 3.3%

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

15. Simplified 3.3%

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

16. Final simplification 3.3%

    \[\leadsto 1 \]

Alternative 11: 3.8% accurate, 415.0× speedup

Math

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

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 a1)

C

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

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 = a1
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 a1;
}

Python

a1 = abs(a1)
a2 = abs(a2)
[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	return a1

Julia

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	return a1
end

MATLAB

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = num2cell(sort([a1, a2])){:}
function tmp = code(a1, a2, th)
	tmp = a1;
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_] := a1

Derivation

1. Initial program 98.9%

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

   1. +-commutative 98.9%

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

   2. distribute-lft-out 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in th around 0 65.4%

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

   1. unpow2 65.4%

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

   2. unpow2 65.4%

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

   3. +-commutative 65.4%

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

6. Simplified 65.4%

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

7. Taylor expanded in a2 around 0 41.1%

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

   1. unpow2 41.1%

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

9. Simplified 41.1%

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

    1. frac-2neg 41.1%

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

    2. div-inv 41.1%

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

11. Applied egg-rr 41.1%

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

13. Applied egg-rr 3.4%

    \[\leadsto \color{blue}{0 + a1} \]
14. Step-by-step derivation

    1. +-lft-identity 3.4%

       \[\leadsto \color{blue}{a1} \]

15. Simplified 3.4%

    \[\leadsto \color{blue}{a1} \]

16. Final simplification 3.4%

    \[\leadsto a1 \]

Alternative 12: 5.5% accurate, 415.0× speedup

Math

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

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)

C

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

Python

a1 = abs(a1)
a2 = abs(a2)
[a1, a2] = sort([a1, a2])
def code(a1, a2, th):
	return a2

Julia

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = sort([a1, a2])
function code(a1, a2, th)
	return a2
end

MATLAB

a1 = abs(a1)
a2 = abs(a2)
a1, a2 = num2cell(sort([a1, a2])){:}
function tmp = code(a1, a2, th)
	tmp = 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_] := a2

Derivation

1. Initial program 98.9%

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

   1. distribute-lft-out 98.9%

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

   2. cos-neg 98.9%

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

   3. associate-*l/ 99.0%

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

   4. cos-neg 99.0%

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

   5. fma-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in a1 around 0 59.2%

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

   1. unpow2 59.2%

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

   2. associate-*l* 59.5%

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

6. Simplified 59.5%

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

   1. div-inv 59.5%

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

   2. pow1/2 59.5%

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

   3. pow-flip 59.5%

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

   4. metadata-eval 59.5%

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

   5. *-commutative 59.5%

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

   6. associate-*l* 59.5%

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

   7. add-sqr-sqrt 59.2%

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

   8. sqrt-unprod 59.5%

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

   9. pow-prod-up 59.5%

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

   10. metadata-eval 59.5%

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

   11. metadata-eval 59.5%

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

8. Applied egg-rr 59.5%

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

9. Taylor expanded in th around 0 39.6%

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

    1. unpow2 39.6%

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

    2. associate-*l* 39.6%

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

11. Simplified 39.6%

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

13. Applied egg-rr 4.8%

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

    1. unpow1 4.8%

       \[\leadsto \left|\color{blue}{{a2}^{1}}\right| \]

    2. *-inverses 4.8%

       \[\leadsto \left|{a2}^{\color{blue}{\left(\frac{a2}{a2}\right)}}\right| \]

    3. sqr-pow 2.4%

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

    4. fabs-sqr 2.4%

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

    5. sqr-pow 3.6%

       \[\leadsto \color{blue}{{a2}^{\left(\frac{a2}{a2}\right)}} \]

    6. *-inverses 3.6%

       \[\leadsto {a2}^{\color{blue}{1}} \]

    7. unpow1 3.6%

       \[\leadsto \color{blue}{a2} \]

15. Simplified 3.6%

    \[\leadsto \color{blue}{a2} \]

16. Final simplification 3.6%

    \[\leadsto a2 \]

Reproduce

herbie shell --seed 2023285 
(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))))