Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 12.5s

Alternatives: 7

Speedup: 2.0×

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around inf 99.6%

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

   1. *-commutative 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 57.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in a2 around inf 54.6%

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

   1. unpow2 54.6%

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

   2. associate-*r/ 54.6%

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

   3. associate-*r* 54.5%

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

6. Simplified 54.5%

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

7. Final simplification 54.5%

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

Alternative 3: 57.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in a2 around inf 54.5%

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

   1. unpow2 54.5%

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

   2. *-commutative 54.5%

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

   3. associate-*l* 54.6%

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

8. Simplified 54.6%

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

9. Final simplification 54.6%

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

Alternative 4: 40.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a1 \leq 2.6 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\\ \mathbf{elif}\;a1 \leq 6.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{a2 \cdot \left(a2 + -0.5 \cdot \left(a2 \cdot \left(th \cdot th\right)\right)\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a1 2.6e-33)
   (* (sqrt 0.5) (* a2 a2))
   (if (<= a1 6.5e+36)
     (/ (* a2 (+ a2 (* -0.5 (* a2 (* th th))))) (sqrt 2.0))
     (* a2 (/ a2 (sqrt 2.0))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= 2.6e-33) {
		tmp = sqrt(0.5) * (a2 * a2);
	} else if (a1 <= 6.5e+36) {
		tmp = (a2 * (a2 + (-0.5 * (a2 * (th * th))))) / sqrt(2.0);
	} else {
		tmp = a2 * (a2 / sqrt(2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (a1 <= 2.6d-33) then
        tmp = sqrt(0.5d0) * (a2 * a2)
    else if (a1 <= 6.5d+36) then
        tmp = (a2 * (a2 + ((-0.5d0) * (a2 * (th * th))))) / sqrt(2.0d0)
    else
        tmp = a2 * (a2 / sqrt(2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= 2.6e-33) {
		tmp = Math.sqrt(0.5) * (a2 * a2);
	} else if (a1 <= 6.5e+36) {
		tmp = (a2 * (a2 + (-0.5 * (a2 * (th * th))))) / Math.sqrt(2.0);
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a1 <= 2.6e-33:
		tmp = math.sqrt(0.5) * (a2 * a2)
	elif a1 <= 6.5e+36:
		tmp = (a2 * (a2 + (-0.5 * (a2 * (th * th))))) / math.sqrt(2.0)
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a1 <= 2.6e-33)
		tmp = Float64(sqrt(0.5) * Float64(a2 * a2));
	elseif (a1 <= 6.5e+36)
		tmp = Float64(Float64(a2 * Float64(a2 + Float64(-0.5 * Float64(a2 * Float64(th * th))))) / sqrt(2.0));
	else
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a1 <= 2.6e-33)
		tmp = sqrt(0.5) * (a2 * a2);
	elseif (a1 <= 6.5e+36)
		tmp = (a2 * (a2 + (-0.5 * (a2 * (th * th))))) / sqrt(2.0);
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a1, 2.6e-33], N[(N[Sqrt[0.5], $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision], If[LessEqual[a1, 6.5e+36], N[(N[(a2 * N[(a2 + N[(-0.5 * N[(a2 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a1 < 2.59999999999999994e-33

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.4%

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

      3. pow1/2 99.4%

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

      4. pow-flip 99.6%

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

      5. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in th around 0 58.3%

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

   7. Taylor expanded in a2 around inf 38.3%

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

      1. unpow2 38.3%

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

      2. *-commutative 38.3%

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

   9. Simplified 38.3%

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

   if 2.59999999999999994e-33 < a1 < 6.4999999999999998e36

   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/ 99.5%

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

      4. cos-neg 99.5%

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

      5. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a1 around 0 39.6%

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

      1. unpow2 39.6%

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

      2. associate-*l* 39.6%

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

   6. Simplified 39.6%

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

   7. Taylor expanded in th around 0 21.7%

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

      1. *-commutative 21.7%

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

      2. unpow2 21.7%

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

   9. Simplified 21.7%

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

   if 6.4999999999999998e36 < a1

   1. Initial program 99.8%

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

      1. distribute-lft-out 99.8%

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

      2. cos-neg 99.8%

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

      3. associate-*l/ 99.7%

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

      4. cos-neg 99.7%

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

      5. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around 0 32.2%

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

      1. unpow2 32.2%

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

      2. associate-*l* 32.2%

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

   6. Simplified 32.2%

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

   7. Taylor expanded in th around 0 26.3%

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

      1. associate-/l* 26.3%

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

      2. associate-/r/ 26.3%

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

   9. Applied egg-rr 26.3%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \leq 2.6 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\\ \mathbf{elif}\;a1 \leq 6.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{a2 \cdot \left(a2 + -0.5 \cdot \left(a2 \cdot \left(th \cdot th\right)\right)\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 5: 40.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 2.7 \cdot 10^{+81}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\\ \mathbf{elif}\;th \leq 2 \cdot 10^{+167}:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(a2 \cdot a2\right) \cdot \left(th \cdot th\right)\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(\sqrt{0.5} \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 2.7e+81)
   (* (sqrt 0.5) (* a2 a2))
   (if (<= th 2e+167)
     (/ (* -0.5 (* (* a2 a2) (* th th))) (sqrt 2.0))
     (* a2 (* (sqrt 0.5) a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 2.7e+81) {
		tmp = sqrt(0.5) * (a2 * a2);
	} else if (th <= 2e+167) {
		tmp = (-0.5 * ((a2 * a2) * (th * th))) / sqrt(2.0);
	} else {
		tmp = a2 * (sqrt(0.5) * a2);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 2.7d+81) then
        tmp = sqrt(0.5d0) * (a2 * a2)
    else if (th <= 2d+167) then
        tmp = ((-0.5d0) * ((a2 * a2) * (th * th))) / sqrt(2.0d0)
    else
        tmp = a2 * (sqrt(0.5d0) * a2)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 2.7e+81) {
		tmp = Math.sqrt(0.5) * (a2 * a2);
	} else if (th <= 2e+167) {
		tmp = (-0.5 * ((a2 * a2) * (th * th))) / Math.sqrt(2.0);
	} else {
		tmp = a2 * (Math.sqrt(0.5) * a2);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 2.7e+81:
		tmp = math.sqrt(0.5) * (a2 * a2)
	elif th <= 2e+167:
		tmp = (-0.5 * ((a2 * a2) * (th * th))) / math.sqrt(2.0)
	else:
		tmp = a2 * (math.sqrt(0.5) * a2)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 2.7e+81)
		tmp = Float64(sqrt(0.5) * Float64(a2 * a2));
	elseif (th <= 2e+167)
		tmp = Float64(Float64(-0.5 * Float64(Float64(a2 * a2) * Float64(th * th))) / sqrt(2.0));
	else
		tmp = Float64(a2 * Float64(sqrt(0.5) * a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 2.7e+81)
		tmp = sqrt(0.5) * (a2 * a2);
	elseif (th <= 2e+167)
		tmp = (-0.5 * ((a2 * a2) * (th * th))) / sqrt(2.0);
	else
		tmp = a2 * (sqrt(0.5) * a2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 2.7e+81], N[(N[Sqrt[0.5], $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 2e+167], N[(N[(-0.5 * N[(N[(a2 * a2), $MachinePrecision] * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(a2 * N[(N[Sqrt[0.5], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 2.6999999999999999e81

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

      3. pow1/2 99.5%

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

      4. pow-flip 99.7%

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

      5. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in th around 0 65.3%

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

   7. Taylor expanded in a2 around inf 38.4%

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

      1. unpow2 38.4%

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

      2. *-commutative 38.4%

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

   9. Simplified 38.4%

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

   if 2.6999999999999999e81 < th < 2.0000000000000001e167

   1. Initial program 99.4%

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

      1. distribute-lft-out 99.4%

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

      2. cos-neg 99.4%

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

      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 43.7%

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

      1. unpow2 43.7%

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

      2. associate-*l* 43.7%

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

   6. Simplified 43.7%

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

   7. Taylor expanded in th around 0 27.6%

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

      1. *-commutative 27.6%

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

      2. unpow2 27.6%

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

   9. Simplified 27.6%

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

   10. Taylor expanded in th around inf 35.3%

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

       1. unpow2 35.3%

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

       2. *-commutative 35.3%

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

       3. unpow2 35.3%

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

   12. Simplified 35.3%

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

   if 2.0000000000000001e167 < th

   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. +-commutative 99.7%

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

      2. distribute-lft-out 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

      3. pow1/2 99.6%

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

      4. pow-flip 99.5%

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

      5. metadata-eval 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in th around inf 99.5%

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

      1. *-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in a2 around inf 60.5%

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

       1. unpow2 60.5%

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

   11. Simplified 60.5%

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

   12. Taylor expanded in th around 0 16.2%

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

       1. unpow2 16.2%

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

       2. associate-*l* 16.2%

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

   14. Simplified 16.2%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 2.7 \cdot 10^{+81}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\\ \mathbf{elif}\;th \leq 2 \cdot 10^{+167}:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(a2 \cdot a2\right) \cdot \left(th \cdot th\right)\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(\sqrt{0.5} \cdot a2\right)\\ \end{array} \]

Alternative 6: 40.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around inf 99.6%

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

   1. *-commutative 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in a2 around inf 54.5%

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

    1. unpow2 54.5%

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

11. Simplified 54.5%

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

12. Taylor expanded in th around 0 34.3%

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

    1. unpow2 34.3%

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

    2. associate-*l* 34.2%

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

14. Simplified 34.2%

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

15. Final simplification 34.2%

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

Alternative 7: 40.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around 0 58.8%

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

7. Taylor expanded in a2 around inf 34.3%

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

   1. unpow2 34.3%

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

   2. *-commutative 34.3%

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

9. Simplified 34.3%

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

10. Final simplification 34.3%

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

Reproduce

herbie shell --seed 2023274 
(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))))