Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 10.1s

Alternatives: 12

Speedup: 2.0×

• 44.3% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\cos th \cdot {\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}{{2}^{0.25}}}{{2}^{0.25}} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (/ (/ (* (cos th) (pow (hypot a1 a2) 2.0)) (pow 2.0 0.25)) (pow 2.0 0.25)))

C

double code(double a1, double a2, double th) {
	return ((cos(th) * pow(hypot(a1, a2), 2.0)) / pow(2.0, 0.25)) / pow(2.0, 0.25);
}

Java

public static double code(double a1, double a2, double th) {
	return ((Math.cos(th) * Math.pow(Math.hypot(a1, a2), 2.0)) / Math.pow(2.0, 0.25)) / Math.pow(2.0, 0.25);
}

Python

def code(a1, a2, th):
	return ((math.cos(th) * math.pow(math.hypot(a1, a2), 2.0)) / math.pow(2.0, 0.25)) / math.pow(2.0, 0.25)

Julia

function code(a1, a2, th)
	return Float64(Float64(Float64(cos(th) * (hypot(a1, a2) ^ 2.0)) / (2.0 ^ 0.25)) / (2.0 ^ 0.25))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = ((cos(th) * (hypot(a1, a2) ^ 2.0)) / (2.0 ^ 0.25)) / (2.0 ^ 0.25);
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(N[Cos[th], $MachinePrecision] * N[Power[N[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision] / N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out 99.3%

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

3. Simplified 99.3%

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

   1. associate-*l/ 99.3%

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

   2. add-sqr-sqrt 99.2%

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

   3. associate-/r* 99.3%

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

   4. add-sqr-sqrt 99.3%

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

   5. pow2 99.3%

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

   6. hypot-def 99.3%

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

   7. pow1/2 99.3%

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

   8. sqrt-pow1 99.3%

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

   9. metadata-eval 99.3%

      \[\leadsto \frac{\frac{\cos th \cdot {\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}{{2}^{\color{blue}{0.25}}}}{\sqrt{\sqrt{2}}} \]

   10. pow1/2 99.3%

       \[\leadsto \frac{\frac{\cos th \cdot {\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}{{2}^{0.25}}}{\sqrt{\color{blue}{{2}^{0.5}}}} \]

   11. sqrt-pow1 99.3%

       \[\leadsto \frac{\frac{\cos th \cdot {\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}{{2}^{0.25}}}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}} \]

   12. metadata-eval 99.3%

       \[\leadsto \frac{\frac{\cos th \cdot {\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}{{2}^{0.25}}}{{2}^{\color{blue}{0.25}}} \]

6. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{\frac{\cos th \cdot {\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}{{2}^{0.25}}}{{2}^{0.25}}} \]

7. Final simplification 99.3%

   \[\leadsto \frac{\frac{\cos th \cdot {\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}{{2}^{0.25}}}{{2}^{0.25}} \]
8. Add Preprocessing

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\cos th}{\frac{\sqrt{2}}{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[N[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out 99.3%

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

3. Simplified 99.3%

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

   1. associate-*l/ 99.3%

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

   2. associate-/l* 99.3%

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

   3. add-sqr-sqrt 99.3%

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

   4. pow2 99.3%

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

   5. hypot-def 99.3%

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

6. Applied egg-rr 99.3%

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

7. Final simplification 99.3%

   \[\leadsto \frac{\cos th}{\frac{\sqrt{2}}{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}} \]
8. Add Preprocessing

Alternative 3: 71.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.6:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \frac{1}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.6)
   (* a2 (* (cos th) a2))
   (* (+ (* a1 a1) (* a2 a2)) (/ 1.0 (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.6) {
		tmp = a2 * (cos(th) * a2);
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * (1.0 / 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 (cos(th) <= 0.6d0) then
        tmp = a2 * (cos(th) * a2)
    else
        tmp = ((a1 * a1) + (a2 * a2)) * (1.0d0 / sqrt(2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.6) {
		tmp = a2 * (Math.cos(th) * a2);
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * (1.0 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.6:
		tmp = a2 * (math.cos(th) * a2)
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * (1.0 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.6)
		tmp = Float64(a2 * Float64(cos(th) * a2));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * Float64(1.0 / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.6)
		tmp = a2 * (cos(th) * a2);
	else
		tmp = ((a1 * a1) + (a2 * a2)) * (1.0 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.6], N[(a2 * N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.599999999999999978

   1. Initial program 98.8%

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

      1. distribute-lft-out 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in a1 around 0 49.4%

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

   7. Applied egg-rr 36.8%

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

   if 0.599999999999999978 < (cos.f64 th)

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in th around 0 93.0%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.6:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \frac{1}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.6:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.6)
   (* a2 (* (cos th) a2))
   (* (sqrt 0.5) (+ (* a1 a1) (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.6) {
		tmp = a2 * (cos(th) * a2);
	} else {
		tmp = sqrt(0.5) * ((a1 * a1) + (a2 * 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 (cos(th) <= 0.6d0) then
        tmp = a2 * (cos(th) * a2)
    else
        tmp = sqrt(0.5d0) * ((a1 * a1) + (a2 * a2))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.6) {
		tmp = a2 * (Math.cos(th) * a2);
	} else {
		tmp = Math.sqrt(0.5) * ((a1 * a1) + (a2 * a2));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.6:
		tmp = a2 * (math.cos(th) * a2)
	else:
		tmp = math.sqrt(0.5) * ((a1 * a1) + (a2 * a2))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.6)
		tmp = Float64(a2 * Float64(cos(th) * a2));
	else
		tmp = Float64(sqrt(0.5) * Float64(Float64(a1 * a1) + Float64(a2 * a2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.6)
		tmp = a2 * (cos(th) * a2);
	else
		tmp = sqrt(0.5) * ((a1 * a1) + (a2 * a2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.6], N[(a2 * N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.599999999999999978

   1. Initial program 98.8%

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

      1. distribute-lft-out 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in a1 around 0 49.4%

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

   7. Applied egg-rr 36.8%

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

   if 0.599999999999999978 < (cos.f64 th)

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

      3. pow1/2 99.6%

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

      4. pow-flip 99.6%

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

      5. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in th around 0 93.0%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.6:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out 99.3%

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

3. Simplified 99.3%

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

   1. clear-num 99.3%

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

   2. associate-/r/ 99.3%

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

   3. pow1/2 99.3%

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

   4. pow-flip 99.3%

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

   5. metadata-eval 99.3%

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

6. Applied egg-rr 99.3%

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

7. Taylor expanded in th around inf 99.3%

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

   1. *-commutative 99.3%

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

9. Simplified 99.3%

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

10. Final simplification 99.3%

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

Alternative 6: 37.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return a2 * (cos(th) * 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 * (cos(th) * a2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in a1 around 0 52.4%

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

7. Applied egg-rr 34.5%

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

8. Final simplification 34.5%

   \[\leadsto a2 \cdot \left(\cos th \cdot a2\right) \]
9. Add Preprocessing

Alternative 7: 46.1% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ t_2 := 0.25 \cdot t_1\\ \mathbf{if}\;th \leq 5 \cdot 10^{+41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;th \leq 1.3 \cdot 10^{+63}:\\ \;\;\;\;t_1 \cdot -0.25\\ \mathbf{elif}\;th \leq 6.5 \cdot 10^{+94} \lor \neg \left(th \leq 3.9 \cdot 10^{+204}\right) \land th \leq 6.6 \cdot 10^{+226}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))) (t_2 (* 0.25 t_1)))
   (if (<= th 5e+41)
     t_2
     (if (<= th 1.3e+63)
       (* t_1 -0.25)
       (if (or (<= th 6.5e+94) (and (not (<= th 3.9e+204)) (<= th 6.6e+226)))
         t_2
         (* t_1 -0.5))))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double t_2 = 0.25 * t_1;
	double tmp;
	if (th <= 5e+41) {
		tmp = t_2;
	} else if (th <= 1.3e+63) {
		tmp = t_1 * -0.25;
	} else if ((th <= 6.5e+94) || (!(th <= 3.9e+204) && (th <= 6.6e+226))) {
		tmp = t_2;
	} else {
		tmp = t_1 * -0.5;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    t_2 = 0.25d0 * t_1
    if (th <= 5d+41) then
        tmp = t_2
    else if (th <= 1.3d+63) then
        tmp = t_1 * (-0.25d0)
    else if ((th <= 6.5d+94) .or. (.not. (th <= 3.9d+204)) .and. (th <= 6.6d+226)) then
        tmp = t_2
    else
        tmp = t_1 * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double t_2 = 0.25 * t_1;
	double tmp;
	if (th <= 5e+41) {
		tmp = t_2;
	} else if (th <= 1.3e+63) {
		tmp = t_1 * -0.25;
	} else if ((th <= 6.5e+94) || (!(th <= 3.9e+204) && (th <= 6.6e+226))) {
		tmp = t_2;
	} else {
		tmp = t_1 * -0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	t_2 = 0.25 * t_1
	tmp = 0
	if th <= 5e+41:
		tmp = t_2
	elif th <= 1.3e+63:
		tmp = t_1 * -0.25
	elif (th <= 6.5e+94) or (not (th <= 3.9e+204) and (th <= 6.6e+226)):
		tmp = t_2
	else:
		tmp = t_1 * -0.5
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	t_2 = Float64(0.25 * t_1)
	tmp = 0.0
	if (th <= 5e+41)
		tmp = t_2;
	elseif (th <= 1.3e+63)
		tmp = Float64(t_1 * -0.25);
	elseif ((th <= 6.5e+94) || (!(th <= 3.9e+204) && (th <= 6.6e+226)))
		tmp = t_2;
	else
		tmp = Float64(t_1 * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	t_2 = 0.25 * t_1;
	tmp = 0.0;
	if (th <= 5e+41)
		tmp = t_2;
	elseif (th <= 1.3e+63)
		tmp = t_1 * -0.25;
	elseif ((th <= 6.5e+94) || (~((th <= 3.9e+204)) && (th <= 6.6e+226)))
		tmp = t_2;
	else
		tmp = t_1 * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.25 * t$95$1), $MachinePrecision]}, If[LessEqual[th, 5e+41], t$95$2, If[LessEqual[th, 1.3e+63], N[(t$95$1 * -0.25), $MachinePrecision], If[Or[LessEqual[th, 6.5e+94], And[N[Not[LessEqual[th, 3.9e+204]], $MachinePrecision], LessEqual[th, 6.6e+226]]], t$95$2, N[(t$95$1 * -0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if th < 5.00000000000000022e41 or 1.3000000000000001e63 < th < 6.49999999999999976e94 or 3.90000000000000017e204 < th < 6.59999999999999956e226

   1. Initial program 99.3%

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

      1. distribute-lft-out 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in th around 0 68.1%

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

   7. Applied egg-rr 43.1%

      \[\leadsto \color{blue}{0.25} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 5.00000000000000022e41 < th < 1.3000000000000001e63

   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%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in th around 0 13.4%

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

   7. Applied egg-rr 50.4%

      \[\leadsto \color{blue}{-0.25} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 6.49999999999999976e94 < th < 3.90000000000000017e204 or 6.59999999999999956e226 < th

   1. Initial program 99.4%

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

      1. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in th around 0 32.1%

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

   7. Applied egg-rr 41.5%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 5 \cdot 10^{+41}:\\ \;\;\;\;0.25 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{elif}\;th \leq 1.3 \cdot 10^{+63}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.25\\ \mathbf{elif}\;th \leq 6.5 \cdot 10^{+94} \lor \neg \left(th \leq 3.9 \cdot 10^{+204}\right) \land th \leq 6.6 \cdot 10^{+226}:\\ \;\;\;\;0.25 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.0% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ t_2 := 0.5 \cdot t_1\\ \mathbf{if}\;th \leq 5 \cdot 10^{+41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;th \leq 1.3 \cdot 10^{+63}:\\ \;\;\;\;t_1 \cdot -0.25\\ \mathbf{elif}\;th \leq 2.65 \cdot 10^{+106} \lor \neg \left(th \leq 3.9 \cdot 10^{+204}\right) \land th \leq 6.6 \cdot 10^{+226}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))) (t_2 (* 0.5 t_1)))
   (if (<= th 5e+41)
     t_2
     (if (<= th 1.3e+63)
       (* t_1 -0.25)
       (if (or (<= th 2.65e+106) (and (not (<= th 3.9e+204)) (<= th 6.6e+226)))
         t_2
         (* t_1 -0.5))))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double t_2 = 0.5 * t_1;
	double tmp;
	if (th <= 5e+41) {
		tmp = t_2;
	} else if (th <= 1.3e+63) {
		tmp = t_1 * -0.25;
	} else if ((th <= 2.65e+106) || (!(th <= 3.9e+204) && (th <= 6.6e+226))) {
		tmp = t_2;
	} else {
		tmp = t_1 * -0.5;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    t_2 = 0.5d0 * t_1
    if (th <= 5d+41) then
        tmp = t_2
    else if (th <= 1.3d+63) then
        tmp = t_1 * (-0.25d0)
    else if ((th <= 2.65d+106) .or. (.not. (th <= 3.9d+204)) .and. (th <= 6.6d+226)) then
        tmp = t_2
    else
        tmp = t_1 * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double t_2 = 0.5 * t_1;
	double tmp;
	if (th <= 5e+41) {
		tmp = t_2;
	} else if (th <= 1.3e+63) {
		tmp = t_1 * -0.25;
	} else if ((th <= 2.65e+106) || (!(th <= 3.9e+204) && (th <= 6.6e+226))) {
		tmp = t_2;
	} else {
		tmp = t_1 * -0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	t_2 = 0.5 * t_1
	tmp = 0
	if th <= 5e+41:
		tmp = t_2
	elif th <= 1.3e+63:
		tmp = t_1 * -0.25
	elif (th <= 2.65e+106) or (not (th <= 3.9e+204) and (th <= 6.6e+226)):
		tmp = t_2
	else:
		tmp = t_1 * -0.5
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	t_2 = Float64(0.5 * t_1)
	tmp = 0.0
	if (th <= 5e+41)
		tmp = t_2;
	elseif (th <= 1.3e+63)
		tmp = Float64(t_1 * -0.25);
	elseif ((th <= 2.65e+106) || (!(th <= 3.9e+204) && (th <= 6.6e+226)))
		tmp = t_2;
	else
		tmp = Float64(t_1 * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	t_2 = 0.5 * t_1;
	tmp = 0.0;
	if (th <= 5e+41)
		tmp = t_2;
	elseif (th <= 1.3e+63)
		tmp = t_1 * -0.25;
	elseif ((th <= 2.65e+106) || (~((th <= 3.9e+204)) && (th <= 6.6e+226)))
		tmp = t_2;
	else
		tmp = t_1 * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * t$95$1), $MachinePrecision]}, If[LessEqual[th, 5e+41], t$95$2, If[LessEqual[th, 1.3e+63], N[(t$95$1 * -0.25), $MachinePrecision], If[Or[LessEqual[th, 2.65e+106], And[N[Not[LessEqual[th, 3.9e+204]], $MachinePrecision], LessEqual[th, 6.6e+226]]], t$95$2, N[(t$95$1 * -0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if th < 5.00000000000000022e41 or 1.3000000000000001e63 < th < 2.65e106 or 3.90000000000000017e204 < th < 6.59999999999999956e226

   1. Initial program 99.3%

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

      1. distribute-lft-out 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in th around 0 67.6%

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

   7. Applied egg-rr 43.6%

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

   if 5.00000000000000022e41 < th < 1.3000000000000001e63

   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%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in th around 0 13.4%

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

   7. Applied egg-rr 50.4%

      \[\leadsto \color{blue}{-0.25} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 2.65e106 < th < 3.90000000000000017e204 or 6.59999999999999956e226 < th

   1. Initial program 99.4%

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

      1. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in th around 0 33.4%

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

   7. Applied egg-rr 40.9%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 5 \cdot 10^{+41}:\\ \;\;\;\;0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{elif}\;th \leq 1.3 \cdot 10^{+63}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.25\\ \mathbf{elif}\;th \leq 2.65 \cdot 10^{+106} \lor \neg \left(th \leq 3.9 \cdot 10^{+204}\right) \land th \leq 6.6 \cdot 10^{+226}:\\ \;\;\;\;0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.5% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ t_2 := t_1 \cdot 0.0625\\ \mathbf{if}\;th \leq 5 \cdot 10^{+41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;th \leq 1.3 \cdot 10^{+63}:\\ \;\;\;\;t_1 \cdot -0.25\\ \mathbf{elif}\;th \leq 6.5 \cdot 10^{+94}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))) (t_2 (* t_1 0.0625)))
   (if (<= th 5e+41)
     t_2
     (if (<= th 1.3e+63)
       (* t_1 -0.25)
       (if (<= th 6.5e+94) t_2 (* t_1 -0.5))))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double t_2 = t_1 * 0.0625;
	double tmp;
	if (th <= 5e+41) {
		tmp = t_2;
	} else if (th <= 1.3e+63) {
		tmp = t_1 * -0.25;
	} else if (th <= 6.5e+94) {
		tmp = t_2;
	} else {
		tmp = t_1 * -0.5;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    t_2 = t_1 * 0.0625d0
    if (th <= 5d+41) then
        tmp = t_2
    else if (th <= 1.3d+63) then
        tmp = t_1 * (-0.25d0)
    else if (th <= 6.5d+94) then
        tmp = t_2
    else
        tmp = t_1 * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double t_2 = t_1 * 0.0625;
	double tmp;
	if (th <= 5e+41) {
		tmp = t_2;
	} else if (th <= 1.3e+63) {
		tmp = t_1 * -0.25;
	} else if (th <= 6.5e+94) {
		tmp = t_2;
	} else {
		tmp = t_1 * -0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	t_2 = t_1 * 0.0625
	tmp = 0
	if th <= 5e+41:
		tmp = t_2
	elif th <= 1.3e+63:
		tmp = t_1 * -0.25
	elif th <= 6.5e+94:
		tmp = t_2
	else:
		tmp = t_1 * -0.5
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	t_2 = Float64(t_1 * 0.0625)
	tmp = 0.0
	if (th <= 5e+41)
		tmp = t_2;
	elseif (th <= 1.3e+63)
		tmp = Float64(t_1 * -0.25);
	elseif (th <= 6.5e+94)
		tmp = t_2;
	else
		tmp = Float64(t_1 * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	t_2 = t_1 * 0.0625;
	tmp = 0.0;
	if (th <= 5e+41)
		tmp = t_2;
	elseif (th <= 1.3e+63)
		tmp = t_1 * -0.25;
	elseif (th <= 6.5e+94)
		tmp = t_2;
	else
		tmp = t_1 * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 0.0625), $MachinePrecision]}, If[LessEqual[th, 5e+41], t$95$2, If[LessEqual[th, 1.3e+63], N[(t$95$1 * -0.25), $MachinePrecision], If[LessEqual[th, 6.5e+94], t$95$2, N[(t$95$1 * -0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if th < 5.00000000000000022e41 or 1.3000000000000001e63 < th < 6.49999999999999976e94

   1. Initial program 99.3%

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

      1. distribute-lft-out 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in th around 0 68.0%

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

   7. Applied egg-rr 41.6%

      \[\leadsto \color{blue}{0.0625} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 5.00000000000000022e41 < th < 1.3000000000000001e63

   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%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in th around 0 13.4%

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

   7. Applied egg-rr 50.4%

      \[\leadsto \color{blue}{-0.25} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 6.49999999999999976e94 < th

   1. Initial program 99.4%

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

      1. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in th around 0 36.7%

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

   7. Applied egg-rr 37.5%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.0625\\ \mathbf{elif}\;th \leq 1.3 \cdot 10^{+63}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.25\\ \mathbf{elif}\;th \leq 6.5 \cdot 10^{+94}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.8% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ t_2 := t_1 \cdot 0.125\\ \mathbf{if}\;th \leq 5 \cdot 10^{+41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;th \leq 1.3 \cdot 10^{+63}:\\ \;\;\;\;t_1 \cdot -0.25\\ \mathbf{elif}\;th \leq 6.5 \cdot 10^{+94}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))) (t_2 (* t_1 0.125)))
   (if (<= th 5e+41)
     t_2
     (if (<= th 1.3e+63)
       (* t_1 -0.25)
       (if (<= th 6.5e+94) t_2 (* t_1 -0.5))))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double t_2 = t_1 * 0.125;
	double tmp;
	if (th <= 5e+41) {
		tmp = t_2;
	} else if (th <= 1.3e+63) {
		tmp = t_1 * -0.25;
	} else if (th <= 6.5e+94) {
		tmp = t_2;
	} else {
		tmp = t_1 * -0.5;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    t_2 = t_1 * 0.125d0
    if (th <= 5d+41) then
        tmp = t_2
    else if (th <= 1.3d+63) then
        tmp = t_1 * (-0.25d0)
    else if (th <= 6.5d+94) then
        tmp = t_2
    else
        tmp = t_1 * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double t_2 = t_1 * 0.125;
	double tmp;
	if (th <= 5e+41) {
		tmp = t_2;
	} else if (th <= 1.3e+63) {
		tmp = t_1 * -0.25;
	} else if (th <= 6.5e+94) {
		tmp = t_2;
	} else {
		tmp = t_1 * -0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	t_2 = t_1 * 0.125
	tmp = 0
	if th <= 5e+41:
		tmp = t_2
	elif th <= 1.3e+63:
		tmp = t_1 * -0.25
	elif th <= 6.5e+94:
		tmp = t_2
	else:
		tmp = t_1 * -0.5
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	t_2 = Float64(t_1 * 0.125)
	tmp = 0.0
	if (th <= 5e+41)
		tmp = t_2;
	elseif (th <= 1.3e+63)
		tmp = Float64(t_1 * -0.25);
	elseif (th <= 6.5e+94)
		tmp = t_2;
	else
		tmp = Float64(t_1 * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	t_2 = t_1 * 0.125;
	tmp = 0.0;
	if (th <= 5e+41)
		tmp = t_2;
	elseif (th <= 1.3e+63)
		tmp = t_1 * -0.25;
	elseif (th <= 6.5e+94)
		tmp = t_2;
	else
		tmp = t_1 * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 0.125), $MachinePrecision]}, If[LessEqual[th, 5e+41], t$95$2, If[LessEqual[th, 1.3e+63], N[(t$95$1 * -0.25), $MachinePrecision], If[LessEqual[th, 6.5e+94], t$95$2, N[(t$95$1 * -0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if th < 5.00000000000000022e41 or 1.3000000000000001e63 < th < 6.49999999999999976e94

   1. Initial program 99.3%

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

      1. distribute-lft-out 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in th around 0 68.0%

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

   7. Applied egg-rr 42.0%

      \[\leadsto \color{blue}{0.125} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 5.00000000000000022e41 < th < 1.3000000000000001e63

   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%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in th around 0 13.4%

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

   7. Applied egg-rr 50.4%

      \[\leadsto \color{blue}{-0.25} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 6.49999999999999976e94 < th

   1. Initial program 99.4%

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

      1. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in th around 0 36.7%

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

   7. Applied egg-rr 37.5%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.125\\ \mathbf{elif}\;th \leq 1.3 \cdot 10^{+63}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.25\\ \mathbf{elif}\;th \leq 6.5 \cdot 10^{+94}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 20.0% accurate, 46.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double a1, double a2, double th) {
	return ((a1 * a1) + (a2 * a2)) * -0.5;
}

Python

def code(a1, a2, th):
	return ((a1 * a1) + (a2 * a2)) * -0.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in th around 0 62.6%

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

7. Applied egg-rr 20.6%

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

8. Final simplification 20.6%

   \[\leadsto \left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5 \]
9. Add Preprocessing

Alternative 12: 19.8% accurate, 51.9× speedup

Math

\[\begin{array}{l} \\ a1 \cdot \left(-a1\right) - a2 \cdot a2 \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (- (* a1 (- a1)) (* a2 a2)))

C

double code(double a1, double a2, double th) {
	return (a1 * -a1) - (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 = (a1 * -a1) - (a2 * a2)
end function

Java

public static double code(double a1, double a2, double th) {
	return (a1 * -a1) - (a2 * a2);
}

Python

def code(a1, a2, th):
	return (a1 * -a1) - (a2 * a2)

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(a1 * (-a1)), $MachinePrecision] - N[(a2 * a2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in th around 0 62.6%

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

7. Applied egg-rr 20.4%

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

8. Final simplification 20.4%

   \[\leadsto a1 \cdot \left(-a1\right) - a2 \cdot a2 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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))))