Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 13.2s

Alternatives: 12

Speedup: 2.0×

• 43.4% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos th \cdot \left({\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2} \cdot {2}^{-0.5}\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (cos th) (* (pow (hypot a2 a1) 2.0) (pow 2.0 -0.5))))

C

double code(double a1, double a2, double th) {
	return cos(th) * (pow(hypot(a2, a1), 2.0) * pow(2.0, -0.5));
}

Java

public static double code(double a1, double a2, double th) {
	return Math.cos(th) * (Math.pow(Math.hypot(a2, a1), 2.0) * Math.pow(2.0, -0.5));
}

Python

def code(a1, a2, th):
	return math.cos(th) * (math.pow(math.hypot(a2, a1), 2.0) * math.pow(2.0, -0.5))

Julia

function code(a1, a2, th)
	return Float64(cos(th) * Float64((hypot(a2, a1) ^ 2.0) * (2.0 ^ -0.5)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = cos(th) * ((hypot(a2, a1) ^ 2.0) * (2.0 ^ -0.5));
end

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[Power[N[Sqrt[a2 ^ 2 + a1 ^ 2], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. cos-neg 99.6%

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

   3. associate-*l/ 99.6%

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

   4. *-commutative 99.6%

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

   5. associate-*l/ 99.6%

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

   6. *-commutative 99.6%

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

   7. cos-neg 99.6%

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

   8. +-commutative 99.6%

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

   9. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.6%

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

   2. add-sqr-sqrt 99.5%

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

   3. pow2 99.5%

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

   4. fma-undefine 99.5%

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

   5. hypot-define 99.5%

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

   6. pow1/2 99.5%

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

   7. pow-flip 99.7%

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

   8. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

   \[\leadsto \cos th \cdot \left({\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2} \cdot {2}^{-0.5}\right) \]
8. Add Preprocessing

Alternative 2: 55.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.7)
   (* (cos th) (+ (* a1 a1) (* a2 a2)))
   (* a2 (/ a2 (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = cos(th) * ((a1 * a1) + (a2 * a2));
	} 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 (cos(th) <= 0.7d0) then
        tmp = cos(th) * ((a1 * a1) + (a2 * a2))
    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 (Math.cos(th) <= 0.7) {
		tmp = Math.cos(th) * ((a1 * a1) + (a2 * a2));
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.7:
		tmp = math.cos(th) * ((a1 * a1) + (a2 * a2))
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.7)
		tmp = Float64(cos(th) * Float64(Float64(a1 * a1) + Float64(a2 * a2)));
	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 (cos(th) <= 0.7)
		tmp = cos(th) * ((a1 * a1) + (a2 * a2));
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.7], N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69999999999999996

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. add-sqr-sqrt 99.6%

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

      3. times-frac 99.5%

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

      4. pow1/2 99.5%

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

      5. sqrt-pow1 99.5%

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

      6. metadata-eval 99.5%

         \[\leadsto \left(\frac{1}{{2}^{\color{blue}{0.25}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      7. pow1/2 99.5%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\sqrt{\color{blue}{{2}^{0.5}}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      8. sqrt-pow1 99.5%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      9. metadata-eval 99.5%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{\color{blue}{0.25}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   6. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{0.25}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   7. Taylor expanded in th around inf 99.7%

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

      1. *-commutative 99.7%

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

   9. Simplified 99.7%

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

       1. expm1-log1p-u 99.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5} \cdot \cos th\right)\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       2. *-commutative 99.6%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos th \cdot \sqrt{0.5}}\right)\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       3. log1p-define 99.3%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \cos th \cdot \sqrt{0.5}\right)}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       4. expm1-define 99.3%

          \[\leadsto \color{blue}{\left(e^{\log \left(1 + \cos th \cdot \sqrt{0.5}\right)} - 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       5. add-exp-log 99.3%

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

       6. associate--l+ 99.3%

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

       7. +-commutative 99.3%

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

       8. *-commutative 99.3%

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

       9. fma-neg 99.2%

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{0.5}, \cos th, -1\right)} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       10. metadata-eval 99.2%

           \[\leadsto \left(\mathsf{fma}\left(\sqrt{0.5}, \cos th, \color{blue}{-1}\right) + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   11. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sqrt{0.5}, \cos th, -1\right) + 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   13. Applied egg-rr 60.2%

       \[\leadsto \color{blue}{\log \left(e^{\cos th}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   14. Step-by-step derivation

       1. rem-log-exp 60.2%

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

   15. Simplified 60.2%

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

   if 0.69999999999999996 < (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 92.6%

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

   6. Taylor expanded in a1 around 0 54.5%

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

      1. pow2 54.5%

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

      2. *-un-lft-identity 54.5%

         \[\leadsto \frac{a2 \cdot a2}{\color{blue}{1 \cdot \sqrt{2}}} \]

      3. times-frac 54.5%

         \[\leadsto \color{blue}{\frac{a2}{1} \cdot \frac{a2}{\sqrt{2}}} \]

   8. Applied egg-rr 54.5%

      \[\leadsto \color{blue}{\frac{a2}{1} \cdot \frac{a2}{\sqrt{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= (cos th) 0.7) (* (cos th) t_1) (* (sqrt 0.5) t_1))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = cos(th) * t_1;
	} else {
		tmp = sqrt(0.5) * t_1;
	}
	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) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (cos(th) <= 0.7d0) then
        tmp = cos(th) * t_1
    else
        tmp = sqrt(0.5d0) * t_1
    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 tmp;
	if (Math.cos(th) <= 0.7) {
		tmp = Math.cos(th) * t_1;
	} else {
		tmp = Math.sqrt(0.5) * t_1;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if math.cos(th) <= 0.7:
		tmp = math.cos(th) * t_1
	else:
		tmp = math.sqrt(0.5) * t_1
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (cos(th) <= 0.7)
		tmp = Float64(cos(th) * t_1);
	else
		tmp = Float64(sqrt(0.5) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (cos(th) <= 0.7)
		tmp = cos(th) * t_1;
	else
		tmp = sqrt(0.5) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[th], $MachinePrecision], 0.7], N[(N[Cos[th], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69999999999999996

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. add-sqr-sqrt 99.6%

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

      3. times-frac 99.5%

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

      4. pow1/2 99.5%

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

      5. sqrt-pow1 99.5%

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

      6. metadata-eval 99.5%

         \[\leadsto \left(\frac{1}{{2}^{\color{blue}{0.25}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      7. pow1/2 99.5%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\sqrt{\color{blue}{{2}^{0.5}}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      8. sqrt-pow1 99.5%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      9. metadata-eval 99.5%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{\color{blue}{0.25}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   6. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{0.25}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   7. Taylor expanded in th around inf 99.7%

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

      1. *-commutative 99.7%

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

   9. Simplified 99.7%

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

       1. expm1-log1p-u 99.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5} \cdot \cos th\right)\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       2. *-commutative 99.6%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos th \cdot \sqrt{0.5}}\right)\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       3. log1p-define 99.3%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \cos th \cdot \sqrt{0.5}\right)}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       4. expm1-define 99.3%

          \[\leadsto \color{blue}{\left(e^{\log \left(1 + \cos th \cdot \sqrt{0.5}\right)} - 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       5. add-exp-log 99.3%

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

       6. associate--l+ 99.3%

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

       7. +-commutative 99.3%

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

       8. *-commutative 99.3%

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

       9. fma-neg 99.2%

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{0.5}, \cos th, -1\right)} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       10. metadata-eval 99.2%

           \[\leadsto \left(\mathsf{fma}\left(\sqrt{0.5}, \cos th, \color{blue}{-1}\right) + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   11. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sqrt{0.5}, \cos th, -1\right) + 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   13. Applied egg-rr 60.2%

       \[\leadsto \color{blue}{\log \left(e^{\cos th}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   14. Step-by-step derivation

       1. rem-log-exp 60.2%

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

   15. Simplified 60.2%

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

   if 0.69999999999999996 < (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. *-un-lft-identity 99.6%

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

      2. add-sqr-sqrt 99.7%

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

      3. times-frac 99.2%

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

      4. pow1/2 99.2%

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

      5. sqrt-pow1 99.2%

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

      6. metadata-eval 99.2%

         \[\leadsto \left(\frac{1}{{2}^{\color{blue}{0.25}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      7. pow1/2 99.2%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\sqrt{\color{blue}{{2}^{0.5}}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      8. sqrt-pow1 99.2%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      9. metadata-eval 99.2%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{\color{blue}{0.25}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   6. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{0.25}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   7. Taylor expanded in th around 0 92.7%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot {2}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) -1e-309)
   (* (+ (* a1 a1) (* a2 a2)) -3.0)
   (* a2 (* a2 (pow 2.0 -0.5)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= -1e-309) {
		tmp = ((a1 * a1) + (a2 * a2)) * -3.0;
	} else {
		tmp = a2 * (a2 * pow(2.0, -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) :: tmp
    if (cos(th) <= (-1d-309)) then
        tmp = ((a1 * a1) + (a2 * a2)) * (-3.0d0)
    else
        tmp = a2 * (a2 * (2.0d0 ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= -1e-309) {
		tmp = ((a1 * a1) + (a2 * a2)) * -3.0;
	} else {
		tmp = a2 * (a2 * Math.pow(2.0, -0.5));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= -1e-309:
		tmp = ((a1 * a1) + (a2 * a2)) * -3.0
	else:
		tmp = a2 * (a2 * math.pow(2.0, -0.5))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= -1e-309)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * -3.0);
	else
		tmp = Float64(a2 * Float64(a2 * (2.0 ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= -1e-309)
		tmp = ((a1 * a1) + (a2 * a2)) * -3.0;
	else
		tmp = a2 * (a2 * (2.0 ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], -1e-309], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision], N[(a2 * N[(a2 * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -1.000000000000002e-309

   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. Step-by-step derivation

      1. *-un-lft-identity 99.4%

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

      2. add-sqr-sqrt 99.6%

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

      3. times-frac 99.6%

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

      4. pow1/2 99.6%

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

      5. sqrt-pow1 99.6%

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

      6. metadata-eval 99.6%

         \[\leadsto \left(\frac{1}{{2}^{\color{blue}{0.25}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      7. pow1/2 99.6%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\sqrt{\color{blue}{{2}^{0.5}}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      8. sqrt-pow1 99.6%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      9. metadata-eval 99.6%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{\color{blue}{0.25}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{0.25}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   7. Taylor expanded in th around inf 99.7%

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

      1. *-commutative 99.7%

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

   9. Simplified 99.7%

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

       1. expm1-log1p-u 99.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5} \cdot \cos th\right)\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       2. *-commutative 99.7%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos th \cdot \sqrt{0.5}}\right)\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       3. log1p-define 99.4%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \cos th \cdot \sqrt{0.5}\right)}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       4. expm1-define 99.4%

          \[\leadsto \color{blue}{\left(e^{\log \left(1 + \cos th \cdot \sqrt{0.5}\right)} - 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       5. add-exp-log 99.4%

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

       6. associate--l+ 99.1%

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

       7. +-commutative 99.1%

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

       8. *-commutative 99.1%

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

       9. fma-neg 99.1%

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{0.5}, \cos th, -1\right)} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       10. metadata-eval 99.1%

           \[\leadsto \left(\mathsf{fma}\left(\sqrt{0.5}, \cos th, \color{blue}{-1}\right) + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   11. Applied egg-rr 99.1%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sqrt{0.5}, \cos th, -1\right) + 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   13. Applied egg-rr 54.5%

       \[\leadsto \left(\color{blue}{\left(\cos th + \left(-4 - \cos th\right)\right)} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   14. Step-by-step derivation

       1. +-commutative 54.5%

          \[\leadsto \left(\color{blue}{\left(\left(-4 - \cos th\right) + \cos th\right)} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       2. associate-+l- 54.5%

          \[\leadsto \left(\color{blue}{\left(-4 - \left(\cos th - \cos th\right)\right)} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       3. +-inverses 54.5%

          \[\leadsto \left(\left(-4 - \color{blue}{0}\right) + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       4. metadata-eval 54.5%

          \[\leadsto \left(\color{blue}{-4} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   15. Simplified 54.5%

       \[\leadsto \left(\color{blue}{-4} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if -1.000000000000002e-309 < (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 88.1%

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

   6. Taylor expanded in a1 around 0 51.4%

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

      1. pow2 51.4%

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

      2. div-inv 51.4%

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

      3. associate-*l* 51.4%

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

      4. pow1/2 51.4%

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

      5. pow-flip 51.4%

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

      6. metadata-eval 51.4%

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

   8. Applied egg-rr 51.4%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot {2}^{-0.5}\right)\\ \end{array} \]
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 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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. *-un-lft-identity 99.6%

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

   2. add-sqr-sqrt 99.6%

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

   3. times-frac 99.3%

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

   4. pow1/2 99.3%

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

   5. sqrt-pow1 99.3%

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

   6. metadata-eval 99.3%

      \[\leadsto \left(\frac{1}{{2}^{\color{blue}{0.25}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   7. pow1/2 99.3%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\sqrt{\color{blue}{{2}^{0.5}}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   8. sqrt-pow1 99.3%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   9. metadata-eval 99.3%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{\color{blue}{0.25}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

6. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{0.25}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

7. Taylor expanded in th around inf 99.7%

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

   1. *-commutative 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 6: 52.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) -1e-309)
   (* (+ (* a1 a1) (* a2 a2)) -3.0)
   (* a2 (/ a2 (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= -1e-309) {
		tmp = ((a1 * a1) + (a2 * a2)) * -3.0;
	} 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 (cos(th) <= (-1d-309)) then
        tmp = ((a1 * a1) + (a2 * a2)) * (-3.0d0)
    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 (Math.cos(th) <= -1e-309) {
		tmp = ((a1 * a1) + (a2 * a2)) * -3.0;
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= -1e-309:
		tmp = ((a1 * a1) + (a2 * a2)) * -3.0
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= -1e-309)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * -3.0);
	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 (cos(th) <= -1e-309)
		tmp = ((a1 * a1) + (a2 * a2)) * -3.0;
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], -1e-309], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -1.000000000000002e-309

   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. Step-by-step derivation

      1. *-un-lft-identity 99.4%

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

      2. add-sqr-sqrt 99.6%

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

      3. times-frac 99.6%

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

      4. pow1/2 99.6%

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

      5. sqrt-pow1 99.6%

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

      6. metadata-eval 99.6%

         \[\leadsto \left(\frac{1}{{2}^{\color{blue}{0.25}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      7. pow1/2 99.6%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\sqrt{\color{blue}{{2}^{0.5}}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      8. sqrt-pow1 99.6%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      9. metadata-eval 99.6%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{\color{blue}{0.25}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{0.25}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   7. Taylor expanded in th around inf 99.7%

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

      1. *-commutative 99.7%

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

   9. Simplified 99.7%

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

       1. expm1-log1p-u 99.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5} \cdot \cos th\right)\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       2. *-commutative 99.7%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos th \cdot \sqrt{0.5}}\right)\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       3. log1p-define 99.4%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \cos th \cdot \sqrt{0.5}\right)}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       4. expm1-define 99.4%

          \[\leadsto \color{blue}{\left(e^{\log \left(1 + \cos th \cdot \sqrt{0.5}\right)} - 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       5. add-exp-log 99.4%

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

       6. associate--l+ 99.1%

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

       7. +-commutative 99.1%

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

       8. *-commutative 99.1%

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

       9. fma-neg 99.1%

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{0.5}, \cos th, -1\right)} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       10. metadata-eval 99.1%

           \[\leadsto \left(\mathsf{fma}\left(\sqrt{0.5}, \cos th, \color{blue}{-1}\right) + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   11. Applied egg-rr 99.1%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sqrt{0.5}, \cos th, -1\right) + 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   13. Applied egg-rr 54.5%

       \[\leadsto \left(\color{blue}{\left(\cos th + \left(-4 - \cos th\right)\right)} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   14. Step-by-step derivation

       1. +-commutative 54.5%

          \[\leadsto \left(\color{blue}{\left(\left(-4 - \cos th\right) + \cos th\right)} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       2. associate-+l- 54.5%

          \[\leadsto \left(\color{blue}{\left(-4 - \left(\cos th - \cos th\right)\right)} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       3. +-inverses 54.5%

          \[\leadsto \left(\left(-4 - \color{blue}{0}\right) + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       4. metadata-eval 54.5%

          \[\leadsto \left(\color{blue}{-4} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   15. Simplified 54.5%

       \[\leadsto \left(\color{blue}{-4} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if -1.000000000000002e-309 < (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 88.1%

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

   6. Taylor expanded in a1 around 0 51.4%

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

      1. pow2 51.4%

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

      2. *-un-lft-identity 51.4%

         \[\leadsto \frac{a2 \cdot a2}{\color{blue}{1 \cdot \sqrt{2}}} \]

      3. times-frac 51.4%

         \[\leadsto \color{blue}{\frac{a2}{1} \cdot \frac{a2}{\sqrt{2}}} \]

   8. Applied egg-rr 51.4%

      \[\leadsto \color{blue}{\frac{a2}{1} \cdot \frac{a2}{\sqrt{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq -1 \cdot 10^{-309}:\\ \;\;\;\;t\_1 \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= (cos th) -1e-309) (* t_1 -3.0) t_1)))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (cos(th) <= -1e-309) {
		tmp = t_1 * -3.0;
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (cos(th) <= (-1d-309)) then
        tmp = t_1 * (-3.0d0)
    else
        tmp = t_1
    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 tmp;
	if (Math.cos(th) <= -1e-309) {
		tmp = t_1 * -3.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if math.cos(th) <= -1e-309:
		tmp = t_1 * -3.0
	else:
		tmp = t_1
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (cos(th) <= -1e-309)
		tmp = Float64(t_1 * -3.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (cos(th) <= -1e-309)
		tmp = t_1 * -3.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[th], $MachinePrecision], -1e-309], N[(t$95$1 * -3.0), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -1.000000000000002e-309

   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. Step-by-step derivation

      1. *-un-lft-identity 99.4%

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

      2. add-sqr-sqrt 99.6%

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

      3. times-frac 99.6%

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

      4. pow1/2 99.6%

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

      5. sqrt-pow1 99.6%

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

      6. metadata-eval 99.6%

         \[\leadsto \left(\frac{1}{{2}^{\color{blue}{0.25}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      7. pow1/2 99.6%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\sqrt{\color{blue}{{2}^{0.5}}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      8. sqrt-pow1 99.6%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      9. metadata-eval 99.6%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{\color{blue}{0.25}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{0.25}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   7. Taylor expanded in th around inf 99.7%

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

      1. *-commutative 99.7%

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

   9. Simplified 99.7%

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

       1. expm1-log1p-u 99.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5} \cdot \cos th\right)\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       2. *-commutative 99.7%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos th \cdot \sqrt{0.5}}\right)\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       3. log1p-define 99.4%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \cos th \cdot \sqrt{0.5}\right)}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       4. expm1-define 99.4%

          \[\leadsto \color{blue}{\left(e^{\log \left(1 + \cos th \cdot \sqrt{0.5}\right)} - 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       5. add-exp-log 99.4%

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

       6. associate--l+ 99.1%

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

       7. +-commutative 99.1%

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

       8. *-commutative 99.1%

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

       9. fma-neg 99.1%

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{0.5}, \cos th, -1\right)} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       10. metadata-eval 99.1%

           \[\leadsto \left(\mathsf{fma}\left(\sqrt{0.5}, \cos th, \color{blue}{-1}\right) + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   11. Applied egg-rr 99.1%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sqrt{0.5}, \cos th, -1\right) + 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   13. Applied egg-rr 54.5%

       \[\leadsto \left(\color{blue}{\left(\cos th + \left(-4 - \cos th\right)\right)} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   14. Step-by-step derivation

       1. +-commutative 54.5%

          \[\leadsto \left(\color{blue}{\left(\left(-4 - \cos th\right) + \cos th\right)} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       2. associate-+l- 54.5%

          \[\leadsto \left(\color{blue}{\left(-4 - \left(\cos th - \cos th\right)\right)} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       3. +-inverses 54.5%

          \[\leadsto \left(\left(-4 - \color{blue}{0}\right) + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       4. metadata-eval 54.5%

          \[\leadsto \left(\color{blue}{-4} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   15. Simplified 54.5%

       \[\leadsto \left(\color{blue}{-4} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if -1.000000000000002e-309 < (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. *-un-lft-identity 99.6%

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

      2. add-sqr-sqrt 99.6%

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

      3. times-frac 99.2%

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

      4. pow1/2 99.2%

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

      5. sqrt-pow1 99.2%

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

      6. metadata-eval 99.2%

         \[\leadsto \left(\frac{1}{{2}^{\color{blue}{0.25}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      7. pow1/2 99.2%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\sqrt{\color{blue}{{2}^{0.5}}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      8. sqrt-pow1 99.2%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      9. metadata-eval 99.2%

         \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{\color{blue}{0.25}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   6. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{0.25}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   7. Taylor expanded in th around inf 99.7%

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

      1. *-commutative 99.7%

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

   9. Simplified 99.7%

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

       1. expm1-log1p-u 99.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5} \cdot \cos th\right)\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       2. *-commutative 99.3%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos th \cdot \sqrt{0.5}}\right)\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       3. log1p-define 99.5%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \cos th \cdot \sqrt{0.5}\right)}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       4. expm1-define 99.5%

          \[\leadsto \color{blue}{\left(e^{\log \left(1 + \cos th \cdot \sqrt{0.5}\right)} - 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       5. add-exp-log 99.5%

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

       6. associate--l+ 99.7%

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

       7. +-commutative 99.7%

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

       8. *-commutative 99.7%

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

       9. fma-neg 99.6%

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{0.5}, \cos th, -1\right)} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

       10. metadata-eval 99.6%

           \[\leadsto \left(\mathsf{fma}\left(\sqrt{0.5}, \cos th, \color{blue}{-1}\right) + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   11. Applied egg-rr 99.6%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sqrt{0.5}, \cos th, -1\right) + 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   13. Applied egg-rr 62.9%

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

       1. *-inverses 62.9%

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

   15. Simplified 62.9%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot a1 + a2 \cdot a2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 29.9% accurate, 41.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 6 \cdot 10^{+31}:\\ \;\;\;\;a1 \cdot a1\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 6e+31) (* a1 a1) (* a1 (* a1 -4.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 6e+31) {
		tmp = a1 * a1;
	} else {
		tmp = a1 * (a1 * -4.0);
	}
	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 (th <= 6d+31) then
        tmp = a1 * a1
    else
        tmp = a1 * (a1 * (-4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 6e+31) {
		tmp = a1 * a1;
	} else {
		tmp = a1 * (a1 * -4.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 6e+31:
		tmp = a1 * a1
	else:
		tmp = a1 * (a1 * -4.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 6e+31)
		tmp = Float64(a1 * a1);
	else
		tmp = Float64(a1 * Float64(a1 * -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 6e+31)
		tmp = a1 * a1;
	else
		tmp = a1 * (a1 * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 6e+31], N[(a1 * a1), $MachinePrecision], N[(a1 * N[(a1 * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 5.99999999999999978e31

   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 81.0%

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

   6. Taylor expanded in a1 around inf 49.1%

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

      1. pow2 49.1%

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

      2. div-inv 49.1%

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

      3. associate-*l* 49.1%

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

      4. pow1/2 49.1%

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

      5. pow-flip 49.1%

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

      6. metadata-eval 49.1%

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

   8. Applied egg-rr 49.1%

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

   10. Applied egg-rr 26.0%

       \[\leadsto a1 \cdot \color{blue}{\left|a1\right|} \]
   11. Step-by-step derivation

       1. unpow1 26.0%

          \[\leadsto a1 \cdot \left|\color{blue}{{a1}^{1}}\right| \]

       2. sqr-pow 19.7%

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

       3. fabs-sqr 19.7%

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

       4. sqr-pow 38.6%

          \[\leadsto a1 \cdot \color{blue}{{a1}^{1}} \]

       5. unpow1 38.6%

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

   12. Simplified 38.6%

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

   if 5.99999999999999978e31 < th

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in th around 0 32.8%

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

   6. Taylor expanded in a1 around inf 14.6%

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

      1. pow2 14.6%

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

      2. *-un-lft-identity 14.6%

         \[\leadsto \frac{a1 \cdot a1}{\color{blue}{1 \cdot \sqrt{2}}} \]

      3. times-frac 14.6%

         \[\leadsto \color{blue}{\frac{a1}{1} \cdot \frac{a1}{\sqrt{2}}} \]

   8. Applied egg-rr 14.6%

      \[\leadsto \color{blue}{\frac{a1}{1} \cdot \frac{a1}{\sqrt{2}}} \]

   10. Applied egg-rr 21.7%

       \[\leadsto \frac{a1}{1} \cdot \color{blue}{\left(-4 \cdot a1\right)} \]
   11. Step-by-step derivation

       1. *-commutative 21.7%

          \[\leadsto \frac{a1}{1} \cdot \color{blue}{\left(a1 \cdot -4\right)} \]

   12. Simplified 21.7%

       \[\leadsto \frac{a1}{1} \cdot \color{blue}{\left(a1 \cdot -4\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 6 \cdot 10^{+31}:\\ \;\;\;\;a1 \cdot a1\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.0% accurate, 46.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 6 \cdot 10^{+31}:\\ \;\;\;\;a1 \cdot a1\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(-a1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 6e+31) (* a1 a1) (* a1 (- a1))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 6e+31) {
		tmp = a1 * a1;
	} else {
		tmp = a1 * -a1;
	}
	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 <= 6d+31) then
        tmp = a1 * a1
    else
        tmp = a1 * -a1
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 6e+31) {
		tmp = a1 * a1;
	} else {
		tmp = a1 * -a1;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 6e+31:
		tmp = a1 * a1
	else:
		tmp = a1 * -a1
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 6e+31)
		tmp = Float64(a1 * a1);
	else
		tmp = Float64(a1 * Float64(-a1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 6e+31)
		tmp = a1 * a1;
	else
		tmp = a1 * -a1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 6e+31], N[(a1 * a1), $MachinePrecision], N[(a1 * (-a1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 5.99999999999999978e31

   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 81.0%

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

   6. Taylor expanded in a1 around inf 49.1%

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

      1. pow2 49.1%

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

      2. div-inv 49.1%

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

      3. associate-*l* 49.1%

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

      4. pow1/2 49.1%

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

      5. pow-flip 49.1%

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

      6. metadata-eval 49.1%

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

   8. Applied egg-rr 49.1%

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

   10. Applied egg-rr 26.0%

       \[\leadsto a1 \cdot \color{blue}{\left|a1\right|} \]
   11. Step-by-step derivation

       1. unpow1 26.0%

          \[\leadsto a1 \cdot \left|\color{blue}{{a1}^{1}}\right| \]

       2. sqr-pow 19.7%

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

       3. fabs-sqr 19.7%

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

       4. sqr-pow 38.6%

          \[\leadsto a1 \cdot \color{blue}{{a1}^{1}} \]

       5. unpow1 38.6%

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

   12. Simplified 38.6%

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

   if 5.99999999999999978e31 < th

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in th around 0 32.8%

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

   6. Taylor expanded in a1 around inf 14.6%

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

      1. pow2 14.6%

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

      2. div-inv 14.6%

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

      3. associate-*l* 14.6%

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

      4. pow1/2 14.6%

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

      5. pow-flip 14.6%

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

      6. metadata-eval 14.6%

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

   8. Applied egg-rr 14.6%

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

   10. Applied egg-rr 22.1%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 6 \cdot 10^{+31}:\\ \;\;\;\;a1 \cdot a1\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(-a1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.1% accurate, 59.3× speedup

Math

\[\begin{array}{l} \\ a1 \cdot a1 + a2 \cdot a2 \end{array} \]

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 * 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 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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. *-un-lft-identity 99.6%

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

   2. add-sqr-sqrt 99.6%

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

   3. times-frac 99.3%

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

   4. pow1/2 99.3%

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

   5. sqrt-pow1 99.3%

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

   6. metadata-eval 99.3%

      \[\leadsto \left(\frac{1}{{2}^{\color{blue}{0.25}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   7. pow1/2 99.3%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\sqrt{\color{blue}{{2}^{0.5}}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   8. sqrt-pow1 99.3%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   9. metadata-eval 99.3%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{\color{blue}{0.25}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

6. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{0.25}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

7. Taylor expanded in th around inf 99.7%

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

   1. *-commutative 99.7%

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

9. Simplified 99.7%

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

    1. expm1-log1p-u 99.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5} \cdot \cos th\right)\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

    2. *-commutative 99.4%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos th \cdot \sqrt{0.5}}\right)\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

    3. log1p-define 99.5%

       \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \cos th \cdot \sqrt{0.5}\right)}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

    4. expm1-define 99.5%

       \[\leadsto \color{blue}{\left(e^{\log \left(1 + \cos th \cdot \sqrt{0.5}\right)} - 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

    5. add-exp-log 99.5%

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

    6. associate--l+ 99.5%

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

    7. +-commutative 99.5%

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

    8. *-commutative 99.5%

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

    9. fma-neg 99.5%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{0.5}, \cos th, -1\right)} + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

    10. metadata-eval 99.5%

        \[\leadsto \left(\mathsf{fma}\left(\sqrt{0.5}, \cos th, \color{blue}{-1}\right) + 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

11. Applied egg-rr 99.5%

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sqrt{0.5}, \cos th, -1\right) + 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

13. Applied egg-rr 49.6%

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

    1. *-inverses 49.6%

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

15. Simplified 49.6%

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

16. Final simplification 49.6%

    \[\leadsto a1 \cdot a1 + a2 \cdot a2 \]
17. Add Preprocessing

Alternative 11: 21.8% accurate, 83.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

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 69.1%

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

6. Taylor expanded in a1 around inf 40.7%

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

   1. pow2 40.7%

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

   2. *-un-lft-identity 40.7%

      \[\leadsto \frac{a1 \cdot a1}{\color{blue}{1 \cdot \sqrt{2}}} \]

   3. times-frac 40.6%

      \[\leadsto \color{blue}{\frac{a1}{1} \cdot \frac{a1}{\sqrt{2}}} \]

8. Applied egg-rr 40.6%

   \[\leadsto \color{blue}{\frac{a1}{1} \cdot \frac{a1}{\sqrt{2}}} \]

10. Applied egg-rr 23.0%

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

11. Final simplification 23.0%

    \[\leadsto a1 \cdot \left(a1 \cdot a1\right) \]
12. Add Preprocessing

Alternative 12: 29.9% accurate, 138.3× speedup

Math

\[\begin{array}{l} \\ a1 \cdot a1 \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

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 69.1%

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

6. Taylor expanded in a1 around inf 40.7%

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

   1. pow2 40.7%

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

   2. div-inv 40.6%

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

   3. associate-*l* 40.6%

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

   4. pow1/2 40.6%

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

   5. pow-flip 40.6%

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

   6. metadata-eval 40.6%

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

8. Applied egg-rr 40.6%

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

10. Applied egg-rr 23.2%

    \[\leadsto a1 \cdot \color{blue}{\left|a1\right|} \]
11. Step-by-step derivation

    1. unpow1 23.2%

       \[\leadsto a1 \cdot \left|\color{blue}{{a1}^{1}}\right| \]

    2. sqr-pow 16.7%

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

    3. fabs-sqr 16.7%

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

    4. sqr-pow 32.7%

       \[\leadsto a1 \cdot \color{blue}{{a1}^{1}} \]

    5. unpow1 32.7%

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

12. Simplified 32.7%

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

13. Final simplification 32.7%

    \[\leadsto a1 \cdot a1 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(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))))