Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 12.3s

Alternatives: 20

Speedup: 2.0×

• 43.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return (pow(2.0, -0.5) * cos(th)) * ((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 = ((2.0d0 ** (-0.5d0)) * cos(th)) * ((a1 * a1) + (a2 * a2))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Power[2.0, -0.5], $MachinePrecision] * N[Cos[th], $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. 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%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq -0.4:\\ \;\;\;\;-0.5 \cdot t_1\\ \mathbf{elif}\;\cos th \leq 0.97:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(\cos th \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= (cos th) -0.4)
     (* -0.5 t_1)
     (if (<= (cos th) 0.97)
       (* a2 (* a2 (* (cos th) (sqrt 0.5))))
       (* t_1 (sqrt 0.5))))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (cos(th) <= -0.4) {
		tmp = -0.5 * t_1;
	} else if (cos(th) <= 0.97) {
		tmp = a2 * (a2 * (cos(th) * sqrt(0.5)));
	} else {
		tmp = t_1 * sqrt(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) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (cos(th) <= (-0.4d0)) then
        tmp = (-0.5d0) * t_1
    else if (cos(th) <= 0.97d0) then
        tmp = a2 * (a2 * (cos(th) * sqrt(0.5d0)))
    else
        tmp = t_1 * sqrt(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 tmp;
	if (Math.cos(th) <= -0.4) {
		tmp = -0.5 * t_1;
	} else if (Math.cos(th) <= 0.97) {
		tmp = a2 * (a2 * (Math.cos(th) * Math.sqrt(0.5)));
	} else {
		tmp = t_1 * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if math.cos(th) <= -0.4:
		tmp = -0.5 * t_1
	elif math.cos(th) <= 0.97:
		tmp = a2 * (a2 * (math.cos(th) * math.sqrt(0.5)))
	else:
		tmp = t_1 * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (cos(th) <= -0.4)
		tmp = Float64(-0.5 * t_1);
	elseif (cos(th) <= 0.97)
		tmp = Float64(a2 * Float64(a2 * Float64(cos(th) * sqrt(0.5))));
	else
		tmp = Float64(t_1 * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (cos(th) <= -0.4)
		tmp = -0.5 * t_1;
	elseif (cos(th) <= 0.97)
		tmp = a2 * (a2 * (cos(th) * sqrt(0.5)));
	else
		tmp = t_1 * sqrt(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]}, If[LessEqual[N[Cos[th], $MachinePrecision], -0.4], N[(-0.5 * t$95$1), $MachinePrecision], If[LessEqual[N[Cos[th], $MachinePrecision], 0.97], N[(a2 * N[(a2 * N[(N[Cos[th], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 th) < -0.40000000000000002

   1. Initial program 99.9%

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

      1. distribute-lft-out 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in th around 0 11.5%

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

   6. Applied egg-rr 75.7%

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

   if -0.40000000000000002 < (cos.f64 th) < 0.96999999999999997

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 99.4%

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

      4. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in a1 around 0 54.3%

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

      1. unpow2 54.3%

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

      2. associate-*l* 54.3%

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

      3. associate-*r/ 54.3%

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

      4. associate-/l* 54.3%

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

   6. Simplified 54.3%

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

      1. *-un-lft-identity 54.3%

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

      2. div-inv 54.2%

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

      3. times-frac 54.2%

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

      4. pow1/2 54.2%

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

      5. pow-flip 54.3%

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

      6. metadata-eval 54.3%

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

      7. add-sqr-sqrt 54.0%

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

      8. sqrt-unprod 54.3%

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

      9. pow-prod-up 54.3%

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

      10. metadata-eval 54.3%

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

      11. metadata-eval 54.3%

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

   8. Applied egg-rr 54.3%

      \[\leadsto a2 \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{a2}{\frac{1}{\cos th}}\right)} \]
   9. Step-by-step derivation

      1. associate-/r/ 54.3%

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

      2. /-rgt-identity 54.3%

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

      3. *-commutative 54.3%

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

      4. associate-*l* 54.3%

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

   10. Simplified 54.3%

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

   if 0.96999999999999997 < (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. 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 99.7%

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

      5. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in th around 0 94.2%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -0.4:\\ \;\;\;\;-0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{elif}\;\cos th \leq 0.97:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(\cos th \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 3: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq -0.4:\\ \;\;\;\;-0.5 \cdot t_1\\ \mathbf{elif}\;\cos th \leq 0.97:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= (cos th) -0.4)
     (* -0.5 t_1)
     (if (<= (cos th) 0.97)
       (* (cos th) (* a2 (* a2 (sqrt 0.5))))
       (* t_1 (sqrt 0.5))))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (cos(th) <= -0.4) {
		tmp = -0.5 * t_1;
	} else if (cos(th) <= 0.97) {
		tmp = cos(th) * (a2 * (a2 * sqrt(0.5)));
	} else {
		tmp = t_1 * sqrt(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) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (cos(th) <= (-0.4d0)) then
        tmp = (-0.5d0) * t_1
    else if (cos(th) <= 0.97d0) then
        tmp = cos(th) * (a2 * (a2 * sqrt(0.5d0)))
    else
        tmp = t_1 * sqrt(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 tmp;
	if (Math.cos(th) <= -0.4) {
		tmp = -0.5 * t_1;
	} else if (Math.cos(th) <= 0.97) {
		tmp = Math.cos(th) * (a2 * (a2 * Math.sqrt(0.5)));
	} else {
		tmp = t_1 * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if math.cos(th) <= -0.4:
		tmp = -0.5 * t_1
	elif math.cos(th) <= 0.97:
		tmp = math.cos(th) * (a2 * (a2 * math.sqrt(0.5)))
	else:
		tmp = t_1 * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (cos(th) <= -0.4)
		tmp = Float64(-0.5 * t_1);
	elseif (cos(th) <= 0.97)
		tmp = Float64(cos(th) * Float64(a2 * Float64(a2 * sqrt(0.5))));
	else
		tmp = Float64(t_1 * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (cos(th) <= -0.4)
		tmp = -0.5 * t_1;
	elseif (cos(th) <= 0.97)
		tmp = cos(th) * (a2 * (a2 * sqrt(0.5)));
	else
		tmp = t_1 * sqrt(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]}, If[LessEqual[N[Cos[th], $MachinePrecision], -0.4], N[(-0.5 * t$95$1), $MachinePrecision], If[LessEqual[N[Cos[th], $MachinePrecision], 0.97], N[(N[Cos[th], $MachinePrecision] * N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 th) < -0.40000000000000002

   1. Initial program 99.9%

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

      1. distribute-lft-out 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in th around 0 11.5%

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

   6. Applied egg-rr 75.7%

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

   if -0.40000000000000002 < (cos.f64 th) < 0.96999999999999997

   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. Step-by-step derivation

      1. clear-num 99.4%

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

      2. associate-/r/ 99.4%

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

      3. pow1/2 99.4%

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

      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%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in a1 around 0 54.4%

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

      1. unpow2 54.4%

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

      2. associate-*r* 54.4%

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

      3. *-commutative 54.4%

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

      4. *-commutative 54.4%

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

      5. associate-*l* 54.4%

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

   8. Simplified 54.4%

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

   if 0.96999999999999997 < (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. 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 99.7%

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

      5. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in th around 0 94.2%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -0.4:\\ \;\;\;\;-0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{elif}\;\cos th \leq 0.97:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 4: 79.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq -0.02:\\ \;\;\;\;-0.5 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= (cos th) -0.02) (* -0.5 t_1) (* t_1 (sqrt 0.5)))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (cos(th) <= -0.02) {
		tmp = -0.5 * t_1;
	} else {
		tmp = t_1 * sqrt(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) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (cos(th) <= (-0.02d0)) then
        tmp = (-0.5d0) * t_1
    else
        tmp = t_1 * sqrt(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 tmp;
	if (Math.cos(th) <= -0.02) {
		tmp = -0.5 * t_1;
	} else {
		tmp = t_1 * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if math.cos(th) <= -0.02:
		tmp = -0.5 * t_1
	else:
		tmp = t_1 * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (cos(th) <= -0.02)
		tmp = Float64(-0.5 * t_1);
	else
		tmp = Float64(t_1 * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (cos(th) <= -0.02)
		tmp = -0.5 * t_1;
	else
		tmp = t_1 * sqrt(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]}, If[LessEqual[N[Cos[th], $MachinePrecision], -0.02], N[(-0.5 * t$95$1), $MachinePrecision], N[(t$95$1 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -0.0200000000000000004

   1. Initial program 99.8%

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

      1. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in th around 0 8.4%

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

   6. Applied egg-rr 70.0%

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

   if -0.0200000000000000004 < (cos.f64 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. Step-by-step derivation

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

      3. pow1/2 99.5%

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

      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%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in th around 0 81.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -0.02:\\ \;\;\;\;-0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 5: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 6: 68.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a1 \leq -7.8 \cdot 10^{-59}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a1 -7.8e-59)
   (* (cos th) (/ a1 (/ (sqrt 2.0) a1)))
   (* (cos th) (* a2 (* a2 (sqrt 0.5))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= -7.8e-59) {
		tmp = cos(th) * (a1 / (sqrt(2.0) / a1));
	} else {
		tmp = cos(th) * (a2 * (a2 * sqrt(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 (a1 <= (-7.8d-59)) then
        tmp = cos(th) * (a1 / (sqrt(2.0d0) / a1))
    else
        tmp = cos(th) * (a2 * (a2 * sqrt(0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= -7.8e-59) {
		tmp = Math.cos(th) * (a1 / (Math.sqrt(2.0) / a1));
	} else {
		tmp = Math.cos(th) * (a2 * (a2 * Math.sqrt(0.5)));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a1 <= -7.8e-59:
		tmp = math.cos(th) * (a1 / (math.sqrt(2.0) / a1))
	else:
		tmp = math.cos(th) * (a2 * (a2 * math.sqrt(0.5)))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a1 <= -7.8e-59)
		tmp = Float64(cos(th) * Float64(a1 / Float64(sqrt(2.0) / a1)));
	else
		tmp = Float64(cos(th) * Float64(a2 * Float64(a2 * sqrt(0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a1 <= -7.8e-59)
		tmp = cos(th) * (a1 / (sqrt(2.0) / a1));
	else
		tmp = cos(th) * (a2 * (a2 * sqrt(0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a1, -7.8e-59], N[(N[Cos[th], $MachinePrecision] * N[(a1 / N[(N[Sqrt[2.0], $MachinePrecision] / a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[th], $MachinePrecision] * N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a1 < -7.80000000000000038e-59

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around inf 80.9%

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

      1. unpow2 80.9%

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

      2. associate-/l* 81.0%

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

   6. Simplified 81.0%

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

   if -7.80000000000000038e-59 < a1

   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. Step-by-step derivation

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

      3. pow1/2 99.5%

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

      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%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in a1 around 0 60.4%

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

      1. unpow2 60.4%

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

      2. associate-*r* 60.4%

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

      3. *-commutative 60.4%

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

      4. *-commutative 60.4%

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

      5. associate-*l* 60.4%

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

   8. Simplified 60.4%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \leq -7.8 \cdot 10^{-59}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\right)\\ \end{array} \]

Alternative 7: 68.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a1 \leq -8.2 \cdot 10^{-59}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a1 -8.2e-59)
   (* (cos th) (/ a1 (/ (sqrt 2.0) a1)))
   (* (cos th) (/ a2 (/ (sqrt 2.0) a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= -8.2e-59) {
		tmp = cos(th) * (a1 / (sqrt(2.0) / a1));
	} else {
		tmp = cos(th) * (a2 / (sqrt(2.0) / 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 (a1 <= (-8.2d-59)) then
        tmp = cos(th) * (a1 / (sqrt(2.0d0) / a1))
    else
        tmp = cos(th) * (a2 / (sqrt(2.0d0) / a2))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= -8.2e-59) {
		tmp = Math.cos(th) * (a1 / (Math.sqrt(2.0) / a1));
	} else {
		tmp = Math.cos(th) * (a2 / (Math.sqrt(2.0) / a2));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a1 <= -8.2e-59:
		tmp = math.cos(th) * (a1 / (math.sqrt(2.0) / a1))
	else:
		tmp = math.cos(th) * (a2 / (math.sqrt(2.0) / a2))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a1 <= -8.2e-59)
		tmp = Float64(cos(th) * Float64(a1 / Float64(sqrt(2.0) / a1)));
	else
		tmp = Float64(cos(th) * Float64(a2 / Float64(sqrt(2.0) / a2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a1 <= -8.2e-59)
		tmp = cos(th) * (a1 / (sqrt(2.0) / a1));
	else
		tmp = cos(th) * (a2 / (sqrt(2.0) / a2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a1, -8.2e-59], N[(N[Cos[th], $MachinePrecision] * N[(a1 / N[(N[Sqrt[2.0], $MachinePrecision] / a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[th], $MachinePrecision] * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a1 < -8.1999999999999991e-59

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around inf 80.9%

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

      1. unpow2 80.9%

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

      2. associate-/l* 81.0%

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

   6. Simplified 81.0%

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

   if -8.1999999999999991e-59 < a1

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 60.4%

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

      1. unpow2 60.4%

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

      2. associate-/l* 60.4%

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

   6. Simplified 60.4%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \leq -8.2 \cdot 10^{-59}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 8: 68.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a1 \leq -1.1 \cdot 10^{-59}:\\ \;\;\;\;\frac{a1 \cdot a1}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a1 -1.1e-59)
   (/ (* a1 a1) (/ (sqrt 2.0) (cos th)))
   (* (cos th) (/ a2 (/ (sqrt 2.0) a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= -1.1e-59) {
		tmp = (a1 * a1) / (sqrt(2.0) / cos(th));
	} else {
		tmp = cos(th) * (a2 / (sqrt(2.0) / 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 (a1 <= (-1.1d-59)) then
        tmp = (a1 * a1) / (sqrt(2.0d0) / cos(th))
    else
        tmp = cos(th) * (a2 / (sqrt(2.0d0) / a2))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= -1.1e-59) {
		tmp = (a1 * a1) / (Math.sqrt(2.0) / Math.cos(th));
	} else {
		tmp = Math.cos(th) * (a2 / (Math.sqrt(2.0) / a2));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a1 <= -1.1e-59:
		tmp = (a1 * a1) / (math.sqrt(2.0) / math.cos(th))
	else:
		tmp = math.cos(th) * (a2 / (math.sqrt(2.0) / a2))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a1 <= -1.1e-59)
		tmp = Float64(Float64(a1 * a1) / Float64(sqrt(2.0) / cos(th)));
	else
		tmp = Float64(cos(th) * Float64(a2 / Float64(sqrt(2.0) / a2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a1 <= -1.1e-59)
		tmp = (a1 * a1) / (sqrt(2.0) / cos(th));
	else
		tmp = cos(th) * (a2 / (sqrt(2.0) / a2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a1, -1.1e-59], N[(N[(a1 * a1), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[th], $MachinePrecision] * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a1 < -1.0999999999999999e-59

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around inf 80.9%

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

      1. unpow2 80.9%

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

      2. associate-/l* 80.9%

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

   6. Simplified 80.9%

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

   if -1.0999999999999999e-59 < a1

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 60.4%

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

      1. unpow2 60.4%

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

      2. associate-/l* 60.4%

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

   6. Simplified 60.4%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \leq -1.1 \cdot 10^{-59}:\\ \;\;\;\;\frac{a1 \cdot a1}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 9: 51.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 1.45:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;a2 \leq 1.05 \cdot 10^{+150}:\\ \;\;\;\;a2 \cdot \sqrt{\frac{a2 \cdot a2}{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 1.45)
   (* (* a1 a1) (sqrt 0.5))
   (if (<= a2 1.05e+150)
     (* a2 (sqrt (/ (* a2 a2) 2.0)))
     (* a2 (* (cos th) a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 1.45) {
		tmp = (a1 * a1) * sqrt(0.5);
	} else if (a2 <= 1.05e+150) {
		tmp = a2 * sqrt(((a2 * a2) / 2.0));
	} else {
		tmp = a2 * (cos(th) * 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 (a2 <= 1.45d0) then
        tmp = (a1 * a1) * sqrt(0.5d0)
    else if (a2 <= 1.05d+150) then
        tmp = a2 * sqrt(((a2 * a2) / 2.0d0))
    else
        tmp = a2 * (cos(th) * a2)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 1.45) {
		tmp = (a1 * a1) * Math.sqrt(0.5);
	} else if (a2 <= 1.05e+150) {
		tmp = a2 * Math.sqrt(((a2 * a2) / 2.0));
	} else {
		tmp = a2 * (Math.cos(th) * a2);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 1.45:
		tmp = (a1 * a1) * math.sqrt(0.5)
	elif a2 <= 1.05e+150:
		tmp = a2 * math.sqrt(((a2 * a2) / 2.0))
	else:
		tmp = a2 * (math.cos(th) * a2)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 1.45)
		tmp = Float64(Float64(a1 * a1) * sqrt(0.5));
	elseif (a2 <= 1.05e+150)
		tmp = Float64(a2 * sqrt(Float64(Float64(a2 * a2) / 2.0)));
	else
		tmp = Float64(a2 * Float64(cos(th) * a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 1.45)
		tmp = (a1 * a1) * sqrt(0.5);
	elseif (a2 <= 1.05e+150)
		tmp = a2 * sqrt(((a2 * a2) / 2.0));
	else
		tmp = a2 * (cos(th) * a2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 1.45], N[(N[(a1 * a1), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[a2, 1.05e+150], N[(a2 * N[Sqrt[N[(N[(a2 * a2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(a2 * N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a2 < 1.44999999999999996

   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. Step-by-step derivation

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

      3. pow1/2 99.5%

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

      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%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in th around 0 65.5%

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

   7. Taylor expanded in a1 around inf 46.3%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot {a1}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 46.3%

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

   9. Simplified 46.3%

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

   if 1.44999999999999996 < a2 < 1.04999999999999999e150

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a1 around 0 55.2%

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

      1. unpow2 55.2%

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

      2. associate-*l* 55.1%

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

      3. associate-*r/ 55.1%

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

      4. associate-/l* 55.2%

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

   6. Simplified 55.2%

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

   7. Taylor expanded in th around 0 28.7%

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

      1. add-sqr-sqrt 28.6%

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

      2. sqrt-unprod 28.7%

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

      3. frac-times 28.6%

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

      4. add-sqr-sqrt 28.6%

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

   9. Applied egg-rr 28.6%

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

   if 1.04999999999999999e150 < a2

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a1 around 0 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*l* 100.0%

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

      3. associate-*r/ 100.0%

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

      4. associate-/l* 100.0%

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

   6. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 100.0%

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

      3. times-frac 100.0%

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

      4. pow1/2 100.0%

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

      5. pow-flip 100.0%

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

      6. metadata-eval 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. sqrt-unprod 100.0%

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

      9. pow-prod-up 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

      \[\leadsto a2 \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{a2}{\frac{1}{\cos th}}\right)} \]
   9. Step-by-step derivation

      1. associate-/r/ 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

   10. Simplified 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. rem-log-exp 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 1.45:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;a2 \leq 1.05 \cdot 10^{+150}:\\ \;\;\;\;a2 \cdot \sqrt{\frac{a2 \cdot a2}{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \end{array} \]

Alternative 10: 60.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-0.5 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= (cos th) -1e-309) (* -0.5 t_1) (* t_1 0.5))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (cos(th) <= -1e-309) {
		tmp = -0.5 * t_1;
	} 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) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (cos(th) <= (-1d-309)) then
        tmp = (-0.5d0) * t_1
    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 tmp;
	if (Math.cos(th) <= -1e-309) {
		tmp = -0.5 * t_1;
	} else {
		tmp = t_1 * 0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if math.cos(th) <= -1e-309:
		tmp = -0.5 * t_1
	else:
		tmp = t_1 * 0.5
	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(-0.5 * t_1);
	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);
	tmp = 0.0;
	if (cos(th) <= -1e-309)
		tmp = -0.5 * t_1;
	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]}, If[LessEqual[N[Cos[th], $MachinePrecision], -1e-309], N[(-0.5 * t$95$1), $MachinePrecision], N[(t$95$1 * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in th around 0 8.4%

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

   6. Applied egg-rr 70.0%

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

   if -1.000000000000002e-309 < (cos.f64 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. Taylor expanded in th around 0 81.3%

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

   6. Applied egg-rr 56.8%

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

4. Final simplification 60.1%

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

Alternative 11: 51.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 0.00035:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;a2 \leq 10^{+150}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 0.00035)
   (* (* a1 a1) (sqrt 0.5))
   (if (<= a2 1e+150) (* a2 (* a2 (sqrt 0.5))) (* a2 (* (cos th) a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 0.00035) {
		tmp = (a1 * a1) * sqrt(0.5);
	} else if (a2 <= 1e+150) {
		tmp = a2 * (a2 * sqrt(0.5));
	} else {
		tmp = a2 * (cos(th) * 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 (a2 <= 0.00035d0) then
        tmp = (a1 * a1) * sqrt(0.5d0)
    else if (a2 <= 1d+150) then
        tmp = a2 * (a2 * sqrt(0.5d0))
    else
        tmp = a2 * (cos(th) * a2)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 0.00035) {
		tmp = (a1 * a1) * Math.sqrt(0.5);
	} else if (a2 <= 1e+150) {
		tmp = a2 * (a2 * Math.sqrt(0.5));
	} else {
		tmp = a2 * (Math.cos(th) * a2);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 0.00035:
		tmp = (a1 * a1) * math.sqrt(0.5)
	elif a2 <= 1e+150:
		tmp = a2 * (a2 * math.sqrt(0.5))
	else:
		tmp = a2 * (math.cos(th) * a2)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 0.00035)
		tmp = Float64(Float64(a1 * a1) * sqrt(0.5));
	elseif (a2 <= 1e+150)
		tmp = Float64(a2 * Float64(a2 * sqrt(0.5)));
	else
		tmp = Float64(a2 * Float64(cos(th) * a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 0.00035)
		tmp = (a1 * a1) * sqrt(0.5);
	elseif (a2 <= 1e+150)
		tmp = a2 * (a2 * sqrt(0.5));
	else
		tmp = a2 * (cos(th) * a2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 0.00035], N[(N[(a1 * a1), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[a2, 1e+150], N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a2 < 3.49999999999999996e-4

   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. Step-by-step derivation

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

      3. pow1/2 99.5%

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

      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%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in th around 0 65.5%

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

   7. Taylor expanded in a1 around inf 46.3%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot {a1}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 46.3%

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

   9. Simplified 46.3%

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

   if 3.49999999999999996e-4 < a2 < 9.99999999999999981e149

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a1 around 0 55.2%

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

      1. unpow2 55.2%

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

      2. associate-*l* 55.1%

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

      3. associate-*r/ 55.1%

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

      4. associate-/l* 55.2%

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

   6. Simplified 55.2%

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

      1. *-un-lft-identity 55.2%

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

      2. div-inv 55.2%

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

      3. times-frac 55.1%

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

      4. pow1/2 55.1%

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

      5. pow-flip 55.2%

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

      6. metadata-eval 55.2%

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

      7. add-sqr-sqrt 54.6%

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

      8. sqrt-unprod 55.2%

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

      9. pow-prod-up 55.2%

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

      10. metadata-eval 55.2%

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

      11. metadata-eval 55.2%

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

   8. Applied egg-rr 55.2%

      \[\leadsto a2 \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{a2}{\frac{1}{\cos th}}\right)} \]
   9. Step-by-step derivation

      1. associate-/r/ 55.3%

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

      2. /-rgt-identity 55.3%

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

      3. *-commutative 55.3%

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

      4. associate-*l* 55.2%

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

   10. Simplified 55.2%

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

   11. Taylor expanded in th around 0 28.6%

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

   if 9.99999999999999981e149 < a2

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a1 around 0 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*l* 100.0%

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

      3. associate-*r/ 100.0%

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

      4. associate-/l* 100.0%

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

   6. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 100.0%

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

      3. times-frac 100.0%

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

      4. pow1/2 100.0%

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

      5. pow-flip 100.0%

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

      6. metadata-eval 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. sqrt-unprod 100.0%

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

      9. pow-prod-up 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

      \[\leadsto a2 \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{a2}{\frac{1}{\cos th}}\right)} \]
   9. Step-by-step derivation

      1. associate-/r/ 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

   10. Simplified 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. rem-log-exp 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 0.00035:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;a2 \leq 10^{+150}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \end{array} \]

Alternative 12: 47.2% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.5 \cdot 10^{+80} \lor \neg \left(th \leq 2.6 \cdot 10^{+254}\right) \land th \leq 6.8 \cdot 10^{+289}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (or (<= th 1.5e+80) (and (not (<= th 2.6e+254)) (<= th 6.8e+289)))
   (* (+ a1 a2) (+ a1 a2))
   (* -0.5 (+ (* a1 a1) (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 1.5e+80) || (!(th <= 2.6e+254) && (th <= 6.8e+289))) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = -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 ((th <= 1.5d+80) .or. (.not. (th <= 2.6d+254)) .and. (th <= 6.8d+289)) then
        tmp = (a1 + a2) * (a1 + a2)
    else
        tmp = (-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 ((th <= 1.5e+80) || (!(th <= 2.6e+254) && (th <= 6.8e+289))) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = -0.5 * ((a1 * a1) + (a2 * a2));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (th <= 1.5e+80) or (not (th <= 2.6e+254) and (th <= 6.8e+289)):
		tmp = (a1 + a2) * (a1 + a2)
	else:
		tmp = -0.5 * ((a1 * a1) + (a2 * a2))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if ((th <= 1.5e+80) || (!(th <= 2.6e+254) && (th <= 6.8e+289)))
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	else
		tmp = Float64(-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 ((th <= 1.5e+80) || (~((th <= 2.6e+254)) && (th <= 6.8e+289)))
		tmp = (a1 + a2) * (a1 + a2);
	else
		tmp = -0.5 * ((a1 * a1) + (a2 * a2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[Or[LessEqual[th, 1.5e+80], And[N[Not[LessEqual[th, 2.6e+254]], $MachinePrecision], LessEqual[th, 6.8e+289]]], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 1.49999999999999993e80 or 2.6000000000000001e254 < th < 6.7999999999999997e289

   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. Taylor expanded in th around 0 71.6%

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

   6. Applied egg-rr 45.7%

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

      1. distribute-lft-out 49.4%

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

      2. +-commutative 49.4%

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

      3. +-commutative 49.4%

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

   8. Simplified 49.4%

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

   if 1.49999999999999993e80 < th < 2.6000000000000001e254 or 6.7999999999999997e289 < 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. Taylor expanded in th around 0 20.8%

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

   6. Applied egg-rr 57.2%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.5 \cdot 10^{+80} \lor \neg \left(th \leq 2.6 \cdot 10^{+254}\right) \land th \leq 6.8 \cdot 10^{+289}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]

Alternative 13: 47.2% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 1.5 \cdot 10^{+80}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{elif}\;th \leq 2.6 \cdot 10^{+254} \lor \neg \left(th \leq 6.8 \cdot 10^{+289}\right):\\ \;\;\;\;-0.5 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot 0.125\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= th 1.5e+80)
     (* (+ a1 a2) (+ a1 a2))
     (if (or (<= th 2.6e+254) (not (<= th 6.8e+289)))
       (* -0.5 t_1)
       (* t_1 0.125)))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 1.5e+80) {
		tmp = (a1 + a2) * (a1 + a2);
	} else if ((th <= 2.6e+254) || !(th <= 6.8e+289)) {
		tmp = -0.5 * t_1;
	} else {
		tmp = t_1 * 0.125;
	}
	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 (th <= 1.5d+80) then
        tmp = (a1 + a2) * (a1 + a2)
    else if ((th <= 2.6d+254) .or. (.not. (th <= 6.8d+289))) then
        tmp = (-0.5d0) * t_1
    else
        tmp = t_1 * 0.125d0
    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 (th <= 1.5e+80) {
		tmp = (a1 + a2) * (a1 + a2);
	} else if ((th <= 2.6e+254) || !(th <= 6.8e+289)) {
		tmp = -0.5 * t_1;
	} else {
		tmp = t_1 * 0.125;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if th <= 1.5e+80:
		tmp = (a1 + a2) * (a1 + a2)
	elif (th <= 2.6e+254) or not (th <= 6.8e+289):
		tmp = -0.5 * t_1
	else:
		tmp = t_1 * 0.125
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (th <= 1.5e+80)
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	elseif ((th <= 2.6e+254) || !(th <= 6.8e+289))
		tmp = Float64(-0.5 * t_1);
	else
		tmp = Float64(t_1 * 0.125);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (th <= 1.5e+80)
		tmp = (a1 + a2) * (a1 + a2);
	elseif ((th <= 2.6e+254) || ~((th <= 6.8e+289)))
		tmp = -0.5 * t_1;
	else
		tmp = t_1 * 0.125;
	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[th, 1.5e+80], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[th, 2.6e+254], N[Not[LessEqual[th, 6.8e+289]], $MachinePrecision]], N[(-0.5 * t$95$1), $MachinePrecision], N[(t$95$1 * 0.125), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if th < 1.49999999999999993e80

   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. Taylor expanded in th around 0 72.9%

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

   6. Applied egg-rr 45.6%

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

      1. distribute-lft-out 49.5%

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

      2. +-commutative 49.5%

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

      3. +-commutative 49.5%

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

   8. Simplified 49.5%

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

   if 1.49999999999999993e80 < th < 2.6000000000000001e254 or 6.7999999999999997e289 < 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. Taylor expanded in th around 0 20.8%

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

   6. Applied egg-rr 57.2%

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

   if 2.6000000000000001e254 < th < 6.7999999999999997e289

   1. Initial program 99.1%

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

      1. distribute-lft-out 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in th around 0 49.4%

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

   6. Applied egg-rr 48.0%

      \[\leadsto \color{blue}{0.125} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.5 \cdot 10^{+80}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{elif}\;th \leq 2.6 \cdot 10^{+254} \lor \neg \left(th \leq 6.8 \cdot 10^{+289}\right):\\ \;\;\;\;-0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.125\\ \end{array} \]

Alternative 14: 47.2% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 1.5 \cdot 10^{+80}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{elif}\;th \leq 2.6 \cdot 10^{+254} \lor \neg \left(th \leq 6.8 \cdot 10^{+289}\right):\\ \;\;\;\;-0.5 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= th 1.5e+80)
     (* (+ a1 a2) (+ a1 a2))
     (if (or (<= th 2.6e+254) (not (<= th 6.8e+289)))
       (* -0.5 t_1)
       (* t_1 0.25)))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 1.5e+80) {
		tmp = (a1 + a2) * (a1 + a2);
	} else if ((th <= 2.6e+254) || !(th <= 6.8e+289)) {
		tmp = -0.5 * t_1;
	} else {
		tmp = t_1 * 0.25;
	}
	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 (th <= 1.5d+80) then
        tmp = (a1 + a2) * (a1 + a2)
    else if ((th <= 2.6d+254) .or. (.not. (th <= 6.8d+289))) then
        tmp = (-0.5d0) * t_1
    else
        tmp = t_1 * 0.25d0
    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 (th <= 1.5e+80) {
		tmp = (a1 + a2) * (a1 + a2);
	} else if ((th <= 2.6e+254) || !(th <= 6.8e+289)) {
		tmp = -0.5 * t_1;
	} else {
		tmp = t_1 * 0.25;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if th <= 1.5e+80:
		tmp = (a1 + a2) * (a1 + a2)
	elif (th <= 2.6e+254) or not (th <= 6.8e+289):
		tmp = -0.5 * t_1
	else:
		tmp = t_1 * 0.25
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (th <= 1.5e+80)
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	elseif ((th <= 2.6e+254) || !(th <= 6.8e+289))
		tmp = Float64(-0.5 * t_1);
	else
		tmp = Float64(t_1 * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (th <= 1.5e+80)
		tmp = (a1 + a2) * (a1 + a2);
	elseif ((th <= 2.6e+254) || ~((th <= 6.8e+289)))
		tmp = -0.5 * t_1;
	else
		tmp = t_1 * 0.25;
	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[th, 1.5e+80], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[th, 2.6e+254], N[Not[LessEqual[th, 6.8e+289]], $MachinePrecision]], N[(-0.5 * t$95$1), $MachinePrecision], N[(t$95$1 * 0.25), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if th < 1.49999999999999993e80

   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. Taylor expanded in th around 0 72.9%

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

   6. Applied egg-rr 45.6%

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

      1. distribute-lft-out 49.5%

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

      2. +-commutative 49.5%

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

      3. +-commutative 49.5%

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

   8. Simplified 49.5%

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

   if 1.49999999999999993e80 < th < 2.6000000000000001e254 or 6.7999999999999997e289 < 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. Taylor expanded in th around 0 20.8%

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

   6. Applied egg-rr 57.2%

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

   if 2.6000000000000001e254 < th < 6.7999999999999997e289

   1. Initial program 99.1%

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

      1. distribute-lft-out 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in th around 0 49.4%

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

   6. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{0.25} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.5 \cdot 10^{+80}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{elif}\;th \leq 2.6 \cdot 10^{+254} \lor \neg \left(th \leq 6.8 \cdot 10^{+289}\right):\\ \;\;\;\;-0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.25\\ \end{array} \]

Alternative 15: 46.9% accurate, 59.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + 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. Taylor expanded in th around 0 63.1%

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

6. Applied egg-rr 41.3%

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

   1. distribute-lft-out 44.4%

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

   2. +-commutative 44.4%

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

   3. +-commutative 44.4%

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

8. Simplified 44.4%

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

9. Final simplification 44.4%

   \[\leadsto \left(a1 + a2\right) \cdot \left(a1 + a2\right) \]

Alternative 16: 36.4% accurate, 82.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 0.03:\\ \;\;\;\;a1 \cdot a1\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot a2\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (if (<= a2 0.03) (* a1 a1) (* a2 a2)))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 0.03) {
		tmp = a1 * a1;
	} else {
		tmp = 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 (a2 <= 0.03d0) then
        tmp = a1 * a1
    else
        tmp = a2 * a2
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 0.03) {
		tmp = a1 * a1;
	} else {
		tmp = a2 * a2;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 0.03:
		tmp = a1 * a1
	else:
		tmp = a2 * a2
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 0.03)
		tmp = Float64(a1 * a1);
	else
		tmp = Float64(a2 * a2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 0.03)
		tmp = a1 * a1;
	else
		tmp = a2 * a2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 0.03], N[(a1 * a1), $MachinePrecision], N[(a2 * a2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 0.029999999999999999

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 65.5%

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

      1. unpow2 65.5%

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

      2. unpow2 65.5%

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

      3. +-commutative 65.5%

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

   6. Simplified 65.5%

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

   7. Taylor expanded in a1 around inf 46.3%

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

      1. unpow2 46.3%

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

   9. Simplified 46.3%

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

   11. Applied egg-rr 35.1%

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

   if 0.029999999999999999 < a2

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around 0 74.3%

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

      1. unpow2 74.3%

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

      2. associate-*l* 74.2%

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

      3. associate-*r/ 74.2%

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

      4. associate-/l* 74.3%

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

   6. Simplified 74.3%

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

   7. Taylor expanded in th around 0 42.4%

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

      1. frac-2neg 42.4%

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

      2. div-inv 42.4%

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

   9. Applied egg-rr 42.4%

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

   11. Applied egg-rr 31.2%

       \[\leadsto a2 \cdot \color{blue}{\left(0 + a2\right)} \]
   12. Step-by-step derivation

       1. +-lft-identity 31.2%

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

   13. Simplified 31.2%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 0.03:\\ \;\;\;\;a1 \cdot a1\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot a2\\ \end{array} \]

Alternative 17: 3.7% accurate, 138.3× speedup

Math

\[\begin{array}{l} \\ a1 \cdot -2 \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* a1 -2.0))

C

double code(double a1, double a2, double th) {
	return a1 * -2.0;
}

Fortran

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

Java

public static double code(double a1, double a2, double th) {
	return a1 * -2.0;
}

Python

def code(a1, a2, th):
	return a1 * -2.0

Julia

function code(a1, a2, th)
	return Float64(a1 * -2.0)
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = a1 * -2.0;
end

Wolfram

code[a1_, a2_, th_] := N[(a1 * -2.0), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 63.1%

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

   1. unpow2 63.1%

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

   2. unpow2 63.1%

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

   3. +-commutative 63.1%

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

6. Simplified 63.1%

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

7. Taylor expanded in a1 around inf 40.2%

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

   1. unpow2 40.2%

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

9. Simplified 40.2%

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

11. Applied egg-rr 3.3%

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

    1. *-commutative 3.3%

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

13. Simplified 3.3%

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

14. Final simplification 3.3%

    \[\leadsto a1 \cdot -2 \]

Alternative 18: 30.5% 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%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 63.1%

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

   1. unpow2 63.1%

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

   2. unpow2 63.1%

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

   3. +-commutative 63.1%

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

6. Simplified 63.1%

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

7. Taylor expanded in a1 around inf 40.2%

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

   1. unpow2 40.2%

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

9. Simplified 40.2%

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

11. Applied egg-rr 31.0%

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

12. Final simplification 31.0%

    \[\leadsto a1 \cdot a1 \]

Alternative 19: 3.7% accurate, 207.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := (-a1)

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 63.1%

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

   1. unpow2 63.1%

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

   2. unpow2 63.1%

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

   3. +-commutative 63.1%

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

6. Simplified 63.1%

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

7. Taylor expanded in a1 around inf 40.2%

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

   1. unpow2 40.2%

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

9. Simplified 40.2%

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

11. Applied egg-rr 3.3%

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

    1. neg-sub0 3.3%

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

13. Simplified 3.3%

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

14. Final simplification 3.3%

    \[\leadsto -a1 \]

Alternative 20: 3.6% accurate, 415.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := a1

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 63.1%

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

   1. unpow2 63.1%

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

   2. unpow2 63.1%

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

   3. +-commutative 63.1%

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

6. Simplified 63.1%

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

7. Taylor expanded in a1 around inf 40.2%

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

   1. unpow2 40.2%

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

9. Simplified 40.2%

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

11. Applied egg-rr 4.7%

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

    1. unpow1 4.7%

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

    2. metadata-eval 4.7%

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

    3. sqr-pow 2.9%

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

    4. fabs-sqr 2.9%

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

    5. sqr-pow 3.9%

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

    6. metadata-eval 3.9%

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

    7. unpow1 3.9%

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

13. Simplified 3.9%

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

14. Final simplification 3.9%

    \[\leadsto a1 \]

Reproduce

herbie shell --seed 2023187 
(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))))