Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 15.0s

Alternatives: 22

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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. 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. Step-by-step derivation

   1. fma-def 99.6%

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

   2. div-inv 99.6%

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

   3. add-sqr-sqrt 99.6%

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

   4. associate-*l* 99.6%

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

   5. hypot-def 99.6%

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

   6. hypot-def 99.6%

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

   7. pow1/2 99.6%

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

   8. pow-flip 99.7%

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

   9. metadata-eval 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return cos(th) * ((a2 / (sqrt(2.0) / a2)) + ((a1 * a1) / 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 = cos(th) * ((a2 / (sqrt(2.0d0) / a2)) + ((a1 * a1) / sqrt(2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. 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 99.6%

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

   1. unpow2 99.6%

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

   2. associate-/l* 99.7%

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

   3. unpow2 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   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. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. 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 inf 99.6%

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

   1. associate-/l* 99.7%

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

   2. unpow2 99.7%

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

   3. unpow2 99.7%

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

   4. fma-def 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 68.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 2.4 \cdot 10^{-104}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 7.5 \cdot 10^{-39} \lor \neg \left(a2 \leq 8.1 \cdot 10^{+55}\right):\\ \;\;\;\;a2 \cdot \frac{\cos th \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left({2}^{-0.5} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 2.4e-104)
   (* (cos th) (* (* a1 a1) (sqrt 0.5)))
   (if (or (<= a2 7.5e-39) (not (<= a2 8.1e+55)))
     (* a2 (/ (* (cos th) a2) (sqrt 2.0)))
     (*
      (+ (* a1 a1) (* a2 a2))
      (* (pow 2.0 -0.5) (+ 1.0 (* -0.5 (* th th))))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 2.4e-104) {
		tmp = cos(th) * ((a1 * a1) * sqrt(0.5));
	} else if ((a2 <= 7.5e-39) || !(a2 <= 8.1e+55)) {
		tmp = a2 * ((cos(th) * a2) / sqrt(2.0));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * (pow(2.0, -0.5) * (1.0 + (-0.5 * (th * th))));
	}
	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 <= 2.4d-104) then
        tmp = cos(th) * ((a1 * a1) * sqrt(0.5d0))
    else if ((a2 <= 7.5d-39) .or. (.not. (a2 <= 8.1d+55))) then
        tmp = a2 * ((cos(th) * a2) / sqrt(2.0d0))
    else
        tmp = ((a1 * a1) + (a2 * a2)) * ((2.0d0 ** (-0.5d0)) * (1.0d0 + ((-0.5d0) * (th * th))))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 2.4e-104) {
		tmp = Math.cos(th) * ((a1 * a1) * Math.sqrt(0.5));
	} else if ((a2 <= 7.5e-39) || !(a2 <= 8.1e+55)) {
		tmp = a2 * ((Math.cos(th) * a2) / Math.sqrt(2.0));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * (Math.pow(2.0, -0.5) * (1.0 + (-0.5 * (th * th))));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 2.4e-104:
		tmp = math.cos(th) * ((a1 * a1) * math.sqrt(0.5))
	elif (a2 <= 7.5e-39) or not (a2 <= 8.1e+55):
		tmp = a2 * ((math.cos(th) * a2) / math.sqrt(2.0))
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * (math.pow(2.0, -0.5) * (1.0 + (-0.5 * (th * th))))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 2.4e-104)
		tmp = Float64(cos(th) * Float64(Float64(a1 * a1) * sqrt(0.5)));
	elseif ((a2 <= 7.5e-39) || !(a2 <= 8.1e+55))
		tmp = Float64(a2 * Float64(Float64(cos(th) * a2) / sqrt(2.0)));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * Float64((2.0 ^ -0.5) * Float64(1.0 + Float64(-0.5 * Float64(th * th)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 2.4e-104)
		tmp = cos(th) * ((a1 * a1) * sqrt(0.5));
	elseif ((a2 <= 7.5e-39) || ~((a2 <= 8.1e+55)))
		tmp = a2 * ((cos(th) * a2) / sqrt(2.0));
	else
		tmp = ((a1 * a1) + (a2 * a2)) * ((2.0 ^ -0.5) * (1.0 + (-0.5 * (th * th))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 2.4e-104], N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[a2, 7.5e-39], N[Not[LessEqual[a2, 8.1e+55]], $MachinePrecision]], N[(a2 * N[(N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Power[2.0, -0.5], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a2 < 2.4000000000000001e-104

   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/ 99.7%

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

      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. Step-by-step derivation

      1. fma-def 99.7%

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

      2. div-inv 99.6%

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

      3. add-sqr-sqrt 99.6%

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

      4. associate-*l* 99.6%

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

      5. hypot-def 99.6%

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

      6. hypot-def 99.6%

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

      7. pow1/2 99.6%

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

      8. pow-flip 99.7%

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

      9. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in a1 around inf 69.5%

      \[\leadsto \cos th \cdot \color{blue}{\left(\sqrt{0.5} \cdot {a1}^{2}\right)} \]
   7. Step-by-step derivation

      1. unpow2 69.5%

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

   8. Simplified 69.5%

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

   if 2.4000000000000001e-104 < a2 < 7.49999999999999971e-39 or 8.0999999999999999e55 < 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/ 99.7%

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

      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 83.9%

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

      1. unpow2 83.9%

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

      2. associate-*l* 83.9%

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

      3. associate-*r/ 83.9%

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

   6. Simplified 83.9%

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

   if 7.49999999999999971e-39 < a2 < 8.0999999999999999e55

   1. Initial program 98.9%

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

      1. distribute-lft-out 98.9%

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

   3. Simplified 98.9%

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

      1. clear-num 99.0%

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

      2. associate-/r/ 98.9%

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

      3. pow1/2 98.9%

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

      4. pow-flip 99.4%

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

      5. metadata-eval 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in th around 0 69.1%

      \[\leadsto \left({2}^{-0.5} \cdot \color{blue}{\left(1 + -0.5 \cdot {th}^{2}\right)}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   7. Step-by-step derivation

      1. unpow2 39.2%

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

   8. Simplified 69.1%

      \[\leadsto \left({2}^{-0.5} \cdot \color{blue}{\left(1 + -0.5 \cdot \left(th \cdot th\right)\right)}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 2.4 \cdot 10^{-104}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 7.5 \cdot 10^{-39} \lor \neg \left(a2 \leq 8.1 \cdot 10^{+55}\right):\\ \;\;\;\;a2 \cdot \frac{\cos th \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left({2}^{-0.5} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \end{array} \]

Alternative 6: 68.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 2.55 \cdot 10^{-104}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 7.6 \cdot 10^{-39} \lor \neg \left(a2 \leq 7.9 \cdot 10^{+49}\right):\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left({2}^{-0.5} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 2.55e-104)
   (* (cos th) (* (* a1 a1) (sqrt 0.5)))
   (if (or (<= a2 7.6e-39) (not (<= a2 7.9e+49)))
     (* (cos th) (* (* a2 a2) (sqrt 0.5)))
     (*
      (+ (* a1 a1) (* a2 a2))
      (* (pow 2.0 -0.5) (+ 1.0 (* -0.5 (* th th))))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 2.55e-104) {
		tmp = cos(th) * ((a1 * a1) * sqrt(0.5));
	} else if ((a2 <= 7.6e-39) || !(a2 <= 7.9e+49)) {
		tmp = cos(th) * ((a2 * a2) * sqrt(0.5));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * (pow(2.0, -0.5) * (1.0 + (-0.5 * (th * th))));
	}
	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 <= 2.55d-104) then
        tmp = cos(th) * ((a1 * a1) * sqrt(0.5d0))
    else if ((a2 <= 7.6d-39) .or. (.not. (a2 <= 7.9d+49))) then
        tmp = cos(th) * ((a2 * a2) * sqrt(0.5d0))
    else
        tmp = ((a1 * a1) + (a2 * a2)) * ((2.0d0 ** (-0.5d0)) * (1.0d0 + ((-0.5d0) * (th * th))))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 2.55e-104) {
		tmp = Math.cos(th) * ((a1 * a1) * Math.sqrt(0.5));
	} else if ((a2 <= 7.6e-39) || !(a2 <= 7.9e+49)) {
		tmp = Math.cos(th) * ((a2 * a2) * Math.sqrt(0.5));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * (Math.pow(2.0, -0.5) * (1.0 + (-0.5 * (th * th))));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 2.55e-104:
		tmp = math.cos(th) * ((a1 * a1) * math.sqrt(0.5))
	elif (a2 <= 7.6e-39) or not (a2 <= 7.9e+49):
		tmp = math.cos(th) * ((a2 * a2) * math.sqrt(0.5))
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * (math.pow(2.0, -0.5) * (1.0 + (-0.5 * (th * th))))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 2.55e-104)
		tmp = Float64(cos(th) * Float64(Float64(a1 * a1) * sqrt(0.5)));
	elseif ((a2 <= 7.6e-39) || !(a2 <= 7.9e+49))
		tmp = Float64(cos(th) * Float64(Float64(a2 * a2) * sqrt(0.5)));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * Float64((2.0 ^ -0.5) * Float64(1.0 + Float64(-0.5 * Float64(th * th)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 2.55e-104)
		tmp = cos(th) * ((a1 * a1) * sqrt(0.5));
	elseif ((a2 <= 7.6e-39) || ~((a2 <= 7.9e+49)))
		tmp = cos(th) * ((a2 * a2) * sqrt(0.5));
	else
		tmp = ((a1 * a1) + (a2 * a2)) * ((2.0 ^ -0.5) * (1.0 + (-0.5 * (th * th))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 2.55e-104], N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[a2, 7.6e-39], N[Not[LessEqual[a2, 7.9e+49]], $MachinePrecision]], N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Power[2.0, -0.5], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a2 < 2.54999999999999996e-104

   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/ 99.7%

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

      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. Step-by-step derivation

      1. fma-def 99.7%

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

      2. div-inv 99.6%

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

      3. add-sqr-sqrt 99.6%

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

      4. associate-*l* 99.6%

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

      5. hypot-def 99.6%

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

      6. hypot-def 99.6%

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

      7. pow1/2 99.6%

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

      8. pow-flip 99.7%

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

      9. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in a1 around inf 69.5%

      \[\leadsto \cos th \cdot \color{blue}{\left(\sqrt{0.5} \cdot {a1}^{2}\right)} \]
   7. Step-by-step derivation

      1. unpow2 69.5%

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

   8. Simplified 69.5%

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

   if 2.54999999999999996e-104 < a2 < 7.6000000000000004e-39 or 7.9000000000000004e49 < 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/ 99.7%

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

      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. Step-by-step derivation

      1. fma-def 99.7%

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

      2. div-inv 99.7%

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

      3. add-sqr-sqrt 99.7%

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

      4. associate-*l* 99.6%

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

      5. hypot-def 99.6%

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

      6. hypot-def 99.6%

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

      7. pow1/2 99.6%

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

      8. pow-flip 99.8%

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

      9. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in a1 around 0 83.9%

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

      1. unpow2 83.9%

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

   8. Simplified 83.9%

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

   if 7.6000000000000004e-39 < a2 < 7.9000000000000004e49

   1. Initial program 98.9%

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

      1. distribute-lft-out 98.9%

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

   3. Simplified 98.9%

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

      1. clear-num 99.0%

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

      2. associate-/r/ 98.9%

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

      3. pow1/2 98.9%

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

      4. pow-flip 99.4%

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

      5. metadata-eval 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in th around 0 69.1%

      \[\leadsto \left({2}^{-0.5} \cdot \color{blue}{\left(1 + -0.5 \cdot {th}^{2}\right)}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   7. Step-by-step derivation

      1. unpow2 39.2%

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

   8. Simplified 69.1%

      \[\leadsto \left({2}^{-0.5} \cdot \color{blue}{\left(1 + -0.5 \cdot \left(th \cdot th\right)\right)}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 2.55 \cdot 10^{-104}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 7.6 \cdot 10^{-39} \lor \neg \left(a2 \leq 7.9 \cdot 10^{+49}\right):\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left({2}^{-0.5} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \end{array} \]

Alternative 7: 68.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 2.55 \cdot 10^{-104}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 7.6 \cdot 10^{-39}:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\right)\\ \mathbf{elif}\;a2 \leq 1.18 \cdot 10^{+58}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left({2}^{-0.5} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 2.55e-104)
   (* (cos th) (* (* a1 a1) (sqrt 0.5)))
   (if (<= a2 7.6e-39)
     (* (cos th) (* a2 (* a2 (sqrt 0.5))))
     (if (<= a2 1.18e+58)
       (*
        (+ (* a1 a1) (* a2 a2))
        (* (pow 2.0 -0.5) (+ 1.0 (* -0.5 (* th th)))))
       (* (cos th) (* (* a2 a2) (sqrt 0.5)))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 2.55e-104) {
		tmp = cos(th) * ((a1 * a1) * sqrt(0.5));
	} else if (a2 <= 7.6e-39) {
		tmp = cos(th) * (a2 * (a2 * sqrt(0.5)));
	} else if (a2 <= 1.18e+58) {
		tmp = ((a1 * a1) + (a2 * a2)) * (pow(2.0, -0.5) * (1.0 + (-0.5 * (th * th))));
	} 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 (a2 <= 2.55d-104) then
        tmp = cos(th) * ((a1 * a1) * sqrt(0.5d0))
    else if (a2 <= 7.6d-39) then
        tmp = cos(th) * (a2 * (a2 * sqrt(0.5d0)))
    else if (a2 <= 1.18d+58) then
        tmp = ((a1 * a1) + (a2 * a2)) * ((2.0d0 ** (-0.5d0)) * (1.0d0 + ((-0.5d0) * (th * th))))
    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 (a2 <= 2.55e-104) {
		tmp = Math.cos(th) * ((a1 * a1) * Math.sqrt(0.5));
	} else if (a2 <= 7.6e-39) {
		tmp = Math.cos(th) * (a2 * (a2 * Math.sqrt(0.5)));
	} else if (a2 <= 1.18e+58) {
		tmp = ((a1 * a1) + (a2 * a2)) * (Math.pow(2.0, -0.5) * (1.0 + (-0.5 * (th * th))));
	} else {
		tmp = Math.cos(th) * ((a2 * a2) * Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 2.55e-104:
		tmp = math.cos(th) * ((a1 * a1) * math.sqrt(0.5))
	elif a2 <= 7.6e-39:
		tmp = math.cos(th) * (a2 * (a2 * math.sqrt(0.5)))
	elif a2 <= 1.18e+58:
		tmp = ((a1 * a1) + (a2 * a2)) * (math.pow(2.0, -0.5) * (1.0 + (-0.5 * (th * th))))
	else:
		tmp = math.cos(th) * ((a2 * a2) * math.sqrt(0.5))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 2.55e-104)
		tmp = Float64(cos(th) * Float64(Float64(a1 * a1) * sqrt(0.5)));
	elseif (a2 <= 7.6e-39)
		tmp = Float64(cos(th) * Float64(a2 * Float64(a2 * sqrt(0.5))));
	elseif (a2 <= 1.18e+58)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * Float64((2.0 ^ -0.5) * Float64(1.0 + Float64(-0.5 * Float64(th * th)))));
	else
		tmp = Float64(cos(th) * Float64(Float64(a2 * a2) * sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 2.55e-104)
		tmp = cos(th) * ((a1 * a1) * sqrt(0.5));
	elseif (a2 <= 7.6e-39)
		tmp = cos(th) * (a2 * (a2 * sqrt(0.5)));
	elseif (a2 <= 1.18e+58)
		tmp = ((a1 * a1) + (a2 * a2)) * ((2.0 ^ -0.5) * (1.0 + (-0.5 * (th * th))));
	else
		tmp = cos(th) * ((a2 * a2) * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 2.55e-104], N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a2, 7.6e-39], N[(N[Cos[th], $MachinePrecision] * N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a2, 1.18e+58], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Power[2.0, -0.5], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a2 < 2.54999999999999996e-104

   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/ 99.7%

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

      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. Step-by-step derivation

      1. fma-def 99.7%

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

      2. div-inv 99.6%

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

      3. add-sqr-sqrt 99.6%

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

      4. associate-*l* 99.6%

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

      5. hypot-def 99.6%

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

      6. hypot-def 99.6%

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

      7. pow1/2 99.6%

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

      8. pow-flip 99.7%

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

      9. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in a1 around inf 69.5%

      \[\leadsto \cos th \cdot \color{blue}{\left(\sqrt{0.5} \cdot {a1}^{2}\right)} \]
   7. Step-by-step derivation

      1. unpow2 69.5%

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

   8. Simplified 69.5%

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

   if 2.54999999999999996e-104 < a2 < 7.6000000000000004e-39

   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 99.5%

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

      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/ 99.5%

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

      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 53.8%

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

      1. unpow2 53.8%

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

      2. associate-/l* 53.8%

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

   6. Simplified 53.8%

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

      1. associate-/r/ 53.9%

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

      2. div-inv 53.6%

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

      3. add-sqr-sqrt 53.6%

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

      4. sqrt-unprod 53.6%

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

      5. frac-times 53.6%

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

      6. metadata-eval 53.6%

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

      7. add-sqr-sqrt 53.9%

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

      8. metadata-eval 53.9%

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

   8. Applied egg-rr 53.9%

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

   if 7.6000000000000004e-39 < a2 < 1.18000000000000003e58

   1. Initial program 98.9%

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

      1. distribute-lft-out 98.9%

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

   3. Simplified 98.9%

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

      1. clear-num 99.0%

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

      2. associate-/r/ 98.9%

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

      3. pow1/2 98.9%

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

      4. pow-flip 99.4%

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

      5. metadata-eval 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in th around 0 69.1%

      \[\leadsto \left({2}^{-0.5} \cdot \color{blue}{\left(1 + -0.5 \cdot {th}^{2}\right)}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   7. Step-by-step derivation

      1. unpow2 39.2%

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

   8. Simplified 69.1%

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

   if 1.18000000000000003e58 < a2

   1. Initial program 99.8%

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

      1. distribute-lft-out 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

      4. fma-def 99.8%

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

   3. Simplified 99.8%

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

      1. fma-def 99.8%

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

      2. div-inv 99.8%

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

      3. add-sqr-sqrt 99.8%

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

      4. associate-*l* 99.7%

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

      5. hypot-def 99.7%

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

      6. hypot-def 99.7%

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

      7. pow1/2 99.7%

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

      8. pow-flip 99.8%

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

      9. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in a1 around 0 92.2%

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

      1. unpow2 92.2%

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

   8. Simplified 92.2%

      \[\leadsto \cos th \cdot \color{blue}{\left(\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 2.55 \cdot 10^{-104}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 7.6 \cdot 10^{-39}:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\right)\\ \mathbf{elif}\;a2 \leq 1.18 \cdot 10^{+58}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left({2}^{-0.5} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 8: 68.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 2.4 \cdot 10^{-104}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 7.5 \cdot 10^{-39}:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot \frac{a2}{\sqrt{2}}\right)\\ \mathbf{elif}\;a2 \leq 1.26 \cdot 10^{+54}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left({2}^{-0.5} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 2.4e-104)
   (* (cos th) (* (* a1 a1) (sqrt 0.5)))
   (if (<= a2 7.5e-39)
     (* (cos th) (* a2 (/ a2 (sqrt 2.0))))
     (if (<= a2 1.26e+54)
       (*
        (+ (* a1 a1) (* a2 a2))
        (* (pow 2.0 -0.5) (+ 1.0 (* -0.5 (* th th)))))
       (* (cos th) (* (* a2 a2) (sqrt 0.5)))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 2.4e-104) {
		tmp = cos(th) * ((a1 * a1) * sqrt(0.5));
	} else if (a2 <= 7.5e-39) {
		tmp = cos(th) * (a2 * (a2 / sqrt(2.0)));
	} else if (a2 <= 1.26e+54) {
		tmp = ((a1 * a1) + (a2 * a2)) * (pow(2.0, -0.5) * (1.0 + (-0.5 * (th * th))));
	} 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 (a2 <= 2.4d-104) then
        tmp = cos(th) * ((a1 * a1) * sqrt(0.5d0))
    else if (a2 <= 7.5d-39) then
        tmp = cos(th) * (a2 * (a2 / sqrt(2.0d0)))
    else if (a2 <= 1.26d+54) then
        tmp = ((a1 * a1) + (a2 * a2)) * ((2.0d0 ** (-0.5d0)) * (1.0d0 + ((-0.5d0) * (th * th))))
    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 (a2 <= 2.4e-104) {
		tmp = Math.cos(th) * ((a1 * a1) * Math.sqrt(0.5));
	} else if (a2 <= 7.5e-39) {
		tmp = Math.cos(th) * (a2 * (a2 / Math.sqrt(2.0)));
	} else if (a2 <= 1.26e+54) {
		tmp = ((a1 * a1) + (a2 * a2)) * (Math.pow(2.0, -0.5) * (1.0 + (-0.5 * (th * th))));
	} else {
		tmp = Math.cos(th) * ((a2 * a2) * Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 2.4e-104:
		tmp = math.cos(th) * ((a1 * a1) * math.sqrt(0.5))
	elif a2 <= 7.5e-39:
		tmp = math.cos(th) * (a2 * (a2 / math.sqrt(2.0)))
	elif a2 <= 1.26e+54:
		tmp = ((a1 * a1) + (a2 * a2)) * (math.pow(2.0, -0.5) * (1.0 + (-0.5 * (th * th))))
	else:
		tmp = math.cos(th) * ((a2 * a2) * math.sqrt(0.5))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 2.4e-104)
		tmp = Float64(cos(th) * Float64(Float64(a1 * a1) * sqrt(0.5)));
	elseif (a2 <= 7.5e-39)
		tmp = Float64(cos(th) * Float64(a2 * Float64(a2 / sqrt(2.0))));
	elseif (a2 <= 1.26e+54)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * Float64((2.0 ^ -0.5) * Float64(1.0 + Float64(-0.5 * Float64(th * th)))));
	else
		tmp = Float64(cos(th) * Float64(Float64(a2 * a2) * sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 2.4e-104)
		tmp = cos(th) * ((a1 * a1) * sqrt(0.5));
	elseif (a2 <= 7.5e-39)
		tmp = cos(th) * (a2 * (a2 / sqrt(2.0)));
	elseif (a2 <= 1.26e+54)
		tmp = ((a1 * a1) + (a2 * a2)) * ((2.0 ^ -0.5) * (1.0 + (-0.5 * (th * th))));
	else
		tmp = cos(th) * ((a2 * a2) * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 2.4e-104], N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a2, 7.5e-39], N[(N[Cos[th], $MachinePrecision] * N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a2, 1.26e+54], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Power[2.0, -0.5], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a2 < 2.4000000000000001e-104

   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/ 99.7%

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

      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. Step-by-step derivation

      1. fma-def 99.7%

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

      2. div-inv 99.6%

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

      3. add-sqr-sqrt 99.6%

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

      4. associate-*l* 99.6%

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

      5. hypot-def 99.6%

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

      6. hypot-def 99.6%

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

      7. pow1/2 99.6%

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

      8. pow-flip 99.7%

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

      9. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in a1 around inf 69.5%

      \[\leadsto \cos th \cdot \color{blue}{\left(\sqrt{0.5} \cdot {a1}^{2}\right)} \]
   7. Step-by-step derivation

      1. unpow2 69.5%

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

   8. Simplified 69.5%

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

   if 2.4000000000000001e-104 < a2 < 7.49999999999999971e-39

   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 99.5%

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

      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/ 99.5%

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

      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 53.8%

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

      1. unpow2 53.8%

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

      2. associate-/l* 53.8%

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

      3. associate-/r/ 53.9%

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

   6. Simplified 53.9%

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

   if 7.49999999999999971e-39 < a2 < 1.25999999999999995e54

   1. Initial program 98.9%

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

      1. distribute-lft-out 98.9%

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

   3. Simplified 98.9%

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

      1. clear-num 99.0%

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

      2. associate-/r/ 98.9%

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

      3. pow1/2 98.9%

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

      4. pow-flip 99.4%

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

      5. metadata-eval 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in th around 0 69.1%

      \[\leadsto \left({2}^{-0.5} \cdot \color{blue}{\left(1 + -0.5 \cdot {th}^{2}\right)}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   7. Step-by-step derivation

      1. unpow2 39.2%

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

   8. Simplified 69.1%

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

   if 1.25999999999999995e54 < a2

   1. Initial program 99.8%

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

      1. distribute-lft-out 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

      4. fma-def 99.8%

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

   3. Simplified 99.8%

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

      1. fma-def 99.8%

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

      2. div-inv 99.8%

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

      3. add-sqr-sqrt 99.8%

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

      4. associate-*l* 99.7%

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

      5. hypot-def 99.7%

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

      6. hypot-def 99.7%

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

      7. pow1/2 99.7%

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

      8. pow-flip 99.8%

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

      9. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in a1 around 0 92.2%

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

      1. unpow2 92.2%

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

   8. Simplified 92.2%

      \[\leadsto \cos th \cdot \color{blue}{\left(\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 2.4 \cdot 10^{-104}:\\ \;\;\;\;\cos th \cdot \left(\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 7.5 \cdot 10^{-39}:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot \frac{a2}{\sqrt{2}}\right)\\ \mathbf{elif}\;a2 \leq 1.26 \cdot 10^{+54}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left({2}^{-0.5} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 9: 68.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 2.55 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\\ \mathbf{elif}\;a2 \leq 7.5 \cdot 10^{-39}:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot \frac{a2}{\sqrt{2}}\right)\\ \mathbf{elif}\;a2 \leq 1.85 \cdot 10^{+54}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left({2}^{-0.5} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 2.55e-104)
   (* (sqrt 0.5) (* (cos th) (* a1 a1)))
   (if (<= a2 7.5e-39)
     (* (cos th) (* a2 (/ a2 (sqrt 2.0))))
     (if (<= a2 1.85e+54)
       (*
        (+ (* a1 a1) (* a2 a2))
        (* (pow 2.0 -0.5) (+ 1.0 (* -0.5 (* th th)))))
       (* (cos th) (* (* a2 a2) (sqrt 0.5)))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 2.55e-104) {
		tmp = sqrt(0.5) * (cos(th) * (a1 * a1));
	} else if (a2 <= 7.5e-39) {
		tmp = cos(th) * (a2 * (a2 / sqrt(2.0)));
	} else if (a2 <= 1.85e+54) {
		tmp = ((a1 * a1) + (a2 * a2)) * (pow(2.0, -0.5) * (1.0 + (-0.5 * (th * th))));
	} 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 (a2 <= 2.55d-104) then
        tmp = sqrt(0.5d0) * (cos(th) * (a1 * a1))
    else if (a2 <= 7.5d-39) then
        tmp = cos(th) * (a2 * (a2 / sqrt(2.0d0)))
    else if (a2 <= 1.85d+54) then
        tmp = ((a1 * a1) + (a2 * a2)) * ((2.0d0 ** (-0.5d0)) * (1.0d0 + ((-0.5d0) * (th * th))))
    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 (a2 <= 2.55e-104) {
		tmp = Math.sqrt(0.5) * (Math.cos(th) * (a1 * a1));
	} else if (a2 <= 7.5e-39) {
		tmp = Math.cos(th) * (a2 * (a2 / Math.sqrt(2.0)));
	} else if (a2 <= 1.85e+54) {
		tmp = ((a1 * a1) + (a2 * a2)) * (Math.pow(2.0, -0.5) * (1.0 + (-0.5 * (th * th))));
	} else {
		tmp = Math.cos(th) * ((a2 * a2) * Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 2.55e-104:
		tmp = math.sqrt(0.5) * (math.cos(th) * (a1 * a1))
	elif a2 <= 7.5e-39:
		tmp = math.cos(th) * (a2 * (a2 / math.sqrt(2.0)))
	elif a2 <= 1.85e+54:
		tmp = ((a1 * a1) + (a2 * a2)) * (math.pow(2.0, -0.5) * (1.0 + (-0.5 * (th * th))))
	else:
		tmp = math.cos(th) * ((a2 * a2) * math.sqrt(0.5))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 2.55e-104)
		tmp = Float64(sqrt(0.5) * Float64(cos(th) * Float64(a1 * a1)));
	elseif (a2 <= 7.5e-39)
		tmp = Float64(cos(th) * Float64(a2 * Float64(a2 / sqrt(2.0))));
	elseif (a2 <= 1.85e+54)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * Float64((2.0 ^ -0.5) * Float64(1.0 + Float64(-0.5 * Float64(th * th)))));
	else
		tmp = Float64(cos(th) * Float64(Float64(a2 * a2) * sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 2.55e-104)
		tmp = sqrt(0.5) * (cos(th) * (a1 * a1));
	elseif (a2 <= 7.5e-39)
		tmp = cos(th) * (a2 * (a2 / sqrt(2.0)));
	elseif (a2 <= 1.85e+54)
		tmp = ((a1 * a1) + (a2 * a2)) * ((2.0 ^ -0.5) * (1.0 + (-0.5 * (th * th))));
	else
		tmp = cos(th) * ((a2 * a2) * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 2.55e-104], N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a2, 7.5e-39], N[(N[Cos[th], $MachinePrecision] * N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a2, 1.85e+54], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Power[2.0, -0.5], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a2 < 2.54999999999999996e-104

   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/ 99.7%

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

      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. Step-by-step derivation

      1. fma-def 99.7%

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

      2. div-inv 99.6%

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

      3. add-sqr-sqrt 99.6%

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

      4. associate-*l* 99.6%

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

      5. hypot-def 99.6%

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

      6. hypot-def 99.6%

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

      7. pow1/2 99.6%

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

      8. pow-flip 99.7%

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

      9. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in a1 around inf 69.5%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left({a1}^{2} \cdot \cos th\right)} \]
   7. Step-by-step derivation

      1. unpow2 69.5%

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

      2. *-commutative 69.5%

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

   8. Simplified 69.5%

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

   if 2.54999999999999996e-104 < a2 < 7.49999999999999971e-39

   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 99.5%

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

      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/ 99.5%

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

      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 53.8%

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

      1. unpow2 53.8%

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

      2. associate-/l* 53.8%

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

      3. associate-/r/ 53.9%

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

   6. Simplified 53.9%

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

   if 7.49999999999999971e-39 < a2 < 1.8500000000000001e54

   1. Initial program 98.9%

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

      1. distribute-lft-out 98.9%

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

   3. Simplified 98.9%

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

      1. clear-num 99.0%

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

      2. associate-/r/ 98.9%

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

      3. pow1/2 98.9%

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

      4. pow-flip 99.4%

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

      5. metadata-eval 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in th around 0 69.1%

      \[\leadsto \left({2}^{-0.5} \cdot \color{blue}{\left(1 + -0.5 \cdot {th}^{2}\right)}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   7. Step-by-step derivation

      1. unpow2 39.2%

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

   8. Simplified 69.1%

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

   if 1.8500000000000001e54 < a2

   1. Initial program 99.8%

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

      1. distribute-lft-out 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

      4. fma-def 99.8%

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

   3. Simplified 99.8%

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

      1. fma-def 99.8%

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

      2. div-inv 99.8%

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

      3. add-sqr-sqrt 99.8%

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

      4. associate-*l* 99.7%

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

      5. hypot-def 99.7%

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

      6. hypot-def 99.7%

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

      7. pow1/2 99.7%

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

      8. pow-flip 99.8%

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

      9. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in a1 around 0 92.2%

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

      1. unpow2 92.2%

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

   8. Simplified 92.2%

      \[\leadsto \cos th \cdot \color{blue}{\left(\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 2.55 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\\ \mathbf{elif}\;a2 \leq 7.5 \cdot 10^{-39}:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot \frac{a2}{\sqrt{2}}\right)\\ \mathbf{elif}\;a2 \leq 1.85 \cdot 10^{+54}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left({2}^{-0.5} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 10: 68.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 2.55 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\\ \mathbf{elif}\;a2 \leq 7.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{\cos th \cdot a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{elif}\;a2 \leq 2.82 \cdot 10^{+48}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left({2}^{-0.5} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 2.55e-104)
   (* (sqrt 0.5) (* (cos th) (* a1 a1)))
   (if (<= a2 7.5e-39)
     (/ (* (cos th) a2) (/ (sqrt 2.0) a2))
     (if (<= a2 2.82e+48)
       (*
        (+ (* a1 a1) (* a2 a2))
        (* (pow 2.0 -0.5) (+ 1.0 (* -0.5 (* th th)))))
       (* (cos th) (* (* a2 a2) (sqrt 0.5)))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 2.55e-104) {
		tmp = sqrt(0.5) * (cos(th) * (a1 * a1));
	} else if (a2 <= 7.5e-39) {
		tmp = (cos(th) * a2) / (sqrt(2.0) / a2);
	} else if (a2 <= 2.82e+48) {
		tmp = ((a1 * a1) + (a2 * a2)) * (pow(2.0, -0.5) * (1.0 + (-0.5 * (th * th))));
	} 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 (a2 <= 2.55d-104) then
        tmp = sqrt(0.5d0) * (cos(th) * (a1 * a1))
    else if (a2 <= 7.5d-39) then
        tmp = (cos(th) * a2) / (sqrt(2.0d0) / a2)
    else if (a2 <= 2.82d+48) then
        tmp = ((a1 * a1) + (a2 * a2)) * ((2.0d0 ** (-0.5d0)) * (1.0d0 + ((-0.5d0) * (th * th))))
    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 (a2 <= 2.55e-104) {
		tmp = Math.sqrt(0.5) * (Math.cos(th) * (a1 * a1));
	} else if (a2 <= 7.5e-39) {
		tmp = (Math.cos(th) * a2) / (Math.sqrt(2.0) / a2);
	} else if (a2 <= 2.82e+48) {
		tmp = ((a1 * a1) + (a2 * a2)) * (Math.pow(2.0, -0.5) * (1.0 + (-0.5 * (th * th))));
	} else {
		tmp = Math.cos(th) * ((a2 * a2) * Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 2.55e-104:
		tmp = math.sqrt(0.5) * (math.cos(th) * (a1 * a1))
	elif a2 <= 7.5e-39:
		tmp = (math.cos(th) * a2) / (math.sqrt(2.0) / a2)
	elif a2 <= 2.82e+48:
		tmp = ((a1 * a1) + (a2 * a2)) * (math.pow(2.0, -0.5) * (1.0 + (-0.5 * (th * th))))
	else:
		tmp = math.cos(th) * ((a2 * a2) * math.sqrt(0.5))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 2.55e-104)
		tmp = Float64(sqrt(0.5) * Float64(cos(th) * Float64(a1 * a1)));
	elseif (a2 <= 7.5e-39)
		tmp = Float64(Float64(cos(th) * a2) / Float64(sqrt(2.0) / a2));
	elseif (a2 <= 2.82e+48)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * Float64((2.0 ^ -0.5) * Float64(1.0 + Float64(-0.5 * Float64(th * th)))));
	else
		tmp = Float64(cos(th) * Float64(Float64(a2 * a2) * sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 2.55e-104)
		tmp = sqrt(0.5) * (cos(th) * (a1 * a1));
	elseif (a2 <= 7.5e-39)
		tmp = (cos(th) * a2) / (sqrt(2.0) / a2);
	elseif (a2 <= 2.82e+48)
		tmp = ((a1 * a1) + (a2 * a2)) * ((2.0 ^ -0.5) * (1.0 + (-0.5 * (th * th))));
	else
		tmp = cos(th) * ((a2 * a2) * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 2.55e-104], N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a2, 7.5e-39], N[(N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision], If[LessEqual[a2, 2.82e+48], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Power[2.0, -0.5], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a2 < 2.54999999999999996e-104

   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/ 99.7%

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

      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. Step-by-step derivation

      1. fma-def 99.7%

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

      2. div-inv 99.6%

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

      3. add-sqr-sqrt 99.6%

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

      4. associate-*l* 99.6%

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

      5. hypot-def 99.6%

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

      6. hypot-def 99.6%

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

      7. pow1/2 99.6%

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

      8. pow-flip 99.7%

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

      9. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in a1 around inf 69.5%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left({a1}^{2} \cdot \cos th\right)} \]
   7. Step-by-step derivation

      1. unpow2 69.5%

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

      2. *-commutative 69.5%

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

   8. Simplified 69.5%

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

   if 2.54999999999999996e-104 < a2 < 7.49999999999999971e-39

   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 99.5%

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

      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/ 99.5%

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

      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 53.8%

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

      1. unpow2 53.8%

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

      2. associate-/l* 53.8%

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

   6. Simplified 53.8%

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

      1. associate-*r/ 53.9%

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

   8. Applied egg-rr 53.9%

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

   if 7.49999999999999971e-39 < a2 < 2.8199999999999999e48

   1. Initial program 98.9%

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

      1. distribute-lft-out 98.9%

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

   3. Simplified 98.9%

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

      1. clear-num 99.0%

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

      2. associate-/r/ 98.9%

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

      3. pow1/2 98.9%

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

      4. pow-flip 99.4%

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

      5. metadata-eval 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in th around 0 69.1%

      \[\leadsto \left({2}^{-0.5} \cdot \color{blue}{\left(1 + -0.5 \cdot {th}^{2}\right)}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   7. Step-by-step derivation

      1. unpow2 39.2%

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

   8. Simplified 69.1%

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

   if 2.8199999999999999e48 < a2

   1. Initial program 99.8%

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

      1. distribute-lft-out 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

      4. fma-def 99.8%

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

   3. Simplified 99.8%

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

      1. fma-def 99.8%

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

      2. div-inv 99.8%

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

      3. add-sqr-sqrt 99.8%

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

      4. associate-*l* 99.7%

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

      5. hypot-def 99.7%

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

      6. hypot-def 99.7%

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

      7. pow1/2 99.7%

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

      8. pow-flip 99.8%

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

      9. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in a1 around 0 92.2%

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

      1. unpow2 92.2%

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

   8. Simplified 92.2%

      \[\leadsto \cos th \cdot \color{blue}{\left(\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 2.55 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\\ \mathbf{elif}\;a2 \leq 7.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{\cos th \cdot a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{elif}\;a2 \leq 2.82 \cdot 10^{+48}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left({2}^{-0.5} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 11: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (cos(th) * (2.0d0 ** (-0.5d0))) * ((a1 * a1) + (a2 * a2))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

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

Alternative 12: 65.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;a1 \leq -1 \cdot 10^{+231}:\\ \;\;\;\;t_1 \cdot \frac{1 + -0.5 \cdot \left(th \cdot th\right)}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -4.9 \cdot 10^{-104}:\\ \;\;\;\;t_1 \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{\cos th \cdot a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= a1 -1e+231)
     (* t_1 (/ (+ 1.0 (* -0.5 (* th th))) (sqrt 2.0)))
     (if (<= a1 -4.9e-104)
       (* t_1 (sqrt 0.5))
       (* a2 (/ (* (cos th) a2) (sqrt 2.0)))))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (a1 <= -1e+231) {
		tmp = t_1 * ((1.0 + (-0.5 * (th * th))) / sqrt(2.0));
	} else if (a1 <= -4.9e-104) {
		tmp = t_1 * sqrt(0.5);
	} else {
		tmp = a2 * ((cos(th) * a2) / sqrt(2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (a1 <= (-1d+231)) then
        tmp = t_1 * ((1.0d0 + ((-0.5d0) * (th * th))) / sqrt(2.0d0))
    else if (a1 <= (-4.9d-104)) then
        tmp = t_1 * sqrt(0.5d0)
    else
        tmp = a2 * ((cos(th) * a2) / sqrt(2.0d0))
    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 (a1 <= -1e+231) {
		tmp = t_1 * ((1.0 + (-0.5 * (th * th))) / Math.sqrt(2.0));
	} else if (a1 <= -4.9e-104) {
		tmp = t_1 * Math.sqrt(0.5);
	} else {
		tmp = a2 * ((Math.cos(th) * a2) / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if a1 <= -1e+231:
		tmp = t_1 * ((1.0 + (-0.5 * (th * th))) / math.sqrt(2.0))
	elif a1 <= -4.9e-104:
		tmp = t_1 * math.sqrt(0.5)
	else:
		tmp = a2 * ((math.cos(th) * a2) / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (a1 <= -1e+231)
		tmp = Float64(t_1 * Float64(Float64(1.0 + Float64(-0.5 * Float64(th * th))) / sqrt(2.0)));
	elseif (a1 <= -4.9e-104)
		tmp = Float64(t_1 * sqrt(0.5));
	else
		tmp = Float64(a2 * Float64(Float64(cos(th) * a2) / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (a1 <= -1e+231)
		tmp = t_1 * ((1.0 + (-0.5 * (th * th))) / sqrt(2.0));
	elseif (a1 <= -4.9e-104)
		tmp = t_1 * sqrt(0.5);
	else
		tmp = a2 * ((cos(th) * a2) / sqrt(2.0));
	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[a1, -1e+231], N[(t$95$1 * N[(N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a1, -4.9e-104], N[(t$95$1 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(a2 * N[(N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a1 < -1.0000000000000001e231

   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%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in th around 0 66.7%

      \[\leadsto \frac{\color{blue}{1 + -0.5 \cdot {th}^{2}}}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   5. Step-by-step derivation

      1. unpow2 34.6%

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

   6. Simplified 66.7%

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

   if -1.0000000000000001e231 < a1 < -4.9000000000000003e-104

   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 99.4%

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

      5. metadata-eval 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in th around 0 62.1%

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

   if -4.9000000000000003e-104 < 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 71.1%

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

      1. unpow2 71.1%

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

      2. associate-*l* 71.1%

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

      3. associate-*r/ 71.1%

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

   6. Simplified 71.1%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \leq -1 \cdot 10^{+231}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \frac{1 + -0.5 \cdot \left(th \cdot th\right)}{\sqrt{2}}\\ \mathbf{elif}\;a1 \leq -4.9 \cdot 10^{-104}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{\cos th \cdot a2}{\sqrt{2}}\\ \end{array} \]

Alternative 13: 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 14: 66.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;a1 \leq -2 \cdot 10^{+227}:\\ \;\;\;\;t_1 \cdot \frac{1 + -0.5 \cdot \left(th \cdot th\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= a1 -2e+227)
     (* t_1 (/ (+ 1.0 (* -0.5 (* th th))) (sqrt 2.0)))
     (* t_1 (sqrt 0.5)))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (a1 <= -2e+227) {
		tmp = t_1 * ((1.0 + (-0.5 * (th * th))) / sqrt(2.0));
	} 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 (a1 <= (-2d+227)) then
        tmp = t_1 * ((1.0d0 + ((-0.5d0) * (th * th))) / sqrt(2.0d0))
    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 (a1 <= -2e+227) {
		tmp = t_1 * ((1.0 + (-0.5 * (th * th))) / Math.sqrt(2.0));
	} 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 a1 <= -2e+227:
		tmp = t_1 * ((1.0 + (-0.5 * (th * th))) / math.sqrt(2.0))
	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 (a1 <= -2e+227)
		tmp = Float64(t_1 * Float64(Float64(1.0 + Float64(-0.5 * Float64(th * th))) / sqrt(2.0)));
	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 (a1 <= -2e+227)
		tmp = t_1 * ((1.0 + (-0.5 * (th * th))) / sqrt(2.0));
	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[a1, -2e+227], N[(t$95$1 * N[(N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a1 < -2.0000000000000002e227

   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%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in th around 0 66.7%

      \[\leadsto \frac{\color{blue}{1 + -0.5 \cdot {th}^{2}}}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   5. Step-by-step derivation

      1. unpow2 34.6%

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

   6. Simplified 66.7%

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

   if -2.0000000000000002e227 < 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 th around 0 66.7%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \leq -2 \cdot 10^{+227}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \frac{1 + -0.5 \cdot \left(th \cdot th\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 15: 46.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 3.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{a1}{\sqrt{2} \cdot \frac{1}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 3.8e-106)
   (/ a1 (* (sqrt 2.0) (/ 1.0 a1)))
   (* (* a2 a2) (sqrt 0.5))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 3.8e-106) {
		tmp = a1 / (sqrt(2.0) * (1.0 / a1));
	} else {
		tmp = (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 (a2 <= 3.8d-106) then
        tmp = a1 / (sqrt(2.0d0) * (1.0d0 / a1))
    else
        tmp = (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 (a2 <= 3.8e-106) {
		tmp = a1 / (Math.sqrt(2.0) * (1.0 / a1));
	} else {
		tmp = (a2 * a2) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 3.8e-106:
		tmp = a1 / (math.sqrt(2.0) * (1.0 / a1))
	else:
		tmp = (a2 * a2) * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 3.8e-106)
		tmp = Float64(a1 / Float64(sqrt(2.0) * Float64(1.0 / a1)));
	else
		tmp = Float64(Float64(a2 * a2) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 3.8e-106)
		tmp = a1 / (sqrt(2.0) * (1.0 / a1));
	else
		tmp = (a2 * a2) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 3.8e-106], N[(a1 / N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 / a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 3.7999999999999999e-106

   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 67.1%

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

   5. Taylor expanded in a1 around inf 51.6%

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

      1. unpow2 51.6%

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

      2. associate-*r/ 51.6%

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

   7. Simplified 51.6%

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

      1. clear-num 51.6%

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

      2. un-div-inv 51.6%

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

   9. Applied egg-rr 51.6%

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

       1. div-inv 51.6%

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

   11. Applied egg-rr 51.6%

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

   if 3.7999999999999999e-106 < a2

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   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 68.3%

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

   7. Taylor expanded in a1 around 0 55.6%

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

      1. unpow2 55.6%

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

   9. Simplified 55.6%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 3.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{a1}{\sqrt{2} \cdot \frac{1}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 16: 66.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $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. Taylor expanded in th around 0 67.5%

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

7. Final simplification 67.5%

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

Alternative 17: 46.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 3.8 \cdot 10^{-106}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot {2}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 3.8e-106) (* a1 (* a1 (pow 2.0 -0.5))) (* (* a2 a2) (sqrt 0.5))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 3.8e-106) {
		tmp = a1 * (a1 * pow(2.0, -0.5));
	} else {
		tmp = (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 (a2 <= 3.8d-106) then
        tmp = a1 * (a1 * (2.0d0 ** (-0.5d0)))
    else
        tmp = (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 (a2 <= 3.8e-106) {
		tmp = a1 * (a1 * Math.pow(2.0, -0.5));
	} else {
		tmp = (a2 * a2) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 3.8e-106:
		tmp = a1 * (a1 * math.pow(2.0, -0.5))
	else:
		tmp = (a2 * a2) * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 3.8e-106)
		tmp = Float64(a1 * Float64(a1 * (2.0 ^ -0.5)));
	else
		tmp = Float64(Float64(a2 * a2) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 3.8e-106)
		tmp = a1 * (a1 * (2.0 ^ -0.5));
	else
		tmp = (a2 * a2) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 3.8e-106], N[(a1 * N[(a1 * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 3.7999999999999999e-106

   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 67.1%

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

   5. Taylor expanded in a1 around inf 51.6%

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

      1. unpow2 51.6%

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

      2. associate-*r/ 51.6%

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

   7. Simplified 51.6%

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

      1. div-inv 51.6%

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

      2. pow1/2 51.6%

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

      3. pow-flip 51.6%

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

      4. metadata-eval 51.6%

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

   9. Applied egg-rr 51.6%

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

   if 3.7999999999999999e-106 < a2

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   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 68.3%

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

   7. Taylor expanded in a1 around 0 55.6%

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

      1. unpow2 55.6%

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

   9. Simplified 55.6%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 3.8 \cdot 10^{-106}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot {2}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 18: 46.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 3.8 \cdot 10^{-106}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 3.8e-106) (* a1 (/ a1 (sqrt 2.0))) (* a2 (/ a2 (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 3.8e-106) {
		tmp = a1 * (a1 / sqrt(2.0));
	} else {
		tmp = a2 * (a2 / sqrt(2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (a2 <= 3.8d-106) then
        tmp = a1 * (a1 / sqrt(2.0d0))
    else
        tmp = a2 * (a2 / sqrt(2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 3.8e-106) {
		tmp = a1 * (a1 / Math.sqrt(2.0));
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 3.8e-106:
		tmp = a1 * (a1 / math.sqrt(2.0))
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 3.8e-106)
		tmp = Float64(a1 * Float64(a1 / sqrt(2.0)));
	else
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 3.8e-106)
		tmp = a1 * (a1 / sqrt(2.0));
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 3.8e-106], N[(a1 * N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 3.7999999999999999e-106

   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 67.1%

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

   5. Taylor expanded in a1 around inf 51.6%

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

      1. unpow2 51.6%

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

      2. associate-*r/ 51.6%

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

   7. Simplified 51.6%

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

   if 3.7999999999999999e-106 < a2

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. 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 72.7%

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

      1. unpow2 72.7%

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

      2. associate-/l* 72.7%

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

   6. Simplified 72.7%

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

      1. associate-*r/ 72.8%

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

   8. Applied egg-rr 72.8%

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

   9. Taylor expanded in th around 0 51.5%

      \[\leadsto \frac{\color{blue}{\left(1 + -0.5 \cdot {th}^{2}\right)} \cdot a2}{\frac{\sqrt{2}}{a2}} \]
   10. Step-by-step derivation

       1. unpow2 51.5%

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

   11. Simplified 51.5%

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

   12. Taylor expanded in th around 0 55.6%

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

       1. unpow2 55.6%

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

       2. associate-*l/ 55.6%

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

       3. *-commutative 55.6%

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

   14. Simplified 55.6%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 3.8 \cdot 10^{-106}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 19: 46.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 2.2 \cdot 10^{-106}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 2.2e-106) (* (* a1 a1) (sqrt 0.5)) (* a2 (/ a2 (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 2.2e-106) {
		tmp = (a1 * a1) * sqrt(0.5);
	} else {
		tmp = a2 * (a2 / sqrt(2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (a2 <= 2.2d-106) then
        tmp = (a1 * a1) * sqrt(0.5d0)
    else
        tmp = a2 * (a2 / sqrt(2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 2.2e-106) {
		tmp = (a1 * a1) * Math.sqrt(0.5);
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 2.2e-106:
		tmp = (a1 * a1) * math.sqrt(0.5)
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 2.2e-106)
		tmp = Float64(Float64(a1 * a1) * sqrt(0.5));
	else
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 2.2e-106)
		tmp = (a1 * a1) * sqrt(0.5);
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 2.2e-106], N[(N[(a1 * a1), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 2.19999999999999994e-106

   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 67.1%

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

   7. Taylor expanded in a1 around inf 51.6%

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

      1. unpow2 51.6%

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

   9. Simplified 51.6%

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

   if 2.19999999999999994e-106 < a2

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. 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 72.7%

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

      1. unpow2 72.7%

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

      2. associate-/l* 72.7%

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

   6. Simplified 72.7%

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

      1. associate-*r/ 72.8%

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

   8. Applied egg-rr 72.8%

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

   9. Taylor expanded in th around 0 51.5%

      \[\leadsto \frac{\color{blue}{\left(1 + -0.5 \cdot {th}^{2}\right)} \cdot a2}{\frac{\sqrt{2}}{a2}} \]
   10. Step-by-step derivation

       1. unpow2 51.5%

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

   11. Simplified 51.5%

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

   12. Taylor expanded in th around 0 55.6%

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

       1. unpow2 55.6%

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

       2. associate-*l/ 55.6%

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

       3. *-commutative 55.6%

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

   14. Simplified 55.6%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 2.2 \cdot 10^{-106}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 20: 46.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 2.8 \cdot 10^{-106}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 2.8e-106) (* (* a1 a1) (sqrt 0.5)) (* (* a2 a2) (sqrt 0.5))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 2.8e-106) {
		tmp = (a1 * a1) * sqrt(0.5);
	} else {
		tmp = (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 (a2 <= 2.8d-106) then
        tmp = (a1 * a1) * sqrt(0.5d0)
    else
        tmp = (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 (a2 <= 2.8e-106) {
		tmp = (a1 * a1) * Math.sqrt(0.5);
	} else {
		tmp = (a2 * a2) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 2.8e-106:
		tmp = (a1 * a1) * math.sqrt(0.5)
	else:
		tmp = (a2 * a2) * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 2.8e-106)
		tmp = Float64(Float64(a1 * a1) * sqrt(0.5));
	else
		tmp = Float64(Float64(a2 * a2) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 2.8e-106)
		tmp = (a1 * a1) * sqrt(0.5);
	else
		tmp = (a2 * a2) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 2.8e-106], N[(N[(a1 * a1), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 2.79999999999999988e-106

   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 67.1%

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

   7. Taylor expanded in a1 around inf 51.6%

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

      1. unpow2 51.6%

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

   9. Simplified 51.6%

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

   if 2.79999999999999988e-106 < a2

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   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 68.3%

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

   7. Taylor expanded in a1 around 0 55.6%

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

      1. unpow2 55.6%

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

   9. Simplified 55.6%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 2.8 \cdot 10^{-106}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 21: 46.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 3.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 3.8e-106) (/ a1 (/ (sqrt 2.0) a1)) (* (* a2 a2) (sqrt 0.5))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 3.8e-106) {
		tmp = a1 / (sqrt(2.0) / a1);
	} else {
		tmp = (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 (a2 <= 3.8d-106) then
        tmp = a1 / (sqrt(2.0d0) / a1)
    else
        tmp = (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 (a2 <= 3.8e-106) {
		tmp = a1 / (Math.sqrt(2.0) / a1);
	} else {
		tmp = (a2 * a2) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 3.8e-106:
		tmp = a1 / (math.sqrt(2.0) / a1)
	else:
		tmp = (a2 * a2) * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 3.8e-106)
		tmp = Float64(a1 / Float64(sqrt(2.0) / a1));
	else
		tmp = Float64(Float64(a2 * a2) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 3.8e-106)
		tmp = a1 / (sqrt(2.0) / a1);
	else
		tmp = (a2 * a2) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 3.8e-106], N[(a1 / N[(N[Sqrt[2.0], $MachinePrecision] / a1), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 3.7999999999999999e-106

   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 67.1%

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

   5. Taylor expanded in a1 around inf 51.6%

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

      1. unpow2 51.6%

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

      2. associate-*r/ 51.6%

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

   7. Simplified 51.6%

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

      1. clear-num 51.6%

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

      2. un-div-inv 51.6%

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

   9. Applied egg-rr 51.6%

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

   if 3.7999999999999999e-106 < a2

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   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 68.3%

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

   7. Taylor expanded in a1 around 0 55.6%

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

      1. unpow2 55.6%

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

   9. Simplified 55.6%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 3.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 22: 40.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return a1 * (a1 / 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 / sqrt(2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a1 * N[(a1 / 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. Taylor expanded in th around 0 67.4%

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

5. Taylor expanded in a1 around inf 42.4%

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

   1. unpow2 42.4%

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

   2. associate-*r/ 42.4%

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

7. Simplified 42.4%

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

8. Final simplification 42.4%

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

Reproduce

herbie shell --seed 2023182 
(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))))