Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 10.6s

Alternatives: 8

Speedup: 2.0×

• 44.5% 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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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) * cos(th))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.7%

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

   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 inf 99.7%

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

   1. *-commutative 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 57.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

   2. cos-neg 99.5%

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

   3. associate-/r/ 99.3%

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

   4. cos-neg 99.3%

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

   5. fma-def 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in a1 around 0 62.7%

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

   1. *-un-lft-identity 62.7%

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

   2. pow2 62.7%

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

   3. times-frac 62.7%

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

6. Applied egg-rr 62.7%

   \[\leadsto \frac{\cos th}{\color{blue}{\frac{1}{a2} \cdot \frac{\sqrt{2}}{a2}}} \]
7. Step-by-step derivation

   1. associate-*l/ 62.7%

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

   2. *-lft-identity 62.7%

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

8. Simplified 62.7%

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

   1. associate-/r/ 63.0%

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

   2. div-inv 63.0%

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

   3. clear-num 62.9%

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

   4. div-inv 62.9%

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

   5. add-sqr-sqrt 62.9%

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

   6. sqrt-unprod 62.9%

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

   7. frac-times 62.9%

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

   8. metadata-eval 62.9%

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

   9. rem-square-sqrt 63.0%

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

   10. metadata-eval 63.0%

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

10. Applied egg-rr 63.0%

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

11. Final simplification 63.0%

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

Alternative 3: 57.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

   2. cos-neg 99.5%

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

   3. associate-/r/ 99.3%

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

   4. cos-neg 99.3%

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

   5. fma-def 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in a1 around 0 62.7%

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

   1. *-un-lft-identity 62.7%

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

   2. pow2 62.7%

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

   3. times-frac 62.7%

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

6. Applied egg-rr 62.7%

   \[\leadsto \frac{\cos th}{\color{blue}{\frac{1}{a2} \cdot \frac{\sqrt{2}}{a2}}} \]
7. Step-by-step derivation

   1. associate-*l/ 62.7%

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

   2. *-lft-identity 62.7%

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

8. Simplified 62.7%

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

   1. clear-num 62.6%

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

   2. associate-/r/ 62.7%

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

   3. associate-/l/ 62.7%

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

   4. clear-num 62.9%

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

   5. div-inv 62.9%

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

   6. add-sqr-sqrt 62.9%

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

   7. sqrt-unprod 62.9%

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

   8. frac-times 62.9%

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

   9. metadata-eval 62.9%

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

   10. rem-square-sqrt 63.0%

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

   11. metadata-eval 63.0%

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

   12. associate-*r* 63.0%

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

   13. *-commutative 63.0%

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

   14. associate-*r* 63.0%

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

10. Applied egg-rr 63.0%

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

11. Taylor expanded in th around inf 63.0%

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

    1. *-commutative 63.0%

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

    2. *-commutative 63.0%

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

    3. associate-*r* 63.0%

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

    4. *-commutative 63.0%

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

13. Simplified 63.0%

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

14. Final simplification 63.0%

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

Alternative 4: 57.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

   2. cos-neg 99.5%

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

   3. associate-/r/ 99.3%

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

   4. cos-neg 99.3%

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

   5. fma-def 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in a1 around 0 62.7%

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

   1. pow2 62.7%

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

   2. associate-/r/ 62.9%

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

   3. pow1/2 62.9%

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

   4. metadata-eval 62.9%

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

   5. pow-prod-up 63.0%

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

   6. associate-/l/ 63.0%

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

   7. associate-*r* 63.0%

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

6. Applied egg-rr 63.0%

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

7. Final simplification 63.0%

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

Alternative 5: 40.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.8 \cdot 10^{+90}:\\ \;\;\;\;a2 \cdot \left(\sqrt{0.5} \cdot a2\right)\\ \mathbf{elif}\;th \leq 6 \cdot 10^{+205} \lor \neg \left(th \leq 4.8 \cdot 10^{+260}\right):\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot \left(-a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{\sqrt{2}}{a2}}{a2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 1.8e+90)
   (* a2 (* (sqrt 0.5) a2))
   (if (or (<= th 6e+205) (not (<= th 4.8e+260)))
     (* (/ a2 (sqrt 2.0)) (- a2))
     (/ 1.0 (/ (/ (sqrt 2.0) a2) a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.8e+90) {
		tmp = a2 * (sqrt(0.5) * a2);
	} else if ((th <= 6e+205) || !(th <= 4.8e+260)) {
		tmp = (a2 / sqrt(2.0)) * -a2;
	} else {
		tmp = 1.0 / ((sqrt(2.0) / a2) / a2);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 1.8d+90) then
        tmp = a2 * (sqrt(0.5d0) * a2)
    else if ((th <= 6d+205) .or. (.not. (th <= 4.8d+260))) then
        tmp = (a2 / sqrt(2.0d0)) * -a2
    else
        tmp = 1.0d0 / ((sqrt(2.0d0) / a2) / a2)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.8e+90) {
		tmp = a2 * (Math.sqrt(0.5) * a2);
	} else if ((th <= 6e+205) || !(th <= 4.8e+260)) {
		tmp = (a2 / Math.sqrt(2.0)) * -a2;
	} else {
		tmp = 1.0 / ((Math.sqrt(2.0) / a2) / a2);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 1.8e+90:
		tmp = a2 * (math.sqrt(0.5) * a2)
	elif (th <= 6e+205) or not (th <= 4.8e+260):
		tmp = (a2 / math.sqrt(2.0)) * -a2
	else:
		tmp = 1.0 / ((math.sqrt(2.0) / a2) / a2)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 1.8e+90)
		tmp = Float64(a2 * Float64(sqrt(0.5) * a2));
	elseif ((th <= 6e+205) || !(th <= 4.8e+260))
		tmp = Float64(Float64(a2 / sqrt(2.0)) * Float64(-a2));
	else
		tmp = Float64(1.0 / Float64(Float64(sqrt(2.0) / a2) / a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 1.8e+90)
		tmp = a2 * (sqrt(0.5) * a2);
	elseif ((th <= 6e+205) || ~((th <= 4.8e+260)))
		tmp = (a2 / sqrt(2.0)) * -a2;
	else
		tmp = 1.0 / ((sqrt(2.0) / a2) / a2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 1.8e+90], N[(a2 * N[(N[Sqrt[0.5], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[th, 6e+205], N[Not[LessEqual[th, 4.8e+260]], $MachinePrecision]], N[(N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * (-a2)), $MachinePrecision], N[(1.0 / N[(N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 1.8e90

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. cos-neg 99.6%

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

      3. associate-/r/ 99.6%

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

      4. cos-neg 99.6%

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

      5. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 61.3%

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

   5. Taylor expanded in th around 0 43.1%

      \[\leadsto \frac{\color{blue}{1}}{\frac{\sqrt{2}}{{a2}^{2}}} \]
   6. Step-by-step derivation

      1. pow2 43.1%

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

      2. associate-/r/ 43.1%

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

      3. add-sqr-sqrt 43.1%

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

      4. sqrt-unprod 43.1%

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

      5. frac-times 43.1%

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

      6. metadata-eval 43.1%

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

      7. rem-square-sqrt 43.1%

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

      8. metadata-eval 43.1%

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

      9. associate-*r* 43.1%

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

   7. Applied egg-rr 43.1%

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

   if 1.8e90 < th < 5.9999999999999999e205 or 4.8000000000000002e260 < th

   1. Initial program 99.3%

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

      1. distribute-lft-out 99.3%

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

      2. cos-neg 99.3%

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

      3. associate-/r/ 97.7%

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

      4. cos-neg 97.7%

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

      5. fma-def 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in a1 around 0 59.9%

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

   5. Taylor expanded in th around 0 14.9%

      \[\leadsto \frac{\color{blue}{1}}{\frac{\sqrt{2}}{{a2}^{2}}} \]
   6. Step-by-step derivation

      1. pow2 14.9%

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

      2. associate-/l/ 14.9%

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

      3. clear-num 14.8%

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

      4. frac-2neg 14.8%

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

      5. associate-/r/ 14.8%

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

   7. Applied egg-rr 14.8%

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

   9. Applied egg-rr 12.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{a2}{\sqrt{2}}\right)} - 1\right)} \cdot \left(-a2\right) \]
   10. Step-by-step derivation

       1. expm1-def 12.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a2}{\sqrt{2}}\right)\right)} \cdot \left(-a2\right) \]

       2. expm1-log1p 24.2%

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

   11. Simplified 24.2%

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

   if 5.9999999999999999e205 < th < 4.8000000000000002e260

   1. Initial program 99.9%

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

      1. distribute-lft-out 99.9%

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

      2. cos-neg 99.9%

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

      3. associate-/r/ 99.4%

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

      4. cos-neg 99.4%

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

      5. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in a1 around 0 91.8%

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

      1. *-un-lft-identity 91.8%

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

      2. pow2 91.8%

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

      3. times-frac 91.8%

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

   6. Applied egg-rr 91.8%

      \[\leadsto \frac{\cos th}{\color{blue}{\frac{1}{a2} \cdot \frac{\sqrt{2}}{a2}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 91.6%

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

      2. *-lft-identity 91.6%

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

   8. Simplified 91.6%

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

   9. Taylor expanded in th around 0 63.2%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.8 \cdot 10^{+90}:\\ \;\;\;\;a2 \cdot \left(\sqrt{0.5} \cdot a2\right)\\ \mathbf{elif}\;th \leq 6 \cdot 10^{+205} \lor \neg \left(th \leq 4.8 \cdot 10^{+260}\right):\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot \left(-a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{\sqrt{2}}{a2}}{a2}}\\ \end{array} \]

Alternative 6: 64.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.8 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{elif}\;th \leq 6 \cdot 10^{+205} \lor \neg \left(th \leq 4.8 \cdot 10^{+260}\right):\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot \left(-a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{\sqrt{2}}{a2}}{a2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 1.8e+90)
   (* (sqrt 0.5) (+ (* a1 a1) (* a2 a2)))
   (if (or (<= th 6e+205) (not (<= th 4.8e+260)))
     (* (/ a2 (sqrt 2.0)) (- a2))
     (/ 1.0 (/ (/ (sqrt 2.0) a2) a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.8e+90) {
		tmp = sqrt(0.5) * ((a1 * a1) + (a2 * a2));
	} else if ((th <= 6e+205) || !(th <= 4.8e+260)) {
		tmp = (a2 / sqrt(2.0)) * -a2;
	} else {
		tmp = 1.0 / ((sqrt(2.0) / a2) / a2);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 1.8d+90) then
        tmp = sqrt(0.5d0) * ((a1 * a1) + (a2 * a2))
    else if ((th <= 6d+205) .or. (.not. (th <= 4.8d+260))) then
        tmp = (a2 / sqrt(2.0d0)) * -a2
    else
        tmp = 1.0d0 / ((sqrt(2.0d0) / a2) / a2)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.8e+90) {
		tmp = Math.sqrt(0.5) * ((a1 * a1) + (a2 * a2));
	} else if ((th <= 6e+205) || !(th <= 4.8e+260)) {
		tmp = (a2 / Math.sqrt(2.0)) * -a2;
	} else {
		tmp = 1.0 / ((Math.sqrt(2.0) / a2) / a2);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 1.8e+90:
		tmp = math.sqrt(0.5) * ((a1 * a1) + (a2 * a2))
	elif (th <= 6e+205) or not (th <= 4.8e+260):
		tmp = (a2 / math.sqrt(2.0)) * -a2
	else:
		tmp = 1.0 / ((math.sqrt(2.0) / a2) / a2)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 1.8e+90)
		tmp = Float64(sqrt(0.5) * Float64(Float64(a1 * a1) + Float64(a2 * a2)));
	elseif ((th <= 6e+205) || !(th <= 4.8e+260))
		tmp = Float64(Float64(a2 / sqrt(2.0)) * Float64(-a2));
	else
		tmp = Float64(1.0 / Float64(Float64(sqrt(2.0) / a2) / a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 1.8e+90)
		tmp = sqrt(0.5) * ((a1 * a1) + (a2 * a2));
	elseif ((th <= 6e+205) || ~((th <= 4.8e+260)))
		tmp = (a2 / sqrt(2.0)) * -a2;
	else
		tmp = 1.0 / ((sqrt(2.0) / a2) / a2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 1.8e+90], N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[th, 6e+205], N[Not[LessEqual[th, 4.8e+260]], $MachinePrecision]], N[(N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * (-a2)), $MachinePrecision], N[(1.0 / N[(N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 1.8e90

   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/ 99.5%

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

      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 99.7%

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

      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 69.8%

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

   if 1.8e90 < th < 5.9999999999999999e205 or 4.8000000000000002e260 < th

   1. Initial program 99.3%

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

      1. distribute-lft-out 99.3%

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

      2. cos-neg 99.3%

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

      3. associate-/r/ 97.7%

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

      4. cos-neg 97.7%

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

      5. fma-def 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in a1 around 0 59.9%

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

   5. Taylor expanded in th around 0 14.9%

      \[\leadsto \frac{\color{blue}{1}}{\frac{\sqrt{2}}{{a2}^{2}}} \]
   6. Step-by-step derivation

      1. pow2 14.9%

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

      2. associate-/l/ 14.9%

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

      3. clear-num 14.8%

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

      4. frac-2neg 14.8%

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

      5. associate-/r/ 14.8%

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

   7. Applied egg-rr 14.8%

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

   9. Applied egg-rr 12.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{a2}{\sqrt{2}}\right)} - 1\right)} \cdot \left(-a2\right) \]
   10. Step-by-step derivation

       1. expm1-def 12.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a2}{\sqrt{2}}\right)\right)} \cdot \left(-a2\right) \]

       2. expm1-log1p 24.2%

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

   11. Simplified 24.2%

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

   if 5.9999999999999999e205 < th < 4.8000000000000002e260

   1. Initial program 99.9%

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

      1. distribute-lft-out 99.9%

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

      2. cos-neg 99.9%

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

      3. associate-/r/ 99.4%

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

      4. cos-neg 99.4%

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

      5. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in a1 around 0 91.8%

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

      1. *-un-lft-identity 91.8%

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

      2. pow2 91.8%

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

      3. times-frac 91.8%

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

   6. Applied egg-rr 91.8%

      \[\leadsto \frac{\cos th}{\color{blue}{\frac{1}{a2} \cdot \frac{\sqrt{2}}{a2}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 91.6%

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

      2. *-lft-identity 91.6%

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

   8. Simplified 91.6%

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

   9. Taylor expanded in th around 0 63.2%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.8 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{elif}\;th \leq 6 \cdot 10^{+205} \lor \neg \left(th \leq 4.8 \cdot 10^{+260}\right):\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot \left(-a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{\sqrt{2}}{a2}}{a2}}\\ \end{array} \]

Alternative 7: 40.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.8 \cdot 10^{+90} \lor \neg \left(th \leq 6 \cdot 10^{+205}\right) \land th \leq 4.8 \cdot 10^{+260}:\\ \;\;\;\;a2 \cdot \left(\sqrt{0.5} \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot \left(-a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (or (<= th 1.8e+90) (and (not (<= th 6e+205)) (<= th 4.8e+260)))
   (* a2 (* (sqrt 0.5) a2))
   (* (/ a2 (sqrt 2.0)) (- a2))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 1.8e+90) || (!(th <= 6e+205) && (th <= 4.8e+260))) {
		tmp = a2 * (sqrt(0.5) * a2);
	} else {
		tmp = (a2 / sqrt(2.0)) * -a2;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((th <= 1.8d+90) .or. (.not. (th <= 6d+205)) .and. (th <= 4.8d+260)) then
        tmp = a2 * (sqrt(0.5d0) * a2)
    else
        tmp = (a2 / sqrt(2.0d0)) * -a2
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 1.8e+90) || (!(th <= 6e+205) && (th <= 4.8e+260))) {
		tmp = a2 * (Math.sqrt(0.5) * a2);
	} else {
		tmp = (a2 / Math.sqrt(2.0)) * -a2;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (th <= 1.8e+90) or (not (th <= 6e+205) and (th <= 4.8e+260)):
		tmp = a2 * (math.sqrt(0.5) * a2)
	else:
		tmp = (a2 / math.sqrt(2.0)) * -a2
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if ((th <= 1.8e+90) || (!(th <= 6e+205) && (th <= 4.8e+260)))
		tmp = Float64(a2 * Float64(sqrt(0.5) * a2));
	else
		tmp = Float64(Float64(a2 / sqrt(2.0)) * Float64(-a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((th <= 1.8e+90) || (~((th <= 6e+205)) && (th <= 4.8e+260)))
		tmp = a2 * (sqrt(0.5) * a2);
	else
		tmp = (a2 / sqrt(2.0)) * -a2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[Or[LessEqual[th, 1.8e+90], And[N[Not[LessEqual[th, 6e+205]], $MachinePrecision], LessEqual[th, 4.8e+260]]], N[(a2 * N[(N[Sqrt[0.5], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision], N[(N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * (-a2)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 1.8e90 or 5.9999999999999999e205 < th < 4.8000000000000002e260

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. cos-neg 99.6%

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

      3. associate-/r/ 99.6%

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

      4. cos-neg 99.6%

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

      5. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 63.1%

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

   5. Taylor expanded in th around 0 44.3%

      \[\leadsto \frac{\color{blue}{1}}{\frac{\sqrt{2}}{{a2}^{2}}} \]
   6. Step-by-step derivation

      1. pow2 44.3%

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

      2. associate-/r/ 44.2%

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

      3. add-sqr-sqrt 44.2%

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

      4. sqrt-unprod 44.2%

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

      5. frac-times 44.2%

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

      6. metadata-eval 44.2%

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

      7. rem-square-sqrt 44.3%

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

      8. metadata-eval 44.3%

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

      9. associate-*r* 44.3%

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

   7. Applied egg-rr 44.3%

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

   if 1.8e90 < th < 5.9999999999999999e205 or 4.8000000000000002e260 < th

   1. Initial program 99.3%

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

      1. distribute-lft-out 99.3%

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

      2. cos-neg 99.3%

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

      3. associate-/r/ 97.7%

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

      4. cos-neg 97.7%

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

      5. fma-def 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in a1 around 0 59.9%

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

   5. Taylor expanded in th around 0 14.9%

      \[\leadsto \frac{\color{blue}{1}}{\frac{\sqrt{2}}{{a2}^{2}}} \]
   6. Step-by-step derivation

      1. pow2 14.9%

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

      2. associate-/l/ 14.9%

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

      3. clear-num 14.8%

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

      4. frac-2neg 14.8%

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

      5. associate-/r/ 14.8%

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

   7. Applied egg-rr 14.8%

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

   9. Applied egg-rr 12.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{a2}{\sqrt{2}}\right)} - 1\right)} \cdot \left(-a2\right) \]
   10. Step-by-step derivation

       1. expm1-def 12.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a2}{\sqrt{2}}\right)\right)} \cdot \left(-a2\right) \]

       2. expm1-log1p 24.2%

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

   11. Simplified 24.2%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.8 \cdot 10^{+90} \lor \neg \left(th \leq 6 \cdot 10^{+205}\right) \land th \leq 4.8 \cdot 10^{+260}:\\ \;\;\;\;a2 \cdot \left(\sqrt{0.5} \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt{2}} \cdot \left(-a2\right)\\ \end{array} \]

Alternative 8: 40.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

   2. cos-neg 99.5%

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

   3. associate-/r/ 99.3%

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

   4. cos-neg 99.3%

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

   5. fma-def 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in a1 around 0 62.7%

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

5. Taylor expanded in th around 0 40.1%

   \[\leadsto \frac{\color{blue}{1}}{\frac{\sqrt{2}}{{a2}^{2}}} \]
6. Step-by-step derivation

   1. pow2 40.1%

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

   2. associate-/r/ 40.1%

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

   3. add-sqr-sqrt 40.1%

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

   4. sqrt-unprod 40.1%

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

   5. frac-times 40.1%

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

   6. metadata-eval 40.1%

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

   7. rem-square-sqrt 40.1%

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

   8. metadata-eval 40.1%

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

   9. associate-*r* 40.1%

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

7. Applied egg-rr 40.1%

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

8. Final simplification 40.1%

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

Reproduce

herbie shell --seed 2023305 
(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))))