Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 10.3s

Alternatives: 8

Speedup: 2.0×

• 43.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. cos-neg 99.6%

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

   3. associate-*l/ 99.6%

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

   4. cos-neg 99.6%

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

   5. fma-def 99.6%

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

3. Simplified 99.6%

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

   1. fma-udef 99.6%

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

   2. +-commutative 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 57.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return a2 * (a2 * (cos(th) * 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 = a2 * (a2 * (cos(th) * sqrt(0.5d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 * N[(N[Cos[th], $MachinePrecision] * N[Sqrt[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. cos-neg 99.6%

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

   3. associate-*l/ 99.6%

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

   4. cos-neg 99.6%

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

   5. fma-def 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.6%

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

   2. fma-udef 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-*l* 99.6%

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

   5. add-sqr-sqrt 99.6%

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

   6. pow2 99.6%

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

   7. hypot-def 99.6%

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

   8. pow1/2 99.6%

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

   9. pow-flip 99.6%

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

   10. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in a2 around inf 55.4%

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

   1. unpow2 55.4%

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

   2. associate-*l* 55.4%

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

8. Simplified 55.4%

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

9. Final simplification 55.4%

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

Alternative 4: 57.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return a2 * (cos(th) * (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 = a2 * (cos(th) * (a2 * sqrt(0.5d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(N[Cos[th], $MachinePrecision] * N[(a2 * N[Sqrt[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. cos-neg 99.6%

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

   3. associate-*l/ 99.6%

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

   4. cos-neg 99.6%

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

   5. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 55.4%

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

   1. unpow2 55.4%

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

   2. *-commutative 55.4%

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

6. Simplified 55.4%

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

   1. expm1-log1p-u 44.8%

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

   2. expm1-udef 35.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\cos th \cdot \left(a2 \cdot a2\right)}{\sqrt{2}}\right)} - 1} \]

   3. div-inv 35.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\cos th \cdot \left(a2 \cdot a2\right)\right) \cdot \frac{1}{\sqrt{2}}}\right)} - 1 \]

   4. associate-*l* 35.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \frac{1}{\sqrt{2}}\right)}\right)} - 1 \]

   5. add-sqr-sqrt 35.4%

      \[\leadsto e^{\mathsf{log1p}\left(\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \color{blue}{\left(\sqrt{\frac{1}{\sqrt{2}}} \cdot \sqrt{\frac{1}{\sqrt{2}}}\right)}\right)\right)} - 1 \]

   6. sqrt-unprod 35.4%

      \[\leadsto e^{\mathsf{log1p}\left(\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \color{blue}{\sqrt{\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}}}\right)\right)} - 1 \]

   7. frac-times 35.4%

      \[\leadsto e^{\mathsf{log1p}\left(\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{\color{blue}{\frac{1 \cdot 1}{\sqrt{2} \cdot \sqrt{2}}}}\right)\right)} - 1 \]

   8. metadata-eval 35.4%

      \[\leadsto e^{\mathsf{log1p}\left(\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{\frac{\color{blue}{1}}{\sqrt{2} \cdot \sqrt{2}}}\right)\right)} - 1 \]

   9. add-sqr-sqrt 35.4%

      \[\leadsto e^{\mathsf{log1p}\left(\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{\frac{1}{\color{blue}{2}}}\right)\right)} - 1 \]

   10. metadata-eval 35.4%

       \[\leadsto e^{\mathsf{log1p}\left(\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{\color{blue}{0.5}}\right)\right)} - 1 \]

   11. associate-*r* 35.4%

       \[\leadsto e^{\mathsf{log1p}\left(\cos th \cdot \color{blue}{\left(a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\right)}\right)} - 1 \]

8. Applied egg-rr 35.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos th \cdot \left(a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\right)\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 44.7%

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

   2. expm1-log1p 55.3%

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

   3. *-commutative 55.3%

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

   4. associate-*l* 55.4%

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

10. Simplified 55.4%

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

11. Final simplification 55.4%

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

Alternative 5: 57.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / 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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a2 around inf 55.4%

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

   1. unpow2 55.4%

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

   2. associate-*l/ 55.4%

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

6. Simplified 55.4%

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

   1. associate-/l* 55.4%

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

   2. div-inv 55.4%

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

8. Applied egg-rr 55.4%

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

   1. associate-*r/ 55.4%

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

   2. *-rgt-identity 55.4%

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

10. Simplified 55.4%

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

11. Final simplification 55.4%

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

Alternative 6: 41.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a1 \cdot a1 \leq 5 \cdot 10^{+24}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a2 \cdot a2\right) \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (* a1 a1) 5e+24)
   (* a2 (/ a2 (sqrt 2.0)))
   (* (sqrt 0.5) (* (* a2 a2) (+ 1.0 (* -0.5 (* th th)))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((a1 * a1) <= 5e+24) {
		tmp = a2 * (a2 / sqrt(2.0));
	} else {
		tmp = sqrt(0.5) * ((a2 * a2) * (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 ((a1 * a1) <= 5d+24) then
        tmp = a2 * (a2 / sqrt(2.0d0))
    else
        tmp = sqrt(0.5d0) * ((a2 * a2) * (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 ((a1 * a1) <= 5e+24) {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	} else {
		tmp = Math.sqrt(0.5) * ((a2 * a2) * (1.0 + (-0.5 * (th * th))));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (a1 * a1) <= 5e+24:
		tmp = a2 * (a2 / math.sqrt(2.0))
	else:
		tmp = math.sqrt(0.5) * ((a2 * a2) * (1.0 + (-0.5 * (th * th))))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (Float64(a1 * a1) <= 5e+24)
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	else
		tmp = Float64(sqrt(0.5) * Float64(Float64(a2 * a2) * 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 ((a1 * a1) <= 5e+24)
		tmp = a2 * (a2 / sqrt(2.0));
	else
		tmp = sqrt(0.5) * ((a2 * a2) * (1.0 + (-0.5 * (th * th))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[(a1 * a1), $MachinePrecision], 5e+24], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a1 a1) < 5.00000000000000045e24

   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. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 61.4%

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

   5. Taylor expanded in a2 around inf 51.2%

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

      1. unpow2 51.2%

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

      2. associate-*r/ 51.2%

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

   7. Simplified 51.2%

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

   if 5.00000000000000045e24 < (*.f64 a1 a1)

   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. cos-neg 99.7%

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

      3. associate-*l/ 99.7%

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

      4. cos-neg 99.7%

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

      5. fma-def 99.7%

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

   3. Simplified 99.7%

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

      1. div-inv 99.7%

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

      2. fma-udef 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-*l* 99.6%

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

      5. add-sqr-sqrt 99.6%

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

      6. pow2 99.6%

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

      7. hypot-def 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 99.7%

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

      10. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in a2 around inf 32.8%

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

      1. unpow2 32.8%

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

      2. associate-*r* 32.8%

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

      3. *-commutative 32.8%

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

   8. Simplified 32.8%

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

   9. Taylor expanded in th around 0 28.3%

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

       1. unpow2 28.3%

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

   11. Simplified 28.3%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a1 \leq 5 \cdot 10^{+24}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a2 \cdot a2\right) \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \end{array} \]

Alternative 7: 40.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return 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 = a2 * (a2 * sqrt(0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 65.9%

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

5. Taylor expanded in a2 around inf 37.1%

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

   1. unpow2 37.1%

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

   2. associate-*r/ 37.1%

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

7. Simplified 37.1%

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

   1. div-inv 37.1%

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

   2. *-commutative 37.1%

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

   3. add-sqr-sqrt 37.1%

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

   4. sqrt-unprod 37.1%

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

   5. frac-times 37.1%

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

   6. metadata-eval 37.1%

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

   7. add-sqr-sqrt 37.1%

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

   8. metadata-eval 37.1%

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

9. Applied egg-rr 37.1%

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

10. Final simplification 37.1%

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

Alternative 8: 40.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ a2 \cdot \frac{a2}{\sqrt{2}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 / 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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 65.9%

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

5. Taylor expanded in a2 around inf 37.1%

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

   1. unpow2 37.1%

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

   2. associate-*r/ 37.1%

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

7. Simplified 37.1%

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

8. Final simplification 37.1%

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

Reproduce

herbie shell --seed 2023292 
(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))))