Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.5%

Time: 9.2s

Alternatives: 8

Speedup: 1.0×

• 43.3% 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: 99.5% 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]]

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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 75.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \leq -1 \cdot 10^{-225}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \frac{th \cdot \left(th \cdot -0.5\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}} + \frac{a1 \cdot a1}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2))) -1e-225)
     (* (* a2 a2) (/ (* th (* th -0.5)) (sqrt 2.0)))
     (+ (/ (* a2 a2) (sqrt 2.0)) (/ (* a1 a1) (sqrt 2.0))))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + (t_1 * (a2 * a2))) <= -1e-225) {
		tmp = (a2 * a2) * ((th * (th * -0.5)) / sqrt(2.0));
	} else {
		tmp = ((a2 * a2) / sqrt(2.0)) + ((a1 * a1) / 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 = cos(th) / sqrt(2.0d0)
    if (((t_1 * (a1 * a1)) + (t_1 * (a2 * a2))) <= (-1d-225)) then
        tmp = (a2 * a2) * ((th * (th * (-0.5d0))) / sqrt(2.0d0))
    else
        tmp = ((a2 * a2) / sqrt(2.0d0)) + ((a1 * a1) / sqrt(2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + (t_1 * (a2 * a2))) <= -1e-225) {
		tmp = (a2 * a2) * ((th * (th * -0.5)) / Math.sqrt(2.0));
	} else {
		tmp = ((a2 * a2) / Math.sqrt(2.0)) + ((a1 * a1) / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	tmp = 0
	if ((t_1 * (a1 * a1)) + (t_1 * (a2 * a2))) <= -1e-225:
		tmp = (a2 * a2) * ((th * (th * -0.5)) / math.sqrt(2.0))
	else:
		tmp = ((a2 * a2) / math.sqrt(2.0)) + ((a1 * a1) / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2))) <= -1e-225)
		tmp = Float64(Float64(a2 * a2) * Float64(Float64(th * Float64(th * -0.5)) / sqrt(2.0)));
	else
		tmp = Float64(Float64(Float64(a2 * a2) / sqrt(2.0)) + Float64(Float64(a1 * a1) / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = 0.0;
	if (((t_1 * (a1 * a1)) + (t_1 * (a2 * a2))) <= -1e-225)
		tmp = (a2 * a2) * ((th * (th * -0.5)) / sqrt(2.0));
	else
		tmp = ((a2 * a2) / sqrt(2.0)) + ((a1 * a1) / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-225], N[(N[(a2 * a2), $MachinePrecision] * N[(N[(th * N[(th * -0.5), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a1 * a1), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -9.9999999999999996e-226

   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. Add Preprocessing

   3. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 44.5

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

   5. Simplified 44.5%

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

   6. Taylor expanded in a1 around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sqrt.f64 50.3

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

   8. Simplified 50.3%

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

   9. Taylor expanded in th around 0

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-lft-identity N/A

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

       5. associate-*l/ N/A

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

       6. distribute-rgt-out N/A

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

       7. +-commutative N/A

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

       8. remove-double-neg N/A

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

       9. metadata-eval N/A

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

       10. distribute-lft-neg-in N/A

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

       11. associate-/l* N/A

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

       12. distribute-neg-in N/A

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

       13. +-commutative N/A

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

   11. Simplified 28.1%

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

   12. Taylor expanded in th around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-sqrt.f64 28.1

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

   14. Simplified 28.1%

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

   if -9.9999999999999996e-226 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   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. Add Preprocessing

   3. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 92.2

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

   5. Simplified 92.2%

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

   6. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 83.6

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

   8. Simplified 83.6%

      \[\leadsto \color{blue}{\frac{a1 \cdot a1}{\sqrt{2}}} + \frac{a2 \cdot a2}{\sqrt{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \leq -1 \cdot 10^{-225}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \frac{th \cdot \left(th \cdot -0.5\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}} + \frac{a1 \cdot a1}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \leq -1 \cdot 10^{-225}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \frac{th \cdot \left(th \cdot -0.5\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2))) -1e-225)
     (* (* a2 a2) (/ (* th (* th -0.5)) (sqrt 2.0)))
     (/ (fma a1 a1 (* a2 a2)) (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + (t_1 * (a2 * a2))) <= -1e-225) {
		tmp = (a2 * a2) * ((th * (th * -0.5)) / sqrt(2.0));
	} else {
		tmp = fma(a1, a1, (a2 * a2)) / sqrt(2.0);
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2))) <= -1e-225)
		tmp = Float64(Float64(a2 * a2) * Float64(Float64(th * Float64(th * -0.5)) / sqrt(2.0)));
	else
		tmp = Float64(fma(a1, a1, Float64(a2 * a2)) / sqrt(2.0));
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-225], N[(N[(a2 * a2), $MachinePrecision] * N[(N[(th * N[(th * -0.5), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -9.9999999999999996e-226

   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. Add Preprocessing

   3. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 44.5

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

   5. Simplified 44.5%

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

   6. Taylor expanded in a1 around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sqrt.f64 50.3

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

   8. Simplified 50.3%

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

   9. Taylor expanded in th around 0

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-lft-identity N/A

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

       5. associate-*l/ N/A

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

       6. distribute-rgt-out N/A

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

       7. +-commutative N/A

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

       8. remove-double-neg N/A

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

       9. metadata-eval N/A

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

       10. distribute-lft-neg-in N/A

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

       11. associate-/l* N/A

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

       12. distribute-neg-in N/A

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

       13. +-commutative N/A

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

   11. Simplified 28.1%

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

   12. Taylor expanded in th around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-sqrt.f64 28.1

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

   14. Simplified 28.1%

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

   if -9.9999999999999996e-226 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   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. Add Preprocessing

   3. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 90.7

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

   5. Simplified 90.7%

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

   6. Taylor expanded in a1 around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. *-inverses N/A

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

      5. associate-/r* N/A

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

      6. associate-/l* N/A

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

      7. times-frac N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. unpow2 N/A

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

      17. associate-/l* N/A

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

   8. Simplified 66.0%

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

   9. Taylor expanded in th around 0

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

       1. distribute-rgt-in N/A

          \[\leadsto \frac{\color{blue}{1 \cdot {a2}^{2} + \frac{{a1}^{2}}{{a2}^{2}} \cdot {a2}^{2}}}{\sqrt{2}} \]

       2. *-lft-identity N/A

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

       3. associate-*l/ N/A

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

       4. associate-/l* N/A

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

       5. *-inverses N/A

          \[\leadsto \frac{{a2}^{2} + {a1}^{2} \cdot \color{blue}{1}}{\sqrt{2}} \]

       6. *-rgt-identity N/A

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

       7. lower-/.f64 N/A

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

       8. +-commutative N/A

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

       9. unpow2 N/A

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

       10. lower-fma.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-sqrt.f64 83.6

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

   11. Simplified 83.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 57.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in th around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 82.6

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

5. Simplified 82.6%

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

6. Taylor expanded in a1 around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-sqrt.f64 54.7

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

8. Simplified 54.7%

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

Alternative 5: 57.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in th around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 82.6

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

5. Simplified 82.6%

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

6. Taylor expanded in a1 around 0

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

   1. +-commutative N/A

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

   2. *-rgt-identity N/A

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

   3. associate-*r/ N/A

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

   4. *-inverses N/A

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

   5. associate-/r* N/A

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

   6. associate-/l* N/A

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

   7. times-frac N/A

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. distribute-rgt-out N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. unpow2 N/A

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

   17. associate-/l* N/A

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

8. Simplified 62.9%

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

9. Taylor expanded in a1 around 0

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

    1. lower-cos.f64 54.7

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

11. Simplified 54.7%

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

12. Final simplification 54.7%

    \[\leadsto \cos th \cdot \frac{a2 \cdot a2}{\sqrt{2}} \]
13. Add Preprocessing

Alternative 6: 66.3% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

3. Taylor expanded in th around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 82.6

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

5. Simplified 82.6%

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

6. Taylor expanded in a1 around 0

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

   1. +-commutative N/A

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

   2. *-rgt-identity N/A

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

   3. associate-*r/ N/A

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

   4. *-inverses N/A

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

   5. associate-/r* N/A

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

   6. associate-/l* N/A

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

   7. times-frac N/A

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. distribute-rgt-out N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. unpow2 N/A

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

   17. associate-/l* N/A

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

8. Simplified 62.9%

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

9. Taylor expanded in th around 0

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

    1. distribute-rgt-in N/A

       \[\leadsto \frac{\color{blue}{1 \cdot {a2}^{2} + \frac{{a1}^{2}}{{a2}^{2}} \cdot {a2}^{2}}}{\sqrt{2}} \]

    2. *-lft-identity N/A

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

    3. associate-*l/ N/A

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

    4. associate-/l* N/A

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

    5. *-inverses N/A

       \[\leadsto \frac{{a2}^{2} + {a1}^{2} \cdot \color{blue}{1}}{\sqrt{2}} \]

    6. *-rgt-identity N/A

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

    7. lower-/.f64 N/A

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

    8. +-commutative N/A

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

    9. unpow2 N/A

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

    10. lower-fma.f64 N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 N/A

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

    13. lower-sqrt.f64 69.0

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

11. Simplified 69.0%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}} \]
12. Add Preprocessing

Alternative 7: 40.2% accurate, 9.9× speedup

Math

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

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(Float64(a2 * 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[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $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. Add Preprocessing

3. Taylor expanded in th around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 84.9

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

5. Simplified 84.9%

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

6. Taylor expanded in a1 around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 40.2

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

8. Simplified 40.2%

   \[\leadsto \color{blue}{\frac{a2 \cdot a2}{\sqrt{2}}} \]
9. Add Preprocessing

Alternative 8: 39.7% accurate, 9.9× speedup

Math

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

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(Float64(a1 * 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[(N[(a1 * a1), $MachinePrecision] / N[Sqrt[2.0], $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. Add Preprocessing

3. Taylor expanded in th around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 82.6

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

5. Simplified 82.6%

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

6. Taylor expanded in a1 around inf

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 42.0

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

8. Simplified 42.0%

   \[\leadsto \color{blue}{\frac{a1 \cdot a1}{\sqrt{2}}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(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))))