Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 10.7s

Alternatives: 7

Speedup: 1.9×

• 44.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (* (* (cos th) (sqrt 2.0)) (fma a2 a2 (* a1 a1))) 0.5))

C

double code(double a1, double a2, double th) {
	return ((cos(th) * sqrt(2.0)) * fma(a2, a2, (a1 * a1))) * 0.5;
}

Julia

function code(a1, a2, th)
	return Float64(Float64(Float64(cos(th) * sqrt(2.0)) * fma(a2, a2, Float64(a1 * a1))) * 0.5)
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(N[Cos[th], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a2 * a2 + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos th, \left(a2 \cdot a2\right) \cdot \sqrt{2}, \sqrt{2} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\right) \cdot 0.5} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos th \cdot \sqrt{2}, a2 \cdot a2, \sqrt{2} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\right)} \cdot 0.5 \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f64 99.7

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

8. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\left(\cos th \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right)} \cdot 0.5 \]
9. Add Preprocessing

Alternative 2: 48.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1 \cdot a1\right) \cdot t\_1 + \left(a2 \cdot a2\right) \cdot t\_1 \leq -2 \cdot 10^{-161}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left(\left(a2 \cdot a2\right) \cdot \mathsf{fma}\left(th \cdot th, -0.5, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left(a2 \cdot a2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1 a1) t_1) (* (* a2 a2) t_1)) -2e-161)
     (* 0.5 (* (sqrt 2.0) (* (* a2 a2) (fma (* th th) -0.5 1.0))))
     (* 0.5 (* (sqrt 2.0) (* a2 a2))))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -2e-161) {
		tmp = 0.5 * (sqrt(2.0) * ((a2 * a2) * fma((th * th), -0.5, 1.0)));
	} else {
		tmp = 0.5 * (sqrt(2.0) * (a2 * a2));
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1 * a1) * t_1) + Float64(Float64(a2 * a2) * t_1)) <= -2e-161)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * Float64(Float64(a2 * a2) * fma(Float64(th * th), -0.5, 1.0))));
	else
		tmp = Float64(0.5 * Float64(sqrt(2.0) * Float64(a2 * a2)));
	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[(N[(a1 * a1), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -2e-161], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] * N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a2 * a2), $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))) < -2.00000000000000006e-161

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. frac-add N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in a2 around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 59.7

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

   7. Applied rewrites 59.7%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 36.3%

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

   if -2.00000000000000006e-161 < (+.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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. frac-add N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a2 around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 57.2

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

   7. Applied rewrites 57.2%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 50.1%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -2 \cdot 10^{-161}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left(\left(a2 \cdot a2\right) \cdot \mathsf{fma}\left(th \cdot th, -0.5, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left(a2 \cdot a2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (cos th) (* (sqrt 2.0) (* 0.5 (fma a1 a1 (* a2 a2))))))

C

double code(double a1, double a2, double th) {
	return cos(th) * (sqrt(2.0) * (0.5 * fma(a1, a1, (a2 * a2))));
}

Julia

function code(a1, a2, th)
	return Float64(cos(th) * Float64(sqrt(2.0) * Float64(0.5 * fma(a1, a1, Float64(a2 * a2)))))
end

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.5 * N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in th around inf

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-*r* N/A

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

   10. distribute-rgt-out N/A

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

   11. distribute-lft-in N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

7. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\cos th \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right)\right)} \]
8. Add Preprocessing

Alternative 4: 56.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\left(\sqrt{2} \cdot a2\right) \cdot \left(\cos th \cdot a2\right)\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

function code(a1, a2, th)
	return Float64(0.5 * Float64(Float64(sqrt(2.0) * a2) * Float64(cos(th) * a2)))
end

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(0.5 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * a2), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a2 around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 57.7

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

7. Applied rewrites 57.7%

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

9. Applied rewrites 57.8%

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

10. Final simplification 57.8%

    \[\leadsto 0.5 \cdot \left(\left(\sqrt{2} \cdot a2\right) \cdot \left(\cos th \cdot a2\right)\right) \]
11. Add Preprocessing

Alternative 5: 56.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\left(\cos th \cdot \sqrt{2}\right) \cdot \left(a2 \cdot a2\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a1, a2, th)
	return Float64(0.5 * Float64(Float64(cos(th) * sqrt(2.0)) * Float64(a2 * a2)))
end

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(0.5 * N[(N[(N[Cos[th], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a2 around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 57.7

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

7. Applied rewrites 57.7%

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

9. Applied rewrites 57.7%

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

10. Final simplification 57.7%

    \[\leadsto 0.5 \cdot \left(\left(\cos th \cdot \sqrt{2}\right) \cdot \left(a2 \cdot a2\right)\right) \]
11. Add Preprocessing

Alternative 6: 56.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\sqrt{2} \cdot \left(a2 \cdot \left(\cos th \cdot a2\right)\right)\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

function code(a1, a2, th)
	return Float64(0.5 * Float64(sqrt(2.0) * Float64(a2 * Float64(cos(th) * a2))))
end

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a2 * N[(N[Cos[th], $MachinePrecision] * a2), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a2 around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 57.7

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

7. Applied rewrites 57.7%

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

8. Final simplification 57.7%

   \[\leadsto 0.5 \cdot \left(\sqrt{2} \cdot \left(a2 \cdot \left(\cos th \cdot a2\right)\right)\right) \]
9. Add Preprocessing

Alternative 7: 39.6% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\sqrt{2} \cdot \left(a2 \cdot a2\right)\right) \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* 0.5 (* (sqrt 2.0) (* a2 a2))))

C

double code(double a1, double a2, double th) {
	return 0.5 * (sqrt(2.0) * (a2 * a2));
}

Fortran

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

Java

public static double code(double a1, double a2, double th) {
	return 0.5 * (Math.sqrt(2.0) * (a2 * a2));
}

Python

def code(a1, a2, th):
	return 0.5 * (math.sqrt(2.0) * (a2 * a2))

Julia

function code(a1, a2, th)
	return Float64(0.5 * Float64(sqrt(2.0) * Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = 0.5 * (sqrt(2.0) * (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * 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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a2 around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 57.7

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

7. Applied rewrites 57.7%

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

8. Taylor expanded in th around 0

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

10. Applied rewrites 40.3%

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

11. Final simplification 40.3%

    \[\leadsto 0.5 \cdot \left(\sqrt{2} \cdot \left(a2 \cdot a2\right)\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024238 
(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))))