Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 9.0s

Alternatives: 13

Speedup: 1.9×

• 43.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (/
  (fma
   (* (* a2 (cos th)) (- a2))
   (- (sqrt 2.0))
   (* (sqrt 2.0) (* (* a1 (cos th)) a1)))
  2.0))

C

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

Julia

a1, a2, th = sort([a1, a2, th])
function code(a1, a2, th)
	return Float64(fma(Float64(Float64(a2 * cos(th)) * Float64(-a2)), Float64(-sqrt(2.0)), Float64(sqrt(2.0) * Float64(Float64(a1 * cos(th)) * a1))) / 2.0)
end

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(N[(N[(N[(a2 * N[Cos[th], $MachinePrecision]), $MachinePrecision] * (-a2)), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision]) + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(a1 * N[Cos[th], $MachinePrecision]), $MachinePrecision] * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. 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. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. associate-*l/ N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-2neg N/A

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

   10. associate-*l/ N/A

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

   11. frac-add N/A

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

   12. sqr-neg-rev N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. rem-square-sqrt N/A

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [a1, a2, th] = \mathsf{sort}([a1, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \leq -5 \cdot 10^{-274}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.5, 1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2))) -5e-274)
     (* (* a2 a2) (/ (fma (* th th) -0.5 1.0) (sqrt 2.0)))
     (* (* 0.5 (sqrt 2.0)) (fma a1 a1 (* a2 a2))))))

C

assert(a1 < a2 && a2 < th);
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))) <= -5e-274) {
		tmp = (a2 * a2) * (fma((th * th), -0.5, 1.0) / sqrt(2.0));
	} else {
		tmp = (0.5 * sqrt(2.0)) * fma(a1, a1, (a2 * a2));
	}
	return tmp;
}

Julia

a1, a2, th = sort([a1, a2, th])
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))) <= -5e-274)
		tmp = Float64(Float64(a2 * a2) * Float64(fma(Float64(th * th), -0.5, 1.0) / sqrt(2.0)));
	else
		tmp = Float64(Float64(0.5 * sqrt(2.0)) * fma(a1, a1, Float64(a2 * a2)));
	end
	return tmp
end

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
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], -5e-274], N[(N[(a2 * a2), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a1 * a1 + 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))) < -5e-274

   1. Initial program 98.3%

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

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in a1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 47.1

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

   7. Applied rewrites 47.1%

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

   8. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 34.3

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

   10. Applied rewrites 34.3%

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

   if -5e-274 < (+.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 99.6%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. associate-*l/ N/A

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

      11. frac-add N/A

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

      12. sqr-neg-rev N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. rem-square-sqrt N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in th around 0

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

      1. distribute-rgt-out N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.2

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

   7. Applied rewrites 84.2%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [a1, a2, th] = \mathsf{sort}([a1, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t_2 := \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \leq -5 \cdot 10^{-274}:\\ \;\;\;\;\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{2}\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))) (t_2 (fma a1 a1 (* a2 a2))))
   (if (<= (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2))) -5e-274)
     (* (* (sqrt 2.0) (fma -0.25 (* th th) 0.5)) t_2)
     (* (* 0.5 (sqrt 2.0)) t_2))))

C

assert(a1 < a2 && a2 < th);
double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double t_2 = fma(a1, a1, (a2 * a2));
	double tmp;
	if (((t_1 * (a1 * a1)) + (t_1 * (a2 * a2))) <= -5e-274) {
		tmp = (sqrt(2.0) * fma(-0.25, (th * th), 0.5)) * t_2;
	} else {
		tmp = (0.5 * sqrt(2.0)) * t_2;
	}
	return tmp;
}

Julia

a1, a2, th = sort([a1, a2, th])
function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	t_2 = fma(a1, a1, Float64(a2 * a2))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2))) <= -5e-274)
		tmp = Float64(Float64(sqrt(2.0) * fma(-0.25, Float64(th * th), 0.5)) * t_2);
	else
		tmp = Float64(Float64(0.5 * sqrt(2.0)) * t_2);
	end
	return tmp
end

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-274], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $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))) < -5e-274

   1. Initial program 98.3%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. associate-*l/ N/A

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

      11. frac-add N/A

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

      12. sqr-neg-rev N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. rem-square-sqrt N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a1 around 0

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

      1. distribute-lft-in N/A

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

      2. distribute-rgt-out N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in th around 0

      \[\leadsto \left(\frac{-1}{4} \cdot \left({th}^{2} \cdot \sqrt{2}\right) + \frac{1}{2} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\color{blue}{a1}, a1, a2 \cdot a2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 58.5%

       \[\leadsto \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\right) \cdot \mathsf{fma}\left(\color{blue}{a1}, a1, a2 \cdot a2\right) \]

   if -5e-274 < (+.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 99.6%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. associate-*l/ N/A

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

      11. frac-add N/A

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

      12. sqr-neg-rev N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. rem-square-sqrt N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in th around 0

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

      1. distribute-rgt-out N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.2

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

   7. Applied rewrites 84.2%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (/
  (fma (* a2 (cos th)) (* a2 (sqrt 2.0)) (* (* a1 (cos th)) (* a1 (sqrt 2.0))))
  2.0))

C

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

Julia

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

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(N[(N[(a2 * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(a2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a1 * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(a1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. 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. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

Alternative 5: 99.6% accurate, 1.1× speedup

Math

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

FPCore

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (/ (fma (* (cos th) a2) a2 (* (* a1 (cos th)) a1)) (sqrt 2.0)))

C

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

Julia

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

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(N[(N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision] * a2 + N[(N[(a1 * N[Cos[th], $MachinePrecision]), $MachinePrecision] * a1), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. 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. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. associate-*l/ N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-2neg N/A

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

   10. associate-*l/ N/A

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

   11. frac-add N/A

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

   12. sqr-neg-rev N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. rem-square-sqrt N/A

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

4. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. rem-square-sqrt N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. sqr-neg-rev N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. frac-add N/A

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

   11. lift-*.f64 N/A

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

6. Applied rewrites 99.6%

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

Alternative 6: 99.6% accurate, 1.9× speedup

Math

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

FPCore

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (* (* (fma a1 a1 (* a2 a2)) (* (sqrt 2.0) 0.5)) (cos th)))

C

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

Julia

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

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. 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. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. associate-*l/ N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-2neg N/A

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

   10. associate-*l/ N/A

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

   11. frac-add N/A

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

   12. sqr-neg-rev N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. rem-square-sqrt N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. distribute-lft-in N/A

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

   2. distribute-rgt-out N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.3

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

7. Applied rewrites 99.3%

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

9. Applied rewrites 99.3%

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

Alternative 7: 99.6% accurate, 1.9× speedup

Math

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

FPCore

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (* (* (* 0.5 (sqrt 2.0)) (cos th)) (fma a1 a1 (* a2 a2))))

C

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

Julia

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

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. 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. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. associate-*l/ N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-2neg N/A

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

   10. associate-*l/ N/A

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

   11. frac-add N/A

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

   12. sqr-neg-rev N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. rem-square-sqrt N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. distribute-lft-in N/A

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

   2. distribute-rgt-out N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.3

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

7. Applied rewrites 99.3%

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

Alternative 8: 57.2% accurate, 2.0× speedup

Math

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

FPCore

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th) :precision binary64 (* (* (cos th) a2) (/ a2 (sqrt 2.0))))

C

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

Fortran

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
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

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

Python

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

Julia

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

MATLAB

a1, a2, th = num2cell(sort([a1, a2, th])){:}
function tmp = code(a1, a2, th)
	tmp = (cos(th) * a2) * (a2 / sqrt(2.0));
end

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision] * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in a1 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 58.2

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

5. Applied rewrites 58.2%

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

Alternative 9: 57.2% accurate, 2.0× speedup

Math

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

FPCore

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (* (* 0.5 (* a2 a2)) (* (sqrt 2.0) (cos th))))

C

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

Fortran

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
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 * (a2 * a2)) * (sqrt(2.0d0) * cos(th))
end function

Java

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

Python

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

Julia

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

MATLAB

a1, a2, th = num2cell(sort([a1, a2, th])){:}
function tmp = code(a1, a2, th)
	tmp = (0.5 * (a2 * a2)) * (sqrt(2.0) * cos(th));
end

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(N[(0.5 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. 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. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. associate-*l/ N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-2neg N/A

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

   10. associate-*l/ N/A

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

   11. frac-add N/A

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

   12. sqr-neg-rev N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. rem-square-sqrt N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-cos.f64 57.9

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

7. Applied rewrites 57.9%

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

Alternative 10: 65.9% accurate, 8.3× speedup

Math

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

FPCore

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (* (* 0.5 (sqrt 2.0)) (fma a1 a1 (* a2 a2))))

C

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

Julia

a1, a2, th = sort([a1, a2, th])
function code(a1, a2, th)
	return Float64(Float64(0.5 * sqrt(2.0)) * fma(a1, a1, Float64(a2 * a2)))
end

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. 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. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. associate-*l/ N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-2neg N/A

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

   10. associate-*l/ N/A

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

   11. frac-add N/A

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

   12. sqr-neg-rev N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. rem-square-sqrt N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in th around 0

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

   1. distribute-rgt-out N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 64.3

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

7. Applied rewrites 64.3%

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

Alternative 11: 39.6% accurate, 9.9× speedup

Math

\[\begin{array}{l} [a1, a2, th] = \mathsf{sort}([a1, a2, th])\\ \\ \frac{a2 \cdot a2}{\sqrt{2}} \end{array} \]

FPCore

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th) :precision binary64 (/ (* a2 a2) (sqrt 2.0)))

C

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

Fortran

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
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

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

Python

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

Julia

a1, a2, th = sort([a1, a2, th])
function code(a1, a2, th)
	return Float64(Float64(a2 * a2) / sqrt(2.0))
end

MATLAB

a1, a2, th = num2cell(sort([a1, a2, th])){:}
function tmp = code(a1, a2, th)
	tmp = (a2 * a2) / sqrt(2.0);
end

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in th around 0

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

   1. div-add-rev N/A

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

   2. lower-/.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 64.3

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

5. Applied rewrites 64.3%

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

6. Taylor expanded in a1 around 0

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

8. Applied rewrites 41.9%

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

Alternative 12: 39.6% accurate, 9.9× speedup

Math

\[\begin{array}{l} [a1, a2, th] = \mathsf{sort}([a1, a2, th])\\ \\ a2 \cdot \frac{a2}{\sqrt{2}} \end{array} \]

FPCore

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th) :precision binary64 (* a2 (/ a2 (sqrt 2.0))))

C

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

Fortran

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
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

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

Python

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

Julia

a1, a2, th = sort([a1, a2, th])
function code(a1, a2, th)
	return Float64(a2 * Float64(a2 / sqrt(2.0)))
end

MATLAB

a1, a2, th = num2cell(sort([a1, a2, th])){:}
function tmp = code(a1, a2, th)
	tmp = a2 * (a2 / sqrt(2.0));
end

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in th around 0

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

   1. div-add-rev N/A

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

   2. lower-/.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 64.3

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

5. Applied rewrites 64.3%

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

6. Taylor expanded in a1 around 0

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

8. Applied rewrites 41.9%

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

Alternative 13: 39.4% accurate, 9.9× speedup

Math

\[\begin{array}{l} [a1, a2, th] = \mathsf{sort}([a1, a2, th])\\ \\ a1 \cdot \frac{a1}{\sqrt{2}} \end{array} \]

FPCore

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th) :precision binary64 (* a1 (/ a1 (sqrt 2.0))))

C

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

Fortran

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
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

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

Python

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

Julia

a1, a2, th = sort([a1, a2, th])
function code(a1, a2, th)
	return Float64(a1 * Float64(a1 / sqrt(2.0)))
end

MATLAB

a1, a2, th = num2cell(sort([a1, a2, th])){:}
function tmp = code(a1, a2, th)
	tmp = a1 * (a1 / sqrt(2.0));
end

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(a1 * N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in th around 0

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

   1. div-add-rev N/A

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

   2. lower-/.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 64.3

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

5. Applied rewrites 64.3%

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

6. Taylor expanded in a1 around inf

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

8. Applied rewrites 39.4%

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

Reproduce

herbie shell --seed 2024332 
(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))))