Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 10.8s

Alternatives: 9

Speedup: 1.9×

• 43.9% 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, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 2: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot t\_1 \leq -1 \cdot 10^{-94}:\\ \;\;\;\;\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2 \cdot a2, \sqrt{2}, \sqrt{2} \cdot \left(a1 \cdot a1\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* (* a2 a2) t_1)) -1e-94)
     (* (* (fma a1 a1 (* a2 a2)) (sqrt 2.0)) (fma -0.25 (* th th) 0.5))
     (/ (fma (* a2 a2) (sqrt 2.0) (* (sqrt 2.0) (* a1 a1))) 2.0))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + ((a2 * a2) * t_1)) <= -1e-94) {
		tmp = (fma(a1, a1, (a2 * a2)) * sqrt(2.0)) * fma(-0.25, (th * th), 0.5);
	} else {
		tmp = fma((a2 * a2), sqrt(2.0), (sqrt(2.0) * (a1 * a1))) / 2.0;
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(Float64(a2 * a2) * t_1)) <= -1e-94)
		tmp = Float64(Float64(fma(a1, a1, Float64(a2 * a2)) * sqrt(2.0)) * fma(-0.25, Float64(th * th), 0.5));
	else
		tmp = Float64(fma(Float64(a2 * a2), sqrt(2.0), Float64(sqrt(2.0) * Float64(a1 * a1))) / 2.0);
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -1e-94], N[(N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -9.9999999999999996e-95

   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. 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. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. frac-add N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. rem-square-sqrt N/A

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

      17. lower-/.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in th around 0

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {th}^{2}\right) \cdot \left({a1}^{2} \cdot \sqrt{2} + {a2}^{2} \cdot \sqrt{2}\right)} + \frac{1}{2} \cdot \left({a1}^{2} \cdot \sqrt{2} + {a2}^{2} \cdot \sqrt{2}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\left({a1}^{2} \cdot \sqrt{2} + {a2}^{2} \cdot \sqrt{2}\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({a1}^{2} \cdot \sqrt{2} + {a2}^{2} \cdot \sqrt{2}\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right)} \]

      4. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left({a1}^{2} + {a2}^{2}\right)\right)} \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left({a1}^{2} + {a2}^{2}\right)\right)} \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\sqrt{2} \cdot \left(\color{blue}{a1 \cdot a1} + {a2}^{2}\right)\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right) \]

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, {th}^{2}, \frac{1}{2}\right)} \]

      12. unpow2 N/A

          \[\leadsto \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{th \cdot th}, \frac{1}{2}\right) \]

      13. lower-*.f64 58.0

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

   9. Applied rewrites 58.0%

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

   if -9.9999999999999996e-95 < (+.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. 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. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 92.0

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

   5. Applied rewrites 92.0%

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

   7. Applied rewrites 92.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -1 \cdot 10^{-94}:\\ \;\;\;\;\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2 \cdot a2, \sqrt{2}, \sqrt{2} \cdot \left(a1 \cdot a1\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\\ t_2 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_2 \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot t\_2 \leq -1 \cdot 10^{-94}:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{2} \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (fma a1 a1 (* a2 a2))) (t_2 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_2 (* a1 a1)) (* (* a2 a2) t_2)) -1e-94)
     (* (* t_1 (sqrt 2.0)) (fma -0.25 (* th th) 0.5))
     (* t_1 (* (sqrt 2.0) 0.5)))))

C

double code(double a1, double a2, double th) {
	double t_1 = fma(a1, a1, (a2 * a2));
	double t_2 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_2 * (a1 * a1)) + ((a2 * a2) * t_2)) <= -1e-94) {
		tmp = (t_1 * sqrt(2.0)) * fma(-0.25, (th * th), 0.5);
	} else {
		tmp = t_1 * (sqrt(2.0) * 0.5);
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = fma(a1, a1, Float64(a2 * a2))
	t_2 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_2 * Float64(a1 * a1)) + Float64(Float64(a2 * a2) * t_2)) <= -1e-94)
		tmp = Float64(Float64(t_1 * sqrt(2.0)) * fma(-0.25, Float64(th * th), 0.5));
	else
		tmp = Float64(t_1 * Float64(sqrt(2.0) * 0.5));
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], -1e-94], N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -9.9999999999999996e-95

   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. 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. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. frac-add N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. rem-square-sqrt N/A

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

      17. lower-/.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in th around 0

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {th}^{2}\right) \cdot \left({a1}^{2} \cdot \sqrt{2} + {a2}^{2} \cdot \sqrt{2}\right)} + \frac{1}{2} \cdot \left({a1}^{2} \cdot \sqrt{2} + {a2}^{2} \cdot \sqrt{2}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\left({a1}^{2} \cdot \sqrt{2} + {a2}^{2} \cdot \sqrt{2}\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({a1}^{2} \cdot \sqrt{2} + {a2}^{2} \cdot \sqrt{2}\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right)} \]

      4. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left({a1}^{2} + {a2}^{2}\right)\right)} \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left({a1}^{2} + {a2}^{2}\right)\right)} \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\sqrt{2} \cdot \left(\color{blue}{a1 \cdot a1} + {a2}^{2}\right)\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right) \]

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, {th}^{2}, \frac{1}{2}\right)} \]

      12. unpow2 N/A

          \[\leadsto \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{th \cdot th}, \frac{1}{2}\right) \]

      13. lower-*.f64 58.0

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

   9. Applied rewrites 58.0%

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

   if -9.9999999999999996e-95 < (+.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. 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. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. frac-add N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. rem-square-sqrt N/A

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

      17. lower-/.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a1}^{2} \cdot \sqrt{2} + {a2}^{2} \cdot \sqrt{2}\right)} \]
   8. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 92.1

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

   9. Applied rewrites 92.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -1 \cdot 10^{-94}:\\ \;\;\;\;\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \left(\sqrt{2} \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

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

   6. div-inv N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 5: 57.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-sqrt.f64 62.7

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

5. Applied rewrites 62.7%

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

Alternative 6: 57.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(a2 * a2), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

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

   6. div-inv N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. lift-/.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

   11. lift-/.f64 N/A

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

   12. div-inv N/A

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

   13. frac-add N/A

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

   14. lift-sqrt.f64 N/A

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

   15. lift-sqrt.f64 N/A

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

   16. rem-square-sqrt N/A

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

   17. lower-/.f64 N/A

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

   18. lower-fma.f64 N/A

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

   19. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Taylor expanded in a1 around 0

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. unpow2 N/A

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

    6. lower-*.f64 N/A

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

    7. associate-*r* N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. lower-sqrt.f64 N/A

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

    11. lower-*.f64 N/A

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

    12. lower-cos.f64 62.7

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

11. Applied rewrites 62.7%

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

12. Final simplification 62.7%

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

Alternative 7: 67.1% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a1_, a2_, th_] := N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

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

   6. div-inv N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. lift-/.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

   11. lift-/.f64 N/A

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

   12. div-inv N/A

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

   13. frac-add N/A

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

   14. lift-sqrt.f64 N/A

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

   15. lift-sqrt.f64 N/A

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

   16. rem-square-sqrt N/A

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

   17. lower-/.f64 N/A

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

   18. lower-fma.f64 N/A

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

   19. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a1}^{2} \cdot \sqrt{2} + {a2}^{2} \cdot \sqrt{2}\right)} \]
8. 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. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 70.6

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

9. Applied rewrites 70.6%

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

10. Final simplification 70.6%

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

Alternative 8: 40.4% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. 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. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 70.5

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

5. Applied rewrites 70.5%

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

6. Taylor expanded in a1 around 0

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

8. Applied rewrites 44.9%

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

Alternative 9: 40.0% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. 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. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 70.5

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

5. Applied rewrites 70.5%

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

6. Taylor expanded in a1 around inf

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

8. Applied rewrites 41.4%

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

10. Applied rewrites 41.4%

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

11. Final simplification 41.4%

    \[\leadsto a1 \cdot \frac{a1}{\sqrt{2}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(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))))