Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 6.9s

Alternatives: 11

Speedup: 2.0×

• 43.6% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   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. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a2 \cdot a2\right) + t\_1 \cdot \left(a1 \cdot a1\right) \leq -2 \cdot 10^{-131}:\\ \;\;\;\;\left(\sqrt{0.5} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \mathsf{fma}\left(th \cdot th, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a2}{\sqrt{2}}, a2, \frac{a1}{\sqrt{2}} \cdot a1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a2 a2)) (* t_1 (* a1 a1))) -2e-131)
     (* (* (sqrt 0.5) (fma a1 a1 (* a2 a2))) (fma (* th th) -0.5 1.0))
     (fma (/ a2 (sqrt 2.0)) a2 (* (/ a1 (sqrt 2.0)) a1)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a2 * a2)) + (t_1 * (a1 * a1))) <= -2e-131) {
		tmp = (sqrt(0.5) * fma(a1, a1, (a2 * a2))) * fma((th * th), -0.5, 1.0);
	} else {
		tmp = fma((a2 / sqrt(2.0)), a2, ((a1 / sqrt(2.0)) * a1));
	}
	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(a2 * a2)) + Float64(t_1 * Float64(a1 * a1))) <= -2e-131)
		tmp = Float64(Float64(sqrt(0.5) * fma(a1, a1, Float64(a2 * a2))) * fma(Float64(th * th), -0.5, 1.0));
	else
		tmp = fma(Float64(a2 / sqrt(2.0)), a2, Float64(Float64(a1 / sqrt(2.0)) * a1));
	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[(a2 * a2), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-131], N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a2 + N[(N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a1), $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))) < -2e-131

   1. Initial program 99.4%

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

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

      7. associate-/r/ N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. pow1/2 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 42.7

          \[\leadsto \mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot \left(\sqrt{0.5} \cdot \mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right)\right) \]

   7. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot \left(\sqrt{0.5} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right)} \]

   if -2e-131 < (+.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. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 87.4

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

   5. Applied rewrites 87.4%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) \leq -2 \cdot 10^{-131}:\\ \;\;\;\;\left(\sqrt{0.5} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \mathsf{fma}\left(th \cdot th, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a2}{\sqrt{2}}, a2, \frac{a1}{\sqrt{2}} \cdot a1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a2 \cdot a2\right) + t\_1 \cdot \left(a1 \cdot a1\right) \leq -2 \cdot 10^{-131}:\\ \;\;\;\;\left(\sqrt{0.5} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \mathsf{fma}\left(th \cdot th, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a2 a2)) (* t_1 (* a1 a1))) -2e-131)
     (* (* (sqrt 0.5) (fma a1 a1 (* a2 a2))) (fma (* th th) -0.5 1.0))
     (/ (fma a2 a2 (* a1 a1)) (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a2 * a2)) + (t_1 * (a1 * a1))) <= -2e-131) {
		tmp = (sqrt(0.5) * fma(a1, a1, (a2 * a2))) * fma((th * th), -0.5, 1.0);
	} else {
		tmp = fma(a2, a2, (a1 * a1)) / sqrt(2.0);
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a2 * a2)) + Float64(t_1 * Float64(a1 * a1))) <= -2e-131)
		tmp = Float64(Float64(sqrt(0.5) * fma(a1, a1, Float64(a2 * a2))) * fma(Float64(th * th), -0.5, 1.0));
	else
		tmp = Float64(fma(a2, a2, Float64(a1 * a1)) / sqrt(2.0));
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-131], N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * a2 + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -2e-131

   1. Initial program 99.4%

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

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

      7. associate-/r/ N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. pow1/2 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 42.7

          \[\leadsto \mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot \left(\sqrt{0.5} \cdot \mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right)\right) \]

   7. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot \left(\sqrt{0.5} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right)} \]

   if -2e-131 < (+.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. *-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. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in th around 0

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

      1. lower-sqrt.f64 87.4

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

   7. Applied rewrites 87.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) \leq -2 \cdot 10^{-131}:\\ \;\;\;\;\left(\sqrt{0.5} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \mathsf{fma}\left(th \cdot th, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing
3. 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. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 5: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   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. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 6: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   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. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. pow-flip N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a1 around 0

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

   1. distribute-rgt-out N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-sqrt.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 7: 57.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   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. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. pow-flip N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 59.2

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

7. Applied rewrites 59.2%

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

Alternative 8: 66.7% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in th around 0

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

   1. lower-sqrt.f64 69.0

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

7. Applied rewrites 69.0%

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

Alternative 9: 66.7% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   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. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. pow-flip N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in th around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 69.0

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

7. Applied rewrites 69.0%

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

Alternative 10: 40.6% accurate, 12.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   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. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. pow-flip N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in th around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 69.0

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

7. Applied rewrites 69.0%

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

8. Taylor expanded in a1 around 0

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

10. Applied rewrites 44.0%

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

Alternative 11: 40.1% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5} \cdot \left(a1 \cdot a1\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   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. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. pow-flip N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in th around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 69.0

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

7. Applied rewrites 69.0%

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

8. Taylor expanded in a1 around inf

   \[\leadsto {a1}^{2} \cdot \color{blue}{\sqrt{\frac{1}{2}}} \]
9. Step-by-step derivation

10. Applied rewrites 38.0%

    \[\leadsto \left(\sqrt{0.5} \cdot a1\right) \cdot \color{blue}{a1} \]
11. Step-by-step derivation

12. Applied rewrites 38.0%

    \[\leadsto \left(a1 \cdot a1\right) \cdot \sqrt{0.5} \]

13. Final simplification 38.0%

    \[\leadsto \sqrt{0.5} \cdot \left(a1 \cdot a1\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024331 
(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))))