Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 12.2s

Alternatives: 12

Speedup: 1.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.9× speedup

Math

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

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[th], $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-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. lift-/.f64 N/A

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

   11. div-inv N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

Alternative 2: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot t\_1 \leq -5 \cdot 10^{-118}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.5, 1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* (* a2 a2) t_1)) -5e-118)
     (* (* a2 a2) (/ (fma (* th th) -0.5 1.0) (sqrt 2.0)))
     (/ (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 * (a1 * a1)) + ((a2 * a2) * t_1)) <= -5e-118) {
		tmp = (a2 * a2) * (fma((th * th), -0.5, 1.0) / sqrt(2.0));
	} 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(a1 * a1)) + Float64(Float64(a2 * a2) * t_1)) <= -5e-118)
		tmp = Float64(Float64(a2 * a2) * Float64(fma(Float64(th * th), -0.5, 1.0) / sqrt(2.0)));
	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[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -5e-118], N[(N[(a2 * a2), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(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))) < -5.00000000000000015e-118

   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-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

      \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\color{blue}{1 + {th}^{2} \cdot \left(\frac{1}{24} \cdot {th}^{2} - \frac{1}{2}\right)}}{\sqrt{2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\color{blue}{{th}^{2} \cdot \left(\frac{1}{24} \cdot {th}^{2} - \frac{1}{2}\right) + 1}}{\sqrt{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{1}{24} \cdot {th}^{2} - \frac{1}{2}, 1\right)}}{\sqrt{2}} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{1}{24} \cdot {th}^{2} - \frac{1}{2}, 1\right)}{\sqrt{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{1}{24} \cdot {th}^{2} - \frac{1}{2}, 1\right)}{\sqrt{2}} \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, \color{blue}{\frac{1}{24} \cdot {th}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)}{\sqrt{2}} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, \color{blue}{{th}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}{\sqrt{2}} \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, {th}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right)}{\sqrt{2}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, \color{blue}{\mathsf{fma}\left({th}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right)}{\sqrt{2}} \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{1}{24}, \frac{-1}{2}\right), 1\right)}{\sqrt{2}} \]

      10. lower-*.f64 0.2

          \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(\color{blue}{th \cdot th}, 0.041666666666666664, -0.5\right), 1\right)}{\sqrt{2}} \]

   7. Applied rewrites 0.2%

      \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(th \cdot th, 0.041666666666666664, -0.5\right), 1\right)}}{\sqrt{2}} \]

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 55.3%

       \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, \color{blue}{-0.5}, 1\right)}{\sqrt{2}} \]

   11. Taylor expanded in a1 around 0

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

       1. unpow2 N/A

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

       2. lower-*.f64 40.6

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

   13. Applied rewrites 40.6%

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

   if -5.00000000000000015e-118 < (+.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-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lift-/.f64 N/A

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

      11. div-inv N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 84.8

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

   7. Applied rewrites 84.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -5 \cdot 10^{-118}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.5, 1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot t\_1 \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.5, 1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* (* a2 a2) t_1)) -1e-61)
     (* (* a1 a1) (/ (fma (* th th) -0.5 1.0) (sqrt 2.0)))
     (/ (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 * (a1 * a1)) + ((a2 * a2) * t_1)) <= -1e-61) {
		tmp = (a1 * a1) * (fma((th * th), -0.5, 1.0) / sqrt(2.0));
	} 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(a1 * a1)) + Float64(Float64(a2 * a2) * t_1)) <= -1e-61)
		tmp = Float64(Float64(a1 * a1) * Float64(fma(Float64(th * th), -0.5, 1.0) / sqrt(2.0)));
	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[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -1e-61], N[(N[(a1 * a1), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $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))) < -1e-61

   1. Initial program 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in th around 0

      \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\color{blue}{1 + {th}^{2} \cdot \left(\frac{1}{24} \cdot {th}^{2} - \frac{1}{2}\right)}}{\sqrt{2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\color{blue}{{th}^{2} \cdot \left(\frac{1}{24} \cdot {th}^{2} - \frac{1}{2}\right) + 1}}{\sqrt{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{1}{24} \cdot {th}^{2} - \frac{1}{2}, 1\right)}}{\sqrt{2}} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{1}{24} \cdot {th}^{2} - \frac{1}{2}, 1\right)}{\sqrt{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{1}{24} \cdot {th}^{2} - \frac{1}{2}, 1\right)}{\sqrt{2}} \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, \color{blue}{\frac{1}{24} \cdot {th}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)}{\sqrt{2}} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, \color{blue}{{th}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right)}{\sqrt{2}} \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, {th}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right)}{\sqrt{2}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, \color{blue}{\mathsf{fma}\left({th}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right)}{\sqrt{2}} \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{1}{24}, \frac{-1}{2}\right), 1\right)}{\sqrt{2}} \]

      10. lower-*.f64 0.2

          \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(\color{blue}{th \cdot th}, 0.041666666666666664, -0.5\right), 1\right)}{\sqrt{2}} \]

   7. Applied rewrites 0.2%

      \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(th \cdot th, 0.041666666666666664, -0.5\right), 1\right)}}{\sqrt{2}} \]

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 58.8%

       \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, \color{blue}{-0.5}, 1\right)}{\sqrt{2}} \]

   11. Taylor expanded in a1 around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 47.7

          \[\leadsto \color{blue}{\left(a1 \cdot a1\right)} \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.5, 1\right)}{\sqrt{2}} \]

   13. Applied rewrites 47.7%

       \[\leadsto \color{blue}{\left(a1 \cdot a1\right)} \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.5, 1\right)}{\sqrt{2}} \]

   if -1e-61 < (+.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-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lift-/.f64 N/A

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

      11. div-inv N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 83.6

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

   7. Applied rewrites 83.6%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.5, 1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq -0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(a1 \cdot a1, -\sqrt{2}, a2 \cdot \left(a2 \cdot \sqrt{2}\right)\right)}{-\sqrt{2} \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) -0.02)
   (/
    (fma (* a1 a1) (- (sqrt 2.0)) (* a2 (* a2 (sqrt 2.0))))
    (- (* (sqrt 2.0) (sqrt 2.0))))
   (fma a1 (/ a1 (sqrt 2.0)) (/ 1.0 (/ (sqrt 2.0) (* a2 a2))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= -0.02) {
		tmp = fma((a1 * a1), -sqrt(2.0), (a2 * (a2 * sqrt(2.0)))) / -(sqrt(2.0) * sqrt(2.0));
	} else {
		tmp = fma(a1, (a1 / sqrt(2.0)), (1.0 / (sqrt(2.0) / (a2 * a2))));
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= -0.02)
		tmp = Float64(fma(Float64(a1 * a1), Float64(-sqrt(2.0)), Float64(a2 * Float64(a2 * sqrt(2.0)))) / Float64(-Float64(sqrt(2.0) * sqrt(2.0))));
	else
		tmp = fma(a1, Float64(a1 / sqrt(2.0)), Float64(1.0 / Float64(sqrt(2.0) / Float64(a2 * a2))));
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], -0.02], N[(N[(N[(a1 * a1), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision]) + N[(a2 * N[(a2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(a1 * N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -0.0200000000000000004

   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 11.2

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

   5. Applied rewrites 11.2%

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

      1. lift-sqrt.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. frac-2neg N/A

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

      7. frac-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 11.2

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

   7. Applied rewrites 11.2%

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

   9. Applied rewrites 0.6%

      \[\leadsto \frac{\mathsf{fma}\left(a1 \cdot a1, -\sqrt{2}, \color{blue}{\frac{-2 \cdot \left(a2 \cdot \left(a2 \cdot \left(a2 \cdot a2\right)\right)\right)}{0 + \sqrt{2} \cdot \left(a2 \cdot a2\right)}}\right)}{\sqrt{2} \cdot \left(-\sqrt{2}\right)} \]

   11. Applied rewrites 36.6%

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

   if -0.0200000000000000004 < (cos.f64 th)

   1. Initial program 99.7%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   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 86.3

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

   5. Applied rewrites 86.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 86.3

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

   7. Applied rewrites 86.3%

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \color{blue}{\frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(a1 \cdot a1, -\sqrt{2}, a2 \cdot \left(a2 \cdot \sqrt{2}\right)\right)}{-\sqrt{2} \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq -0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(a1 \cdot a1, -\sqrt{2}, \left(a2 \cdot a2\right) \cdot \sqrt{2}\right)}{-\sqrt{2} \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) -0.02)
   (/
    (fma (* a1 a1) (- (sqrt 2.0)) (* (* a2 a2) (sqrt 2.0)))
    (- (* (sqrt 2.0) (sqrt 2.0))))
   (fma a1 (/ a1 (sqrt 2.0)) (/ 1.0 (/ (sqrt 2.0) (* a2 a2))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= -0.02) {
		tmp = fma((a1 * a1), -sqrt(2.0), ((a2 * a2) * sqrt(2.0))) / -(sqrt(2.0) * sqrt(2.0));
	} else {
		tmp = fma(a1, (a1 / sqrt(2.0)), (1.0 / (sqrt(2.0) / (a2 * a2))));
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= -0.02)
		tmp = Float64(fma(Float64(a1 * a1), Float64(-sqrt(2.0)), Float64(Float64(a2 * a2) * sqrt(2.0))) / Float64(-Float64(sqrt(2.0) * sqrt(2.0))));
	else
		tmp = fma(a1, Float64(a1 / sqrt(2.0)), Float64(1.0 / Float64(sqrt(2.0) / Float64(a2 * a2))));
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], -0.02], N[(N[(N[(a1 * a1), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision]) + N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(a1 * N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -0.0200000000000000004

   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 11.2

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

   5. Applied rewrites 11.2%

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

      1. lift-sqrt.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. frac-2neg N/A

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

      7. frac-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 11.2

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

   7. Applied rewrites 11.2%

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

   9. Applied rewrites 0.6%

      \[\leadsto \frac{\mathsf{fma}\left(a1 \cdot a1, -\sqrt{2}, \color{blue}{\frac{-2 \cdot \left(a2 \cdot \left(a2 \cdot \left(a2 \cdot a2\right)\right)\right)}{0 + \sqrt{2} \cdot \left(a2 \cdot a2\right)}}\right)}{\sqrt{2} \cdot \left(-\sqrt{2}\right)} \]

   11. Applied rewrites 36.6%

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

   if -0.0200000000000000004 < (cos.f64 th)

   1. Initial program 99.7%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   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 86.3

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

   5. Applied rewrites 86.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 86.3

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

   7. Applied rewrites 86.3%

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \color{blue}{\frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(a1 \cdot a1, -\sqrt{2}, \left(a2 \cdot a2\right) \cdot \sqrt{2}\right)}{-\sqrt{2} \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2}, a2 \cdot a2, -a1 \cdot \left(a1 \cdot \sqrt{2}\right)\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) -0.02)
   (* (fma (sqrt 2.0) (* a2 a2) (- (* a1 (* a1 (sqrt 2.0))))) -0.5)
   (fma a1 (/ a1 (sqrt 2.0)) (/ 1.0 (/ (sqrt 2.0) (* a2 a2))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= -0.02) {
		tmp = fma(sqrt(2.0), (a2 * a2), -(a1 * (a1 * sqrt(2.0)))) * -0.5;
	} else {
		tmp = fma(a1, (a1 / sqrt(2.0)), (1.0 / (sqrt(2.0) / (a2 * a2))));
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= -0.02)
		tmp = Float64(fma(sqrt(2.0), Float64(a2 * a2), Float64(-Float64(a1 * Float64(a1 * sqrt(2.0))))) * -0.5);
	else
		tmp = fma(a1, Float64(a1 / sqrt(2.0)), Float64(1.0 / Float64(sqrt(2.0) / Float64(a2 * a2))));
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], -0.02], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(a2 * a2), $MachinePrecision] + (-N[(a1 * N[(a1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * -0.5), $MachinePrecision], N[(a1 * N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -0.0200000000000000004

   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 11.2

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

   5. Applied rewrites 11.2%

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

      1. lift-sqrt.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. frac-2neg N/A

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

      7. frac-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 11.2

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

   7. Applied rewrites 11.2%

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

   9. Applied rewrites 0.6%

      \[\leadsto \frac{\mathsf{fma}\left(a1 \cdot a1, -\sqrt{2}, \color{blue}{\frac{-2 \cdot \left(a2 \cdot \left(a2 \cdot \left(a2 \cdot a2\right)\right)\right)}{0 + \sqrt{2} \cdot \left(a2 \cdot a2\right)}}\right)}{\sqrt{2} \cdot \left(-\sqrt{2}\right)} \]

   11. Applied rewrites 36.6%

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

   if -0.0200000000000000004 < (cos.f64 th)

   1. Initial program 99.7%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   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 86.3

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

   5. Applied rewrites 86.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 86.3

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

   7. Applied rewrites 86.3%

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \color{blue}{\frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2}, a2 \cdot a2, -a1 \cdot \left(a1 \cdot \sqrt{2}\right)\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2}, a2 \cdot a2, -a1 \cdot \left(a1 \cdot \sqrt{2}\right)\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) -0.02)
   (* (fma (sqrt 2.0) (* a2 a2) (- (* a1 (* a1 (sqrt 2.0))))) -0.5)
   (/ (fma a2 a2 (* a1 a1)) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= -0.02) {
		tmp = fma(sqrt(2.0), (a2 * a2), -(a1 * (a1 * sqrt(2.0)))) * -0.5;
	} else {
		tmp = fma(a2, a2, (a1 * a1)) / sqrt(2.0);
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= -0.02)
		tmp = Float64(fma(sqrt(2.0), Float64(a2 * a2), Float64(-Float64(a1 * Float64(a1 * sqrt(2.0))))) * -0.5);
	else
		tmp = Float64(fma(a2, a2, Float64(a1 * a1)) / sqrt(2.0));
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], -0.02], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(a2 * a2), $MachinePrecision] + (-N[(a1 * N[(a1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * -0.5), $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 (cos.f64 th) < -0.0200000000000000004

   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 11.2

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

   5. Applied rewrites 11.2%

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

      1. lift-sqrt.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. frac-2neg N/A

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

      7. frac-add N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 11.2

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

   7. Applied rewrites 11.2%

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

   9. Applied rewrites 0.6%

      \[\leadsto \frac{\mathsf{fma}\left(a1 \cdot a1, -\sqrt{2}, \color{blue}{\frac{-2 \cdot \left(a2 \cdot \left(a2 \cdot \left(a2 \cdot a2\right)\right)\right)}{0 + \sqrt{2} \cdot \left(a2 \cdot a2\right)}}\right)}{\sqrt{2} \cdot \left(-\sqrt{2}\right)} \]

   11. Applied rewrites 36.6%

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

   if -0.0200000000000000004 < (cos.f64 th)

   1. Initial program 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lift-/.f64 N/A

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

      11. div-inv N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 86.3

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

   7. Applied rewrites 86.3%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2}, a2 \cdot a2, -a1 \cdot \left(a1 \cdot \sqrt{2}\right)\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. lift-/.f64 N/A

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

   11. div-inv N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 53.6

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

7. Applied rewrites 53.6%

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

8. Final simplification 53.6%

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

Alternative 9: 56.8% accurate, 2.0× speedup

Math

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

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(a2 * a2) * Float64(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[(a2 * a2), $MachinePrecision] * N[(N[Cos[th], $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-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a1 around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 53.6

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

7. Applied rewrites 53.6%

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

Alternative 10: 66.0% 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 99.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. lift-/.f64 N/A

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

   11. div-inv N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in th around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 69.3

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

7. Applied rewrites 69.3%

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

Alternative 11: 39.5% 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 69.3

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

5. Applied rewrites 69.3%

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

6. Taylor expanded in a1 around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 38.9

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

8. Applied rewrites 38.9%

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

Alternative 12: 39.5% 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 69.3

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

5. Applied rewrites 69.3%

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

6. Taylor expanded in a1 around inf

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 43.4

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

8. Applied rewrites 43.4%

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

   1. lift-sqrt.f64 N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 43.4

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

10. Applied rewrites 43.4%

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

11. Final simplification 43.4%

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

Reproduce

herbie shell --seed 2024219 
(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))))