Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 8.6s

Alternatives: 10

Speedup: 1.9×

• 43.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

Alternative 2: 71.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \leq 10^{-202}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, th \cdot th, 0.5\right) \cdot \left(\left(\sqrt{2} \cdot a2\right) \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2))) 1e-202)
     (* (fma -0.25 (* th th) 0.5) (* (* (sqrt 2.0) a2) a2))
     (/ (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)) + (t_1 * (a2 * a2))) <= 1e-202) {
		tmp = fma(-0.25, (th * th), 0.5) * ((sqrt(2.0) * a2) * a2);
	} 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(t_1 * Float64(a2 * a2))) <= 1e-202)
		tmp = Float64(fma(-0.25, Float64(th * th), 0.5) * Float64(Float64(sqrt(2.0) * a2) * a2));
	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[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-202], N[(N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * a2), $MachinePrecision] * a2), $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-202

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. frac-add N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a1 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 59.2

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

   7. Applied rewrites 59.2%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 49.7%

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

   if 1e-202 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 85.3

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

   5. Applied rewrites 85.3%

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

Alternative 3: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \leq -2 \cdot 10^{-155}:\\ \;\;\;\;\left(-0.25 \cdot \left(th \cdot th\right)\right) \cdot \left(\left(\sqrt{2} \cdot a2\right) \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2))) -2e-155)
     (* (* -0.25 (* th th)) (* (* (sqrt 2.0) a2) a2))
     (* (* 0.5 (sqrt 2.0)) (fma a1 a1 (* a2 a2))))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + (t_1 * (a2 * a2))) <= -2e-155) {
		tmp = (-0.25 * (th * th)) * ((sqrt(2.0) * a2) * a2);
	} else {
		tmp = (0.5 * sqrt(2.0)) * fma(a1, a1, (a2 * a2));
	}
	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(t_1 * Float64(a2 * a2))) <= -2e-155)
		tmp = Float64(Float64(-0.25 * Float64(th * th)) * Float64(Float64(sqrt(2.0) * a2) * a2));
	else
		tmp = Float64(Float64(0.5 * sqrt(2.0)) * fma(a1, a1, Float64(a2 * a2)));
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-155], N[(N[(-0.25 * N[(th * th), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * a2), $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -2.00000000000000003e-155

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. frac-add N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a1 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 49.0

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

   7. Applied rewrites 49.0%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 40.7%

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

   11. Taylor expanded in th around inf

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

   13. Applied rewrites 40.7%

       \[\leadsto \left(-0.25 \cdot \left(th \cdot th\right)\right) \cdot \left(\left(\sqrt{2} \cdot \color{blue}{a2}\right) \cdot a2\right) \]

   if -2.00000000000000003e-155 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. frac-add N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in th around 0

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

      1. distribute-rgt-out N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.5

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

   7. Applied rewrites 84.5%

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

Alternative 4: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. distribute-lft-in N/A

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

   2. distribute-rgt-out N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

9. Applied rewrites 99.7%

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

Alternative 5: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. distribute-lft-in N/A

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

   2. distribute-rgt-out N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

Alternative 6: 56.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sqrt.f64 53.0

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

7. Applied rewrites 53.0%

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

9. Applied rewrites 53.0%

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

Alternative 7: 56.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sqrt.f64 53.0

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

7. Applied rewrites 53.0%

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

Alternative 8: 56.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sqrt.f64 53.0

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

7. Applied rewrites 53.0%

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

9. Applied rewrites 53.0%

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

Alternative 9: 65.9% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in th around 0

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

   1. distribute-rgt-out N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 62.2

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

7. Applied rewrites 62.2%

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

Alternative 10: 39.7% accurate, 10.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sqrt.f64 53.0

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

7. Applied rewrites 53.0%

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

8. Taylor expanded in th around 0

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

10. Applied rewrites 35.0%

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

Reproduce

herbie shell --seed 2024340 
(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))))