Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 8.7s

Alternatives: 15

Speedup: 1.9×

• 45.1% 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   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: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a1, a2, th] = \mathsf{sort}([a1, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \leq -5 \cdot 10^{-117}:\\ \;\;\;\;\left(\left(\sqrt{2} \cdot a2\right) \cdot a2\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2))) -5e-117)
     (* (* (* (sqrt 2.0) a2) a2) (fma -0.25 (* th th) 0.5))
     (/ (fma a2 a2 (* a1 a1)) (sqrt 2.0)))))

C

assert(a1 < a2 && a2 < th);
double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + (t_1 * (a2 * a2))) <= -5e-117) {
		tmp = ((sqrt(2.0) * a2) * a2) * fma(-0.25, (th * th), 0.5);
	} else {
		tmp = fma(a2, a2, (a1 * a1)) / sqrt(2.0);
	}
	return tmp;
}

Julia

a1, a2, th = sort([a1, a2, th])
function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2))) <= -5e-117)
		tmp = Float64(Float64(Float64(sqrt(2.0) * a2) * a2) * fma(-0.25, Float64(th * th), 0.5));
	else
		tmp = Float64(fma(a2, a2, Float64(a1 * a1)) / sqrt(2.0));
	end
	return tmp
end

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-117], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * a2), $MachinePrecision] * a2), $MachinePrecision] * N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $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))) < -5e-117

   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 99.8%

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

   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-cos.f64 49.7

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

   7. Applied rewrites 49.7%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 52.5%

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

   if -5e-117 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.6%

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

   3. Taylor expanded in th around 0

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

      1. 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 86.2

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

   5. Applied rewrites 86.2%

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

Alternative 3: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} [a1, a2, th] = \mathsf{sort}([a1, a2, th])\\ \\ \mathsf{fma}\left(\frac{a1}{a2}, \frac{a1}{a2}, 1\right) \cdot \left(\left(\cos th \cdot a2\right) \cdot \frac{a2}{\sqrt{2}}\right) \end{array} \]

FPCore

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2 th)
 :precision binary64
 (* (fma (/ a1 a2) (/ a1 a2) 1.0) (* (* (cos th) a2) (/ a2 (sqrt 2.0)))))

C

assert(a1 < a2 && a2 < th);
double code(double a1, double a2, double th) {
	return fma((a1 / a2), (a1 / a2), 1.0) * ((cos(th) * a2) * (a2 / sqrt(2.0)));
}

Julia

a1, a2, th = sort([a1, a2, th])
function code(a1, a2, th)
	return Float64(fma(Float64(a1 / a2), Float64(a1 / a2), 1.0) * Float64(Float64(cos(th) * a2) * Float64(a2 / sqrt(2.0))))
end

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(N[(N[(a1 / a2), $MachinePrecision] * N[(a1 / a2), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision] * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $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 a2 around inf

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

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. times-frac N/A

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

   5. associate-*l* N/A

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

   6. associate-*l/ N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt1-in N/A

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

   9. lower-*.f64 N/A

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

5. Applied rewrites 78.7%

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

Alternative 4: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 5: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   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 99.6%

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

Alternative 6: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   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 7: 56.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (((cos(th) * 0.5d0) * sqrt(2.0d0)) * a2) * a2
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   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. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-cos.f64 54.3

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

7. Applied rewrites 54.3%

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

9. Applied rewrites 54.4%

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

Alternative 8: 56.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = ((0.5d0 * a2) * (sqrt(2.0d0) * a2)) * cos(th)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-cos.f64 54.3

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

7. Applied rewrites 54.3%

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

9. Applied rewrites 54.4%

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

Alternative 9: 56.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (0.5d0 * (a2 * a2)) * (sqrt(2.0d0) * cos(th))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   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. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-cos.f64 54.3

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

7. Applied rewrites 54.3%

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

Alternative 10: 66.7% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(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. 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 68.1

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

5. Applied rewrites 68.1%

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

Alternative 11: 66.7% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   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 68.1

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

7. Applied rewrites 68.1%

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

Alternative 12: 38.9% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 * sqrt(((a2 * a2) / 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
code[a1_, a2_, th_] := N[(a2 * N[Sqrt[N[(N[(a2 * a2), $MachinePrecision] / 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. 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 68.1

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

5. Applied rewrites 68.1%

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

6. Taylor expanded in a1 around 0

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

8. Applied rewrites 39.5%

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

10. Applied rewrites 28.7%

    \[\leadsto a2 \cdot \sqrt{\frac{a2 \cdot a2}{2}} \]
11. Add Preprocessing

Alternative 13: 39.9% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (a2 * a2) / sqrt(2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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. 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 68.1

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

5. Applied rewrites 68.1%

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

6. Taylor expanded in a1 around 0

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

8. Applied rewrites 39.5%

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

Alternative 14: 39.9% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 * (a2 / sqrt(2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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. 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 68.1

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

5. Applied rewrites 68.1%

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

6. Taylor expanded in a1 around 0

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

8. Applied rewrites 39.5%

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

Alternative 15: 39.9% accurate, 10.2× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a1, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (0.5d0 * (a2 * a2)) * sqrt(2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-cos.f64 54.3

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

7. Applied rewrites 54.3%

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

8. Taylor expanded in th around 0

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

10. Applied rewrites 39.5%

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

Reproduce

herbie shell --seed 2024326 
(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))))