Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 8.9s

Alternatives: 12

Speedup: 1.9×

• 44.2% 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

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \frac{\frac{\mathsf{fma}\left(\sqrt{2} \cdot a2\_m, a2\_m, \sqrt{2} \cdot \left(a1 \cdot a1\right)\right)}{2}}{{\cos th}^{-1}} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (/
  (/ (fma (* (sqrt 2.0) a2_m) a2_m (* (sqrt 2.0) (* a1 a1))) 2.0)
  (pow (cos th) -1.0)))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	return (fma((sqrt(2.0) * a2_m), a2_m, (sqrt(2.0) * (a1 * a1))) / 2.0) / pow(cos(th), -1.0);
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	return Float64(Float64(fma(Float64(sqrt(2.0) * a2_m), a2_m, Float64(sqrt(2.0) * Float64(a1 * a1))) / 2.0) / (cos(th) ^ -1.0))
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * a2$95$m), $MachinePrecision] * a2$95$m + N[(N[Sqrt[2.0], $MachinePrecision] * N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] / N[Power[N[Cos[th], $MachinePrecision], -1.0], $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

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. div-inv N/A

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

   10. associate-/r* N/A

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

   11. *-lft-identity N/A

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

   12. associate-*l/ N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. unpow-1 N/A

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

   3. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-fma.f64 99.7

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. associate-/r/ N/A

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

   9. lift-/.f64 N/A

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

   10. lift-fma.f64 N/A

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

   11. distribute-rgt-in N/A

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

   12. +-commutative N/A

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

   13. lift-/.f64 N/A

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

   14. un-div-inv N/A

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

   15. lift-/.f64 N/A

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

   16. un-div-inv N/A

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

   17. frac-add N/A

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

   18. lift-sqrt.f64 N/A

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

   19. lift-sqrt.f64 N/A

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

   20. rem-square-sqrt N/A

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2\_m \cdot a2\_m\right) \leq -1 \cdot 10^{-196}:\\ \;\;\;\;\left(\mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot a2\_m\right) \cdot \frac{a2\_m}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(a1, a1, a2\_m \cdot a2\_m\right)\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* t_1 (* a2_m a2_m))) -1e-196)
     (* (* (fma (* th th) -0.5 1.0) a2_m) (/ a2_m (sqrt 2.0)))
     (* (* 0.5 (sqrt 2.0)) (fma a1 a1 (* a2_m a2_m))))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + (t_1 * (a2_m * a2_m))) <= -1e-196) {
		tmp = (fma((th * th), -0.5, 1.0) * a2_m) * (a2_m / sqrt(2.0));
	} else {
		tmp = (0.5 * sqrt(2.0)) * fma(a1, a1, (a2_m * a2_m));
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, 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_m * a2_m))) <= -1e-196)
		tmp = Float64(Float64(fma(Float64(th * th), -0.5, 1.0) * a2_m) * Float64(a2_m / sqrt(2.0)));
	else
		tmp = Float64(Float64(0.5 * sqrt(2.0)) * fma(a1, a1, Float64(a2_m * a2_m)));
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, 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$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-196], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * a2$95$m), $MachinePrecision] * N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a1 * a1 + N[(a2$95$m * a2$95$m), $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))) < -1e-196

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. div-inv N/A

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

      10. associate-/r* N/A

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

      11. *-lft-identity N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a1 around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 53.0

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

   7. Applied rewrites 53.0%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 42.7%

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

   if -1e-196 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. div-inv N/A

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

      10. associate-/r* N/A

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

      11. *-lft-identity N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.6

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

   6. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 99.6

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. distribute-rgt-in N/A

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

      12. +-commutative N/A

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

      13. lift-/.f64 N/A

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

      14. un-div-inv N/A

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

      15. lift-/.f64 N/A

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

      16. un-div-inv N/A

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

      17. frac-add N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lift-sqrt.f64 N/A

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

      20. rem-square-sqrt N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a1}^{2} \cdot \sqrt{2} + {a2}^{2} \cdot \sqrt{2}\right)} \]
   10. 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 88.5

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

   11. Applied rewrites 88.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \leq -1 \cdot 10^{-196}:\\ \;\;\;\;\left(\mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot a2\right) \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2\_m \cdot a2\_m\right) \leq -1 \cdot 10^{-196}:\\ \;\;\;\;\left(\left(\left(th \cdot th\right) \cdot a1\right) \cdot -0.5\right) \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(a1, a1, a2\_m \cdot a2\_m\right)\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* t_1 (* a2_m a2_m))) -1e-196)
     (* (* (* (* th th) a1) -0.5) (/ a1 (sqrt 2.0)))
     (* (* 0.5 (sqrt 2.0)) (fma a1 a1 (* a2_m a2_m))))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + (t_1 * (a2_m * a2_m))) <= -1e-196) {
		tmp = (((th * th) * a1) * -0.5) * (a1 / sqrt(2.0));
	} else {
		tmp = (0.5 * sqrt(2.0)) * fma(a1, a1, (a2_m * a2_m));
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, 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_m * a2_m))) <= -1e-196)
		tmp = Float64(Float64(Float64(Float64(th * th) * a1) * -0.5) * Float64(a1 / sqrt(2.0)));
	else
		tmp = Float64(Float64(0.5 * sqrt(2.0)) * fma(a1, a1, Float64(a2_m * a2_m)));
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, 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$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-196], N[(N[(N[(N[(th * th), $MachinePrecision] * a1), $MachinePrecision] * -0.5), $MachinePrecision] * N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a1 * a1 + N[(a2$95$m * a2$95$m), $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))) < -1e-196

   1. Initial program 99.6%

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

   3. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 0.8

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

   5. Applied rewrites 0.8%

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

   6. Taylor expanded in th around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

   8. Applied rewrites 50.9%

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

   9. Taylor expanded in a1 around inf

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

   11. Applied rewrites 49.2%

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

   12. Taylor expanded in th around inf

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

   14. Applied rewrites 49.2%

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

   if -1e-196 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. div-inv N/A

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

      10. associate-/r* N/A

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

      11. *-lft-identity N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.6

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

   6. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 99.6

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. distribute-rgt-in N/A

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

      12. +-commutative N/A

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

      13. lift-/.f64 N/A

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

      14. un-div-inv N/A

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

      15. lift-/.f64 N/A

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

      16. un-div-inv N/A

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

      17. frac-add N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lift-sqrt.f64 N/A

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

      20. rem-square-sqrt N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a1}^{2} \cdot \sqrt{2} + {a2}^{2} \cdot \sqrt{2}\right)} \]
   10. 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 88.5

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

   11. Applied rewrites 88.5%

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

Alternative 4: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2\_m \cdot a2\_m\right) \leq -1 \cdot 10^{-196}:\\ \;\;\;\;\left(\left(-0.5 \cdot a1\right) \cdot a1\right) \cdot \left(th \cdot \frac{th}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(a1, a1, a2\_m \cdot a2\_m\right)\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* t_1 (* a2_m a2_m))) -1e-196)
     (* (* (* -0.5 a1) a1) (* th (/ th (sqrt 2.0))))
     (* (* 0.5 (sqrt 2.0)) (fma a1 a1 (* a2_m a2_m))))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + (t_1 * (a2_m * a2_m))) <= -1e-196) {
		tmp = ((-0.5 * a1) * a1) * (th * (th / sqrt(2.0)));
	} else {
		tmp = (0.5 * sqrt(2.0)) * fma(a1, a1, (a2_m * a2_m));
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, 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_m * a2_m))) <= -1e-196)
		tmp = Float64(Float64(Float64(-0.5 * a1) * a1) * Float64(th * Float64(th / sqrt(2.0))));
	else
		tmp = Float64(Float64(0.5 * sqrt(2.0)) * fma(a1, a1, Float64(a2_m * a2_m)));
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, 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$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-196], N[(N[(N[(-0.5 * a1), $MachinePrecision] * a1), $MachinePrecision] * N[(th * N[(th / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a1 * a1 + N[(a2$95$m * a2$95$m), $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))) < -1e-196

   1. Initial program 99.6%

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

   3. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 0.8

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

   5. Applied rewrites 0.8%

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

   6. Taylor expanded in th around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

   8. Applied rewrites 50.9%

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

   9. Taylor expanded in a1 around inf

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

   11. Applied rewrites 49.2%

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

   12. Taylor expanded in th around inf

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

   14. Applied rewrites 47.2%

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

   if -1e-196 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. div-inv N/A

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

      10. associate-/r* N/A

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

      11. *-lft-identity N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.6

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

   6. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 99.6

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. distribute-rgt-in N/A

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

      12. +-commutative N/A

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

      13. lift-/.f64 N/A

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

      14. un-div-inv N/A

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

      15. lift-/.f64 N/A

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

      16. un-div-inv N/A

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

      17. frac-add N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lift-sqrt.f64 N/A

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

      20. rem-square-sqrt N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a1}^{2} \cdot \sqrt{2} + {a2}^{2} \cdot \sqrt{2}\right)} \]
   10. 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 88.5

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

   11. Applied rewrites 88.5%

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

Alternative 5: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \frac{\frac{\mathsf{fma}\left(a2\_m, a2\_m, a1 \cdot a1\right)}{\sqrt{2}}}{{\cos th}^{-1}} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (/ (/ (fma a2_m a2_m (* a1 a1)) (sqrt 2.0)) (pow (cos th) -1.0)))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	return (fma(a2_m, a2_m, (a1 * a1)) / sqrt(2.0)) / pow(cos(th), -1.0);
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	return Float64(Float64(fma(a2_m, a2_m, Float64(a1 * a1)) / sqrt(2.0)) / (cos(th) ^ -1.0))
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[(a2$95$m * a2$95$m + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[th], $MachinePrecision], -1.0], $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

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. div-inv N/A

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

   10. associate-/r* N/A

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

   11. *-lft-identity N/A

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

   12. associate-*l/ N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. unpow-1 N/A

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

   3. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 6: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \frac{\mathsf{fma}\left(a1, a1, a2\_m \cdot a2\_m\right) \cdot \cos th}{\sqrt{2}} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (/ (* (fma a1 a1 (* a2_m a2_m)) (cos th)) (sqrt 2.0)))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	return (fma(a1, a1, (a2_m * a2_m)) * cos(th)) / sqrt(2.0);
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	return Float64(Float64(fma(a1, a1, Float64(a2_m * a2_m)) * cos(th)) / sqrt(2.0))
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[(a1 * a1 + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. div-inv N/A

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

   10. associate-/r* N/A

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

   11. *-lft-identity N/A

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

   12. associate-*l/ N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. unpow-1 N/A

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

   3. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-/.f64 N/A

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

   4. remove-double-div N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-fma.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*l/ N/A

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

   12. lower-/.f64 N/A

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

   13. lower-*.f64 99.6

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

8. Applied rewrites 99.6%

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

Alternative 7: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \frac{\mathsf{fma}\left(a2\_m, a2\_m, a1 \cdot a1\right)}{\sqrt{2}} \cdot \cos th \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (* (/ (fma a2_m a2_m (* a1 a1)) (sqrt 2.0)) (cos th)))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	return (fma(a2_m, a2_m, (a1 * a1)) / sqrt(2.0)) * cos(th);
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	return Float64(Float64(fma(a2_m, a2_m, Float64(a1 * a1)) / sqrt(2.0)) * cos(th))
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[(a2$95$m * a2$95$m + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 8: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2\_m \cdot a2\_m\right)\right) \cdot \left(0.5 \cdot \cos th\right) \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (* (* (sqrt 2.0) (fma a1 a1 (* a2_m a2_m))) (* 0.5 (cos th))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	return (sqrt(2.0) * fma(a1, a1, (a2_m * a2_m))) * (0.5 * cos(th));
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	return Float64(Float64(sqrt(2.0) * fma(a1, a1, Float64(a2_m * a2_m))) * Float64(0.5 * cos(th)))
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(a1 * a1 + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. div-inv N/A

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

   10. associate-/r* N/A

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

   11. *-lft-identity N/A

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

   12. associate-*l/ N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. unpow-1 N/A

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

   3. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-fma.f64 99.7

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. associate-/r/ N/A

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

   9. lift-/.f64 N/A

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

   10. lift-fma.f64 N/A

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

   11. distribute-rgt-in N/A

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

   12. +-commutative N/A

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

   13. lift-/.f64 N/A

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

   14. un-div-inv N/A

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

   15. lift-/.f64 N/A

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

   16. un-div-inv N/A

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

   17. frac-add N/A

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

   18. lift-sqrt.f64 N/A

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

   19. lift-sqrt.f64 N/A

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

   20. rem-square-sqrt N/A

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

8. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. /-rgt-identity N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

10. Applied rewrites 99.6%

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

Alternative 9: 77.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \frac{a2\_m \cdot a2\_m}{\sqrt{2}} \cdot \cos th \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (* (/ (* a2_m a2_m) (sqrt 2.0)) (cos th)))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	return ((a2_m * a2_m) / sqrt(2.0)) * cos(th);
}

Fortran

a2_m = abs(a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1, a2_m, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = ((a2_m * a2_m) / sqrt(2.0d0)) * cos(th)
end function

Java

a2_m = Math.abs(a2);
assert a1 < a2_m && a2_m < th;
public static double code(double a1, double a2_m, double th) {
	return ((a2_m * a2_m) / Math.sqrt(2.0)) * Math.cos(th);
}

Python

a2_m = math.fabs(a2)
[a1, a2_m, th] = sort([a1, a2_m, th])
def code(a1, a2_m, th):
	return ((a2_m * a2_m) / math.sqrt(2.0)) * math.cos(th)

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	return Float64(Float64(Float64(a2_m * a2_m) / sqrt(2.0)) * cos(th))
end

MATLAB

a2_m = abs(a2);
a1, a2_m, th = num2cell(sort([a1, a2_m, th])){:}
function tmp = code(a1, a2_m, th)
	tmp = ((a2_m * a2_m) / sqrt(2.0)) * cos(th);
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[(a2$95$m * a2$95$m), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 57.3

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

7. Applied rewrites 57.3%

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

Alternative 10: 77.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \left(0.5 \cdot \left(a2\_m \cdot a2\_m\right)\right) \cdot \left(\sqrt{2} \cdot \cos th\right) \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (* (* 0.5 (* a2_m a2_m)) (* (sqrt 2.0) (cos th))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	return (0.5 * (a2_m * a2_m)) * (sqrt(2.0) * cos(th));
}

Fortran

a2_m = abs(a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1, a2_m, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = (0.5d0 * (a2_m * a2_m)) * (sqrt(2.0d0) * cos(th))
end function

Java

a2_m = Math.abs(a2);
assert a1 < a2_m && a2_m < th;
public static double code(double a1, double a2_m, double th) {
	return (0.5 * (a2_m * a2_m)) * (Math.sqrt(2.0) * Math.cos(th));
}

Python

a2_m = math.fabs(a2)
[a1, a2_m, th] = sort([a1, a2_m, th])
def code(a1, a2_m, th):
	return (0.5 * (a2_m * a2_m)) * (math.sqrt(2.0) * math.cos(th))

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	return Float64(Float64(0.5 * Float64(a2_m * a2_m)) * Float64(sqrt(2.0) * cos(th)))
end

MATLAB

a2_m = abs(a2);
a1, a2_m, th = num2cell(sort([a1, a2_m, th])){:}
function tmp = code(a1, a2_m, th)
	tmp = (0.5 * (a2_m * a2_m)) * (sqrt(2.0) * cos(th));
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(0.5 * N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. div-inv N/A

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

   10. associate-/r* N/A

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

   11. *-lft-identity N/A

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

   12. associate-*l/ N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. unpow-1 N/A

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

   3. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-fma.f64 99.7

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. associate-/r/ N/A

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

   9. lift-/.f64 N/A

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

   10. lift-fma.f64 N/A

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

   11. distribute-rgt-in N/A

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

   12. +-commutative N/A

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

   13. lift-/.f64 N/A

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

   14. un-div-inv N/A

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

   15. lift-/.f64 N/A

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

   16. un-div-inv N/A

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

   17. frac-add N/A

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

   18. lift-sqrt.f64 N/A

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

   19. lift-sqrt.f64 N/A

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

   20. rem-square-sqrt N/A

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in a1 around 0

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

    1. 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 57.3

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

11. Applied rewrites 57.3%

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

Alternative 11: 66.6% accurate, 8.3× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \left(0.5 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(a1, a1, a2\_m \cdot a2\_m\right) \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (* (* 0.5 (sqrt 2.0)) (fma a1 a1 (* a2_m a2_m))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	return (0.5 * sqrt(2.0)) * fma(a1, a1, (a2_m * a2_m));
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	return Float64(Float64(0.5 * sqrt(2.0)) * fma(a1, a1, Float64(a2_m * a2_m)))
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a1 * a1 + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. div-inv N/A

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

   10. associate-/r* N/A

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

   11. *-lft-identity N/A

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

   12. associate-*l/ N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. unpow-1 N/A

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

   3. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-fma.f64 99.7

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. associate-/r/ N/A

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

   9. lift-/.f64 N/A

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

   10. lift-fma.f64 N/A

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

   11. distribute-rgt-in N/A

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

   12. +-commutative N/A

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

   13. lift-/.f64 N/A

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

   14. un-div-inv N/A

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

   15. lift-/.f64 N/A

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

   16. un-div-inv N/A

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

   17. frac-add N/A

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

   18. lift-sqrt.f64 N/A

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

   19. lift-sqrt.f64 N/A

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

   20. rem-square-sqrt N/A

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a1}^{2} \cdot \sqrt{2} + {a2}^{2} \cdot \sqrt{2}\right)} \]
10. 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 68.7

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

11. Applied rewrites 68.7%

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

Alternative 12: 52.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \left(1 \cdot a2\_m\right) \cdot \frac{a2\_m}{\sqrt{2}} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th) :precision binary64 (* (* 1.0 a2_m) (/ a2_m (sqrt 2.0))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	return (1.0 * a2_m) * (a2_m / sqrt(2.0));
}

Fortran

a2_m = abs(a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1, a2_m, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = (1.0d0 * a2_m) * (a2_m / sqrt(2.0d0))
end function

Java

a2_m = Math.abs(a2);
assert a1 < a2_m && a2_m < th;
public static double code(double a1, double a2_m, double th) {
	return (1.0 * a2_m) * (a2_m / Math.sqrt(2.0));
}

Python

a2_m = math.fabs(a2)
[a1, a2_m, th] = sort([a1, a2_m, th])
def code(a1, a2_m, th):
	return (1.0 * a2_m) * (a2_m / math.sqrt(2.0))

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	return Float64(Float64(1.0 * a2_m) * Float64(a2_m / sqrt(2.0)))
end

MATLAB

a2_m = abs(a2);
a1, a2_m, th = num2cell(sort([a1, a2_m, th])){:}
function tmp = code(a1, a2_m, th)
	tmp = (1.0 * a2_m) * (a2_m / sqrt(2.0));
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(1.0 * a2$95$m), $MachinePrecision] * N[(a2$95$m / 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. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. div-inv N/A

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

   10. associate-/r* N/A

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

   11. *-lft-identity N/A

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

   12. associate-*l/ N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 57.3

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

7. Applied rewrites 57.3%

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

8. Taylor expanded in th around 0

   \[\leadsto \left(\left(1 + {th}^{2} \cdot \left({th}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {th}^{2}\right) - \frac{1}{2}\right)\right) \cdot a2\right) \cdot \frac{a2}{\sqrt{2}} \]
9. Step-by-step derivation

10. Applied rewrites 42.2%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, th \cdot th, 0.041666666666666664\right), th \cdot th, -0.5\right), th \cdot th, 1\right) \cdot a2\right) \cdot \frac{a2}{\sqrt{2}} \]

11. Taylor expanded in th around 0

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

13. Applied rewrites 42.7%

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

14. Final simplification 42.7%

    \[\leadsto \left(1 \cdot a2\right) \cdot \frac{a2}{\sqrt{2}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024323 
(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))))