Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 12.3s

Alternatives: 11

Speedup: 1.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \left(\left(\sqrt{2} \cdot \left(a2\_m + \frac{a1\_m \cdot a1\_m}{a2\_m}\right)\right) \cdot \cos th\right) \cdot \left(a2\_m \cdot 0.5\right) \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (* (* (* (sqrt 2.0) (+ a2_m (/ (* a1_m a1_m) a2_m))) (cos th)) (* a2_m 0.5)))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return ((sqrt(2.0) * (a2_m + ((a1_m * a1_m) / a2_m))) * cos(th)) * (a2_m * 0.5);
}

Fortran

a2_m = abs(a2)
a1_m = abs(a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = ((sqrt(2.0d0) * (a2_m + ((a1_m * a1_m) / a2_m))) * cos(th)) * (a2_m * 0.5d0)
end function

Java

a2_m = Math.abs(a2);
a1_m = Math.abs(a1);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return ((Math.sqrt(2.0) * (a2_m + ((a1_m * a1_m) / a2_m))) * Math.cos(th)) * (a2_m * 0.5);
}

Python

a2_m = math.fabs(a2)
a1_m = math.fabs(a1)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return ((math.sqrt(2.0) * (a2_m + ((a1_m * a1_m) / a2_m))) * math.cos(th)) * (a2_m * 0.5)

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(Float64(Float64(sqrt(2.0) * Float64(a2_m + Float64(Float64(a1_m * a1_m) / a2_m))) * cos(th)) * Float64(a2_m * 0.5))
end

MATLAB

a2_m = abs(a2);
a1_m = abs(a1);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = ((sqrt(2.0) * (a2_m + ((a1_m * a1_m) / a2_m))) * cos(th)) * (a2_m * 0.5);
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(a2$95$m + N[(N[(a1$95$m * a1$95$m), $MachinePrecision] / a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(a2$95$m * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   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. lift-*.f64 N/A

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

   14. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-rgt-out N/A

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

   8. div-inv N/A

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

   9. div-inv N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l/ N/A

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

   12. associate-/r/ N/A

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

   13. lift-/.f64 N/A

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

   14. frac-add N/A

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

   15. lift-*.f64 N/A

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

   16. lift-fma.f64 N/A

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

   17. lift-*.f64 N/A

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

8. Applied rewrites 93.8%

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

9. Taylor expanded in th around inf

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 N/A

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

11. Applied rewrites 93.8%

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

12. Final simplification 93.8%

    \[\leadsto \left(\left(\sqrt{2} \cdot \left(a2 + \frac{a1 \cdot a1}{a2}\right)\right) \cdot \cos th\right) \cdot \left(a2 \cdot 0.5\right) \]
13. Add Preprocessing

Alternative 2: 77.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1\_m \cdot a1\_m\right) \cdot t\_1 + \left(a2\_m \cdot a2\_m\right) \cdot t\_1 \leq -5 \cdot 10^{-118}:\\ \;\;\;\;\mathsf{fma}\left(a1\_m, a1\_m, a2\_m \cdot a2\_m\right) \cdot \frac{th \cdot \left(th \cdot -0.5\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1\_m, \frac{a1\_m}{\sqrt{2}}, \frac{1}{\frac{\sqrt{2}}{a2\_m \cdot a2\_m}}\right)\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1_m a1_m) t_1) (* (* a2_m a2_m) t_1)) -5e-118)
     (* (fma a1_m a1_m (* a2_m a2_m)) (/ (* th (* th -0.5)) (sqrt 2.0)))
     (fma a1_m (/ a1_m (sqrt 2.0)) (/ 1.0 (/ (sqrt 2.0) (* a2_m a2_m)))))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1_m * a1_m) * t_1) + ((a2_m * a2_m) * t_1)) <= -5e-118) {
		tmp = fma(a1_m, a1_m, (a2_m * a2_m)) * ((th * (th * -0.5)) / sqrt(2.0));
	} else {
		tmp = fma(a1_m, (a1_m / sqrt(2.0)), (1.0 / (sqrt(2.0) / (a2_m * a2_m))));
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1_m * a1_m) * t_1) + Float64(Float64(a2_m * a2_m) * t_1)) <= -5e-118)
		tmp = Float64(fma(a1_m, a1_m, Float64(a2_m * a2_m)) * Float64(Float64(th * Float64(th * -0.5)) / sqrt(2.0)));
	else
		tmp = fma(a1_m, Float64(a1_m / sqrt(2.0)), Float64(1.0 / Float64(sqrt(2.0) / Float64(a2_m * a2_m))));
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -5e-118], N[(N[(a1$95$m * a1$95$m + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(th * N[(th * -0.5), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a1$95$m * N[(a1$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a2$95$m * a2$95$m), $MachinePrecision]), $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))) < -5.00000000000000015e-118

   1. Initial program 99.6%

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

      1. lift-+.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. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 55.3

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

   7. Applied rewrites 55.3%

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

   8. Taylor expanded in th around inf

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

   10. Applied rewrites 55.3%

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

   if -5.00000000000000015e-118 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.6%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 84.8

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

   5. Applied rewrites 84.8%

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

   7. Applied rewrites 84.8%

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

4. Final simplification 79.4%

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

Alternative 3: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1\_m \cdot a1\_m\right) \cdot t\_1 + \left(a2\_m \cdot a2\_m\right) \cdot t\_1 \leq -5 \cdot 10^{-118}:\\ \;\;\;\;\mathsf{fma}\left(a1\_m, a1\_m, a2\_m \cdot a2\_m\right) \cdot \frac{th \cdot \left(th \cdot -0.5\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2\_m, a2\_m, a1\_m \cdot a1\_m\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1_m a1_m) t_1) (* (* a2_m a2_m) t_1)) -5e-118)
     (* (fma a1_m a1_m (* a2_m a2_m)) (/ (* th (* th -0.5)) (sqrt 2.0)))
     (/ (fma a2_m a2_m (* a1_m a1_m)) (sqrt 2.0)))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1_m * a1_m) * t_1) + ((a2_m * a2_m) * t_1)) <= -5e-118) {
		tmp = fma(a1_m, a1_m, (a2_m * a2_m)) * ((th * (th * -0.5)) / sqrt(2.0));
	} else {
		tmp = fma(a2_m, a2_m, (a1_m * a1_m)) / sqrt(2.0);
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1_m * a1_m) * t_1) + Float64(Float64(a2_m * a2_m) * t_1)) <= -5e-118)
		tmp = Float64(fma(a1_m, a1_m, Float64(a2_m * a2_m)) * Float64(Float64(th * Float64(th * -0.5)) / sqrt(2.0)));
	else
		tmp = Float64(fma(a2_m, a2_m, Float64(a1_m * a1_m)) / sqrt(2.0));
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -5e-118], N[(N[(a1$95$m * a1$95$m + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(th * N[(th * -0.5), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a2$95$m * a2$95$m + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -5.00000000000000015e-118

   1. Initial program 99.6%

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

      1. lift-+.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. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 55.3

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

   7. Applied rewrites 55.3%

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

   8. Taylor expanded in th around inf

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

   10. Applied rewrites 55.3%

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

   if -5.00000000000000015e-118 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.6%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 84.8

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in th around 0

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

      1. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt1-in N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

   8. Applied rewrites 84.8%

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

4. Final simplification 79.4%

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

Alternative 4: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1\_m \cdot a1\_m\right) \cdot t\_1 + \left(a2\_m \cdot a2\_m\right) \cdot t\_1 \leq -5 \cdot 10^{-118}:\\ \;\;\;\;\left(a2\_m \cdot a2\_m\right) \cdot \frac{\mathsf{fma}\left(-0.5, th \cdot th, 1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2\_m, a2\_m, a1\_m \cdot a1\_m\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1_m a1_m) t_1) (* (* a2_m a2_m) t_1)) -5e-118)
     (* (* a2_m a2_m) (/ (fma -0.5 (* th th) 1.0) (sqrt 2.0)))
     (/ (fma a2_m a2_m (* a1_m a1_m)) (sqrt 2.0)))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1_m * a1_m) * t_1) + ((a2_m * a2_m) * t_1)) <= -5e-118) {
		tmp = (a2_m * a2_m) * (fma(-0.5, (th * th), 1.0) / sqrt(2.0));
	} else {
		tmp = fma(a2_m, a2_m, (a1_m * a1_m)) / sqrt(2.0);
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1_m * a1_m) * t_1) + Float64(Float64(a2_m * a2_m) * t_1)) <= -5e-118)
		tmp = Float64(Float64(a2_m * a2_m) * Float64(fma(-0.5, Float64(th * th), 1.0) / sqrt(2.0)));
	else
		tmp = Float64(fma(a2_m, a2_m, Float64(a1_m * a1_m)) / sqrt(2.0));
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -5e-118], N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * N[(N[(-0.5 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a2$95$m * a2$95$m + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -5.00000000000000015e-118

   1. Initial program 99.6%

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

      1. lift-+.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. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 55.3

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

   7. Applied rewrites 55.3%

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

   8. Taylor expanded in a1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 40.6

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

   10. Applied rewrites 40.6%

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

   if -5.00000000000000015e-118 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.6%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 84.8

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in th around 0

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

      1. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt1-in N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

   8. Applied rewrites 84.8%

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

4. Final simplification 76.7%

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

Alternative 5: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1\_m \cdot a1\_m\right) \cdot t\_1 + \left(a2\_m \cdot a2\_m\right) \cdot t\_1 \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, th \cdot th, 1\right)}{\sqrt{2}} \cdot \left(a1\_m \cdot a1\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2\_m, a2\_m, a1\_m \cdot a1\_m\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1_m a1_m) t_1) (* (* a2_m a2_m) t_1)) -1e-61)
     (* (/ (fma -0.5 (* th th) 1.0) (sqrt 2.0)) (* a1_m a1_m))
     (/ (fma a2_m a2_m (* a1_m a1_m)) (sqrt 2.0)))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1_m * a1_m) * t_1) + ((a2_m * a2_m) * t_1)) <= -1e-61) {
		tmp = (fma(-0.5, (th * th), 1.0) / sqrt(2.0)) * (a1_m * a1_m);
	} else {
		tmp = fma(a2_m, a2_m, (a1_m * a1_m)) / sqrt(2.0);
	}
	return tmp;
}

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1_m * a1_m) * t_1) + Float64(Float64(a2_m * a2_m) * t_1)) <= -1e-61)
		tmp = Float64(Float64(fma(-0.5, Float64(th * th), 1.0) / sqrt(2.0)) * Float64(a1_m * a1_m));
	else
		tmp = Float64(fma(a2_m, a2_m, Float64(a1_m * a1_m)) / sqrt(2.0));
	end
	return tmp
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -1e-61], N[(N[(N[(-0.5 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(a2$95$m * a2$95$m + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -1e-61

   1. Initial program 99.7%

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

      1. lift-+.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. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 58.8

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

   7. Applied rewrites 58.8%

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

   8. Taylor expanded in a1 around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 47.7

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

   10. Applied rewrites 47.7%

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

   if -1e-61 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.6%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in th around 0

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

      1. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt1-in N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

   8. Applied rewrites 83.6%

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

4. Final simplification 77.5%

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

Alternative 6: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \cos th \cdot \frac{\mathsf{fma}\left(a1\_m, a1\_m, a2\_m \cdot a2\_m\right)}{\sqrt{2}} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (* (cos th) (/ (fma a1_m a1_m (* a2_m a2_m)) (sqrt 2.0))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return cos(th) * (fma(a1_m, a1_m, (a2_m * a2_m)) / sqrt(2.0));
}

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(cos(th) * Float64(fma(a1_m, a1_m, Float64(a2_m * a2_m)) / sqrt(2.0)))
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a1$95$m * a1$95$m + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lift-/.f64 N/A

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

   6. 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. lift-*.f64 N/A

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

   14. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 7: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \left(a2\_m \cdot a2\_m\right) \cdot \left(\sqrt{2} \cdot \left(\cos th \cdot 0.5\right)\right) \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (* (* a2_m a2_m) (* (sqrt 2.0) (* (cos th) 0.5))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return (a2_m * a2_m) * (sqrt(2.0) * (cos(th) * 0.5));
}

Fortran

a2_m = abs(a2)
a1_m = abs(a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = (a2_m * a2_m) * (sqrt(2.0d0) * (cos(th) * 0.5d0))
end function

Java

a2_m = Math.abs(a2);
a1_m = Math.abs(a1);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return (a2_m * a2_m) * (Math.sqrt(2.0) * (Math.cos(th) * 0.5));
}

Python

a2_m = math.fabs(a2)
a1_m = math.fabs(a1)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return (a2_m * a2_m) * (math.sqrt(2.0) * (math.cos(th) * 0.5))

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(Float64(a2_m * a2_m) * Float64(sqrt(2.0) * Float64(cos(th) * 0.5)))
end

MATLAB

a2_m = abs(a2);
a1_m = abs(a1);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = (a2_m * a2_m) * (sqrt(2.0) * (cos(th) * 0.5));
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lift-/.f64 N/A

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

   6. 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. lift-*.f64 N/A

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

   14. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-rgt-out N/A

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

   8. div-inv N/A

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

   9. div-inv N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l/ N/A

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

   12. associate-/r/ N/A

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

   13. lift-/.f64 N/A

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

   14. frac-add N/A

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

   15. lift-*.f64 N/A

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

   16. lift-fma.f64 N/A

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

   17. lift-*.f64 N/A

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

8. Applied rewrites 93.8%

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

9. Taylor expanded in a2 around inf

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. unpow2 N/A

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

    6. lower-*.f64 N/A

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

    7. associate-*r* N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. lower-sqrt.f64 N/A

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

    11. lower-*.f64 N/A

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

    12. lower-cos.f64 53.7

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

11. Applied rewrites 53.7%

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

12. Final simplification 53.7%

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

Alternative 8: 66.0% accurate, 8.1× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \frac{\mathsf{fma}\left(a2\_m, a2\_m, a1\_m \cdot a1\_m\right)}{\sqrt{2}} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (/ (fma a2_m a2_m (* a1_m a1_m)) (sqrt 2.0)))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return fma(a2_m, a2_m, (a1_m * a1_m)) / sqrt(2.0);
}

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(fma(a2_m, a2_m, Float64(a1_m * a1_m)) / sqrt(2.0))
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(a2$95$m * a2$95$m + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 69.3

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

5. Applied rewrites 69.3%

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

6. Taylor expanded in th around 0

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

   1. *-rgt-identity N/A

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

   2. *-inverses N/A

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

   3. associate-/l* N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt1-in N/A

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

   7. +-commutative N/A

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. lower-/.f64 N/A

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

8. Applied rewrites 69.3%

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

Alternative 9: 66.0% accurate, 8.3× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \sqrt{2} \cdot \left(0.5 \cdot \mathsf{fma}\left(a1\_m, a1\_m, a2\_m \cdot a2\_m\right)\right) \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (* (sqrt 2.0) (* 0.5 (fma a1_m a1_m (* a2_m a2_m)))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return sqrt(2.0) * (0.5 * fma(a1_m, a1_m, (a2_m * a2_m)));
}

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(sqrt(2.0) * Float64(0.5 * fma(a1_m, a1_m, Float64(a2_m * a2_m))))
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.5 * N[(a1$95$m * a1$95$m + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing
3. 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. lift-*.f64 N/A

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

   14. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. associate-*l/ N/A

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

   6. lift-/.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. lift-/.f64 N/A

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

   12. associate-*l/ N/A

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

   13. lift-/.f64 N/A

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

   14. associate-*l/ N/A

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

   15. frac-add N/A

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in th around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. distribute-lft-out N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 69.3

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

9. Applied rewrites 69.3%

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

Alternative 10: 65.7% accurate, 9.9× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \frac{a2\_m \cdot a2\_m}{\sqrt{2}} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th) :precision binary64 (/ (* a2_m a2_m) (sqrt 2.0)))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return (a2_m * a2_m) / sqrt(2.0);
}

Fortran

a2_m = abs(a2)
a1_m = abs(a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = (a2_m * a2_m) / sqrt(2.0d0)
end function

Java

a2_m = Math.abs(a2);
a1_m = Math.abs(a1);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return (a2_m * a2_m) / Math.sqrt(2.0);
}

Python

a2_m = math.fabs(a2)
a1_m = math.fabs(a1)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return (a2_m * a2_m) / math.sqrt(2.0)

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(Float64(a2_m * a2_m) / sqrt(2.0))
end

MATLAB

a2_m = abs(a2);
a1_m = abs(a1);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = (a2_m * a2_m) / sqrt(2.0);
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[(a2$95$m * a2$95$m), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 69.3

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

5. Applied rewrites 69.3%

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

6. Taylor expanded in a1 around 0

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

8. Applied rewrites 38.9%

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

Alternative 11: 13.3% accurate, 9.9× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ a1\_m \cdot \frac{a1\_m}{\sqrt{2}} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
a1_m = (fabs.f64 a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th) :precision binary64 (* a1_m (/ a1_m (sqrt 2.0))))

C

a2_m = fabs(a2);
a1_m = fabs(a1);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return a1_m * (a1_m / sqrt(2.0));
}

Fortran

a2_m = abs(a2)
a1_m = abs(a1)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = a1_m * (a1_m / sqrt(2.0d0))
end function

Java

a2_m = Math.abs(a2);
a1_m = Math.abs(a1);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return a1_m * (a1_m / Math.sqrt(2.0));
}

Python

a2_m = math.fabs(a2)
a1_m = math.fabs(a1)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return a1_m * (a1_m / math.sqrt(2.0))

Julia

a2_m = abs(a2)
a1_m = abs(a1)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(a1_m * Float64(a1_m / sqrt(2.0)))
end

MATLAB

a2_m = abs(a2);
a1_m = abs(a1);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = a1_m * (a1_m / sqrt(2.0));
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(a1$95$m * N[(a1$95$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. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 69.3

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

5. Applied rewrites 69.3%

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

6. Taylor expanded in a1 around inf

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

8. Applied rewrites 43.4%

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

10. Applied rewrites 43.4%

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

11. Final simplification 43.4%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))