Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 11.2s

Alternatives: 10

Speedup: 1.9×

• 43.6% 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

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

FPCore

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

C

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

Julia

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

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(a1$95$m * a1$95$m + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. *-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. clear-num N/A

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

   13. un-div-inv N/A

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

   14. lower-/.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-/.f64 99.7

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

4. Applied egg-rr 99.7%

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

Alternative 2: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1\_m \cdot a1\_m\right) + \left(a2 \cdot a2\right) \cdot t\_1 \leq -5 \cdot 10^{-298}:\\ \;\;\;\;\left(\mathsf{fma}\left(a1\_m, a1\_m, a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1\_m \cdot a1\_m\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1_m a1_m)) (* (* a2 a2) t_1)) -5e-298)
     (* (* (fma a1_m a1_m (* a2 a2)) (sqrt 2.0)) -0.5)
     (/ (fma a2 a2 (* a1_m a1_m)) (sqrt 2.0)))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1_m * a1_m)) + ((a2 * a2) * t_1)) <= -5e-298) {
		tmp = (fma(a1_m, a1_m, (a2 * a2)) * sqrt(2.0)) * -0.5;
	} else {
		tmp = fma(a2, a2, (a1_m * a1_m)) / sqrt(2.0);
	}
	return tmp;
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1_m * a1_m)) + Float64(Float64(a2 * a2) * t_1)) <= -5e-298)
		tmp = Float64(Float64(fma(a1_m, a1_m, Float64(a2 * a2)) * sqrt(2.0)) * -0.5);
	else
		tmp = Float64(fma(a2, a2, Float64(a1_m * a1_m)) / sqrt(2.0));
	end
	return tmp
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -5e-298], N[(N[(N[(a1$95$m * a1$95$m + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(a2 * a2 + 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))) < -5.0000000000000002e-298

   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 0.8

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

   5. Simplified 0.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-*.f64 N/A

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

      11. frac-add N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. rem-square-sqrt N/A

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

      15. lower-/.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 0.8

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

   7. Applied egg-rr 0.8%

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

   9. Applied egg-rr 64.5%

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

   if -5.0000000000000002e-298 < (+.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 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 87.8

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

   7. Simplified 87.8%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -5 \cdot 10^{-298}:\\ \;\;\;\;\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a1\_m, a1\_m, a2 \cdot a2\right) \cdot \sqrt{2}\\ t_2 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_2 \cdot \left(a1\_m \cdot a1\_m\right) + \left(a2 \cdot a2\right) \cdot t\_2 \leq -5 \cdot 10^{-298}:\\ \;\;\;\;t\_1 \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (let* ((t_1 (* (fma a1_m a1_m (* a2 a2)) (sqrt 2.0)))
        (t_2 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_2 (* a1_m a1_m)) (* (* a2 a2) t_2)) -5e-298)
     (* t_1 -0.5)
     (* t_1 0.5))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	double t_1 = fma(a1_m, a1_m, (a2 * a2)) * sqrt(2.0);
	double t_2 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_2 * (a1_m * a1_m)) + ((a2 * a2) * t_2)) <= -5e-298) {
		tmp = t_1 * -0.5;
	} else {
		tmp = t_1 * 0.5;
	}
	return tmp;
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	t_1 = Float64(fma(a1_m, a1_m, Float64(a2 * a2)) * sqrt(2.0))
	t_2 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_2 * Float64(a1_m * a1_m)) + Float64(Float64(a2 * a2) * t_2)) <= -5e-298)
		tmp = Float64(t_1 * -0.5);
	else
		tmp = Float64(t_1 * 0.5);
	end
	return tmp
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := Block[{t$95$1 = N[(N[(a1$95$m * a1$95$m + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 * N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], -5e-298], N[(t$95$1 * -0.5), $MachinePrecision], N[(t$95$1 * 0.5), $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))) < -5.0000000000000002e-298

   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 0.8

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

   5. Simplified 0.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-*.f64 N/A

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

      11. frac-add N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. rem-square-sqrt N/A

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

      15. lower-/.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 0.8

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

   7. Applied egg-rr 0.8%

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

   9. Applied egg-rr 64.5%

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

   if -5.0000000000000002e-298 < (+.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 99.7%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 87.7

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

   5. Simplified 87.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-*.f64 N/A

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

      11. frac-add N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. rem-square-sqrt N/A

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

      15. lower-/.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 87.8

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

   7. Applied egg-rr 87.8%

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

   9. Applied egg-rr 87.8%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -5 \cdot 10^{-298}:\\ \;\;\;\;\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1\_m \cdot a1\_m\right) + \left(a2 \cdot a2\right) \cdot t\_1 \leq -5 \cdot 10^{-298}:\\ \;\;\;\;\left(\mathsf{fma}\left(a1\_m, a1\_m, a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(a2 \cdot \left(a2 \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1_m a1_m)) (* (* a2 a2) t_1)) -5e-298)
     (* (* (fma a1_m a1_m (* a2 a2)) (sqrt 2.0)) -0.5)
     (* 0.5 (* a2 (* a2 (sqrt 2.0)))))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1_m * a1_m)) + ((a2 * a2) * t_1)) <= -5e-298) {
		tmp = (fma(a1_m, a1_m, (a2 * a2)) * sqrt(2.0)) * -0.5;
	} else {
		tmp = 0.5 * (a2 * (a2 * sqrt(2.0)));
	}
	return tmp;
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1_m * a1_m)) + Float64(Float64(a2 * a2) * t_1)) <= -5e-298)
		tmp = Float64(Float64(fma(a1_m, a1_m, Float64(a2 * a2)) * sqrt(2.0)) * -0.5);
	else
		tmp = Float64(0.5 * Float64(a2 * Float64(a2 * sqrt(2.0))));
	end
	return tmp
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -5e-298], N[(N[(N[(a1$95$m * a1$95$m + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], N[(0.5 * N[(a2 * N[(a2 * N[Sqrt[2.0], $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))) < -5.0000000000000002e-298

   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 0.8

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

   5. Simplified 0.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-*.f64 N/A

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

      11. frac-add N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. rem-square-sqrt N/A

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

      15. lower-/.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 0.8

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

   7. Applied egg-rr 0.8%

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

   9. Applied egg-rr 64.5%

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

   if -5.0000000000000002e-298 < (+.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 99.7%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 87.7

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

   5. Simplified 87.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-*.f64 N/A

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

      11. frac-add N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. rem-square-sqrt N/A

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

      15. lower-/.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 87.8

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

   7. Applied egg-rr 87.8%

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

   8. Taylor expanded in a2 around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 50.7

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

   10. Simplified 50.7%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -5 \cdot 10^{-298}:\\ \;\;\;\;\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(a2 \cdot \left(a2 \cdot \sqrt{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} \mathbf{if}\;\cos th \leq -0.004:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a1\_m, a1\_m, a2 \cdot a2\right) \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1\_m \cdot a1\_m\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (if (<= (cos th) -0.004)
   (- (/ (* (fma a1_m a1_m (* a2 a2)) (sqrt 2.0)) (* (sqrt 2.0) (sqrt 2.0))))
   (/ (fma a2 a2 (* a1_m a1_m)) (sqrt 2.0))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	double tmp;
	if (cos(th) <= -0.004) {
		tmp = -((fma(a1_m, a1_m, (a2 * a2)) * sqrt(2.0)) / (sqrt(2.0) * sqrt(2.0)));
	} else {
		tmp = fma(a2, a2, (a1_m * a1_m)) / sqrt(2.0);
	}
	return tmp;
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	tmp = 0.0
	if (cos(th) <= -0.004)
		tmp = Float64(-Float64(Float64(fma(a1_m, a1_m, Float64(a2 * a2)) * sqrt(2.0)) / Float64(sqrt(2.0) * sqrt(2.0))));
	else
		tmp = Float64(fma(a2, a2, Float64(a1_m * a1_m)) / sqrt(2.0));
	end
	return tmp
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], -0.004], (-N[(N[(N[(a1$95$m * a1$95$m + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(a2 * a2 + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -0.0040000000000000001

   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 7.3

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

   5. Simplified 7.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-*.f64 N/A

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

      11. frac-2neg N/A

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

      12. frac-add N/A

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

      13. lower-/.f64 N/A

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

   7. Applied egg-rr 7.3%

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

   9. Applied egg-rr 66.8%

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

   if -0.0040000000000000001 < (cos.f64 th)

   1. Initial program 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 87.5

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

   7. Simplified 87.5%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -0.004:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(a1$95$m * a1$95$m + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 99.7

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

4. Applied egg-rr 99.7%

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

Alternative 7: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a1$95$m * a1$95$m + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. lift-/.f64 N/A

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

   11. div-inv N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

4. Applied egg-rr 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 8: 78.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(a2 * N[(a2 * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. *-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. clear-num N/A

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

   13. un-div-inv N/A

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

   14. lower-/.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-/.f64 99.7

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-sqrt.f64 54.6

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

7. Simplified 54.6%

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

8. Final simplification 54.6%

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

Alternative 9: 78.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(a2 * N[(N[(a2 * N[Cos[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in a1 around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-sqrt.f64 54.6

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

5. Simplified 54.6%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 54.6

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

7. Applied egg-rr 54.6%

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

Alternative 10: 52.9% accurate, 10.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(0.5 * N[(a2 * N[(a2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 68.4

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

5. Simplified 68.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. associate-*r/ N/A

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

   10. lift-*.f64 N/A

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

   11. frac-add N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. rem-square-sqrt N/A

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

   15. lower-/.f64 N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-*.f64 68.4

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

7. Applied egg-rr 68.4%

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

8. Taylor expanded in a2 around inf

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 39.7

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

10. Simplified 39.7%

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

Reproduce

herbie shell --seed 2024208 
(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))))