Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 10.9s

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, 1.0× speedup

Math

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

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
 (fma
  (* (cos th) (/ a2_m (sqrt 2.0)))
  a2_m
  (* (cos th) (* 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((cos(th) * (a2_m / sqrt(2.0))), a2_m, (cos(th) * (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 fma(Float64(cos(th) * Float64(a2_m / sqrt(2.0))), a2_m, Float64(cos(th) * Float64(a1_m * Float64(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[(N[Cos[th], $MachinePrecision] * N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a2$95$m + N[(N[Cos[th], $MachinePrecision] * N[(a1$95$m * N[(a1$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. associate-*l/ N/A

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

   12. *-lft-identity N/A

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

   13. lower-/.f64 99.6

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

   14. lift-*.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. associate-*l/ N/A

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

   17. associate-/l* N/A

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

   18. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

Alternative 2: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1\_m \cdot a1\_m\right) \cdot t\_1 + \left(a2\_m \cdot a2\_m\right) \cdot t\_1 \leq -2 \cdot 10^{-137}:\\ \;\;\;\;\frac{\left(a2\_m \cdot a2\_m\right) \cdot \mathsf{fma}\left(th, th \cdot -0.5, 1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a2\_m, \frac{a2\_m}{\sqrt{2}}, \frac{a1\_m \cdot a1\_m}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

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)) -2e-137)
     (/ (* (* a2_m a2_m) (fma th (* th -0.5) 1.0)) (sqrt 2.0))
     (fma a2_m (/ a2_m (sqrt 2.0)) (/ (* 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)) <= -2e-137) {
		tmp = ((a2_m * a2_m) * fma(th, (th * -0.5), 1.0)) / sqrt(2.0);
	} else {
		tmp = fma(a2_m, (a2_m / sqrt(2.0)), ((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)) <= -2e-137)
		tmp = Float64(Float64(Float64(a2_m * a2_m) * fma(th, Float64(th * -0.5), 1.0)) / sqrt(2.0));
	else
		tmp = fma(a2_m, Float64(a2_m / sqrt(2.0)), Float64(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], -2e-137], N[(N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * N[(th * N[(th * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(a2$95$m * N[(a2$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a1$95$m * a1$95$m), $MachinePrecision] / N[Sqrt[2.0], $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))) < -1.99999999999999996e-137

   1. Initial program 99.5%

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

   3. Taylor expanded in a1 around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*l* N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 57.9%

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

   9. Taylor expanded in a1 around 0

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

   11. Applied rewrites 46.7%

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

   if -1.99999999999999996e-137 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 85.1

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

   5. Applied rewrites 85.1%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -2 \cdot 10^{-137}:\\ \;\;\;\;\frac{\left(a2 \cdot a2\right) \cdot \mathsf{fma}\left(th, th \cdot -0.5, 1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a2, \frac{a2}{\sqrt{2}}, \frac{a1 \cdot a1}{\sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1\_m \cdot a1\_m\right) \cdot t\_1 + \left(a2\_m \cdot a2\_m\right) \cdot t\_1 \leq -2 \cdot 10^{-137}:\\ \;\;\;\;\frac{\left(a2\_m \cdot a2\_m\right) \cdot \mathsf{fma}\left(th, th \cdot -0.5, 1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(a1\_m, a1\_m, a2\_m \cdot a2\_m\right)\right)\\ \end{array} \end{array} \]

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)) -2e-137)
     (/ (* (* a2_m a2_m) (fma th (* th -0.5) 1.0)) (sqrt 2.0))
     (* 0.5 (* (sqrt 2.0) (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) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1_m * a1_m) * t_1) + ((a2_m * a2_m) * t_1)) <= -2e-137) {
		tmp = ((a2_m * a2_m) * fma(th, (th * -0.5), 1.0)) / sqrt(2.0);
	} else {
		tmp = 0.5 * (sqrt(2.0) * fma(a1_m, a1_m, (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)) <= -2e-137)
		tmp = Float64(Float64(Float64(a2_m * a2_m) * fma(th, Float64(th * -0.5), 1.0)) / sqrt(2.0));
	else
		tmp = Float64(0.5 * Float64(sqrt(2.0) * fma(a1_m, a1_m, 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], -2e-137], N[(N[(N[(a2$95$m * a2$95$m), $MachinePrecision] * N[(th * N[(th * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a1$95$m * a1$95$m + 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))) < -1.99999999999999996e-137

   1. Initial program 99.5%

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

   3. Taylor expanded in a1 around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*l* N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 57.9%

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

   9. Taylor expanded in a1 around 0

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

   11. Applied rewrites 46.7%

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

   if -1.99999999999999996e-137 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 85.1%

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

   9. Applied rewrites 85.1%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -2 \cdot 10^{-137}:\\ \;\;\;\;\frac{\left(a2 \cdot a2\right) \cdot \mathsf{fma}\left(th, th \cdot -0.5, 1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1\_m \cdot a1\_m\right) \cdot t\_1 + \left(a2\_m \cdot a2\_m\right) \cdot t\_1 \leq -2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\left(a1\_m \cdot a1\_m\right) \cdot \mathsf{fma}\left(th, th \cdot -0.5, 1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(a1\_m, a1\_m, a2\_m \cdot a2\_m\right)\right)\\ \end{array} \end{array} \]

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)) -2e-48)
     (/ (* (* a1_m a1_m) (fma th (* th -0.5) 1.0)) (sqrt 2.0))
     (* 0.5 (* (sqrt 2.0) (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) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1_m * a1_m) * t_1) + ((a2_m * a2_m) * t_1)) <= -2e-48) {
		tmp = ((a1_m * a1_m) * fma(th, (th * -0.5), 1.0)) / sqrt(2.0);
	} else {
		tmp = 0.5 * (sqrt(2.0) * fma(a1_m, a1_m, (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)) <= -2e-48)
		tmp = Float64(Float64(Float64(a1_m * a1_m) * fma(th, Float64(th * -0.5), 1.0)) / sqrt(2.0));
	else
		tmp = Float64(0.5 * Float64(sqrt(2.0) * fma(a1_m, a1_m, 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], -2e-48], N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] * N[(th * N[(th * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a1$95$m * a1$95$m + 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))) < -1.9999999999999999e-48

   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 a1 around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*l* N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 61.1%

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

   9. Taylor expanded in a1 around inf

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

   11. Applied rewrites 49.3%

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

   if -1.9999999999999999e-48 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 83.9%

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

   9. Applied rewrites 83.9%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\left(a1 \cdot a1\right) \cdot \mathsf{fma}\left(th, th \cdot -0.5, 1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ a1_m = \left|a1\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \frac{\mathsf{fma}\left(a1\_m, a1\_m, a2\_m \cdot a2\_m\right)}{\sqrt{2} \cdot \frac{1}{\cos th}} \end{array} \]

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 a1_m a1_m (* a2_m a2_m)) (* (sqrt 2.0) (/ 1.0 (cos th)))))

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(a1_m, a1_m, (a2_m * a2_m)) / (sqrt(2.0) * (1.0 / cos(th)));
}

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(a1_m, a1_m, Float64(a2_m * a2_m)) / Float64(sqrt(2.0) * Float64(1.0 / cos(th))))
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[(a1$95$m * a1$95$m + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 / N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 6: 99.6% accurate, 1.9× speedup

Math

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

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
 (* (cos th) (* (sqrt 2.0) (* 0.5 (fma a2_m a2_m (* a1_m a1_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 cos(th) * (sqrt(2.0) * (0.5 * fma(a2_m, a2_m, (a1_m * a1_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(cos(th) * Float64(sqrt(2.0) * Float64(0.5 * fma(a2_m, a2_m, Float64(a1_m * a1_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[Cos[th], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.5 * N[(a2$95$m * a2$95$m + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. associate-*l/ N/A

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

   12. *-lft-identity N/A

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

   13. lower-/.f64 99.6

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

   14. lift-*.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. associate-*l/ N/A

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

   17. associate-/l* N/A

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

   18. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*r/ N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. distribute-lft-out N/A

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

   12. lift-fma.f64 N/A

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

   13. lower-*.f64 99.6

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

   14. lift-fma.f64 N/A

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

   15. +-commutative N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in a1 around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. distribute-lft-out N/A

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

   8. +-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.6

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

9. Applied rewrites 99.6%

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

Alternative 7: 99.2% accurate, 2.0× speedup

Math

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

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
 (* (* a2_m (* (cos th) a2_m)) (* (sqrt 2.0) 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 * (cos(th) * a2_m)) * (sqrt(2.0) * 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 * (cos(th) * a2_m)) * (sqrt(2.0d0) * 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 * (Math.cos(th) * a2_m)) * (Math.sqrt(2.0) * 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 * (math.cos(th) * a2_m)) * (math.sqrt(2.0) * 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 * Float64(cos(th) * a2_m)) * Float64(sqrt(2.0) * 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 * (cos(th) * a2_m)) * (sqrt(2.0) * 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 * N[(N[Cos[th], $MachinePrecision] * a2$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. associate-*l/ N/A

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

   12. *-lft-identity N/A

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

   13. lower-/.f64 99.6

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

   14. lift-*.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. associate-*l/ N/A

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

   17. associate-/l* N/A

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

   18. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*r/ N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. distribute-lft-out N/A

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

   12. lift-fma.f64 N/A

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

   13. lower-*.f64 99.6

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

   14. lift-fma.f64 N/A

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

   15. +-commutative N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a2}^{2} \cdot \left(\cos th \cdot \sqrt{2}\right)\right)} \]
8. 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

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sqrt.f64 59.8

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

9. Applied rewrites 59.8%

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

10. Final simplification 59.8%

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

Alternative 8: 66.9% accurate, 8.3× speedup

Math

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

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
 (* 0.5 (* (sqrt 2.0) (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 0.5 * (sqrt(2.0) * 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(0.5 * Float64(sqrt(2.0) * 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[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a1$95$m * a1$95$m + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in th around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 67.3

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

5. Applied rewrites 67.3%

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

7. Applied rewrites 67.2%

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

9. Applied rewrites 67.2%

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

10. Final simplification 67.2%

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

Alternative 9: 66.7% 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 99.5%

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

3. Taylor expanded in th around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 67.3

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

5. Applied rewrites 67.3%

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

6. Taylor expanded in a1 around 0

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

8. Applied rewrites 44.0%

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

Alternative 10: 66.7% accurate, 10.2× speedup

Math

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

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
 (* 0.5 (* 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 0.5 * (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 = 0.5d0 * (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 0.5 * (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 0.5 * (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(0.5 * Float64(a2_m * Float64(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 = 0.5 * (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[(0.5 * N[(a2$95$m * N[(a2$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in th around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 67.3

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

5. Applied rewrites 67.3%

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

7. Applied rewrites 67.2%

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

8. Taylor expanded in a1 around 0

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

10. Applied rewrites 44.0%

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

11. Final simplification 44.0%

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

Alternative 11: 13.3% accurate, 10.2× speedup

Math

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

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 (* a1_m a1_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 * (a1_m * a1_m));
}

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) * (0.5d0 * (a1_m * a1_m))
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) * (0.5 * (a1_m * a1_m));
}

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) * (0.5 * (a1_m * a1_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 * Float64(a1_m * a1_m)))
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) * (0.5 * (a1_m * a1_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), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in th around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 67.3

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

5. Applied rewrites 67.3%

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

7. Applied rewrites 67.2%

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

8. Taylor expanded in a1 around inf

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

10. Applied rewrites 37.4%

    \[\leadsto \left(a1 \cdot \left(a1 \cdot \sqrt{2}\right)\right) \cdot \color{blue}{0.5} \]
11. Step-by-step derivation

12. Applied rewrites 37.4%

    \[\leadsto \left(0.5 \cdot \left(a1 \cdot a1\right)\right) \cdot \sqrt{2} \]

13. Final simplification 37.4%

    \[\leadsto \sqrt{2} \cdot \left(0.5 \cdot \left(a1 \cdot a1\right)\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(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))))