Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.5%

Time: 11.4s

Alternatives: 16

Speedup: 1.9×

• 42.8% 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: 99.5% accurate, 1.9× speedup

Math

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

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
 (* (/ (cos th) (sqrt 2.0)) (fma a2 a2 (* a1_m a1_m))))

C

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

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(cos(th) / sqrt(2.0)) * fma(a2, a2, Float64(a1_m * a1_m)))
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[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a2 * a2 + N[(a1$95$m * a1$95$m), $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-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a2 \cdot a2\right) \cdot t\_1 + t\_1 \cdot \left(a1\_m \cdot a1\_m\right) \leq -5 \cdot 10^{-122}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(a1\_m, a1\_m, a2 \cdot a2\right) \cdot th\right) \cdot th\right) \cdot \left(-0.5 \cdot \sqrt{2}\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1\_m \cdot a1\_m\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

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 (<= (+ (* (* a2 a2) t_1) (* t_1 (* a1_m a1_m))) -5e-122)
     (* (* (* (* (fma a1_m a1_m (* a2 a2)) th) th) (* -0.5 (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 ((((a2 * a2) * t_1) + (t_1 * (a1_m * a1_m))) <= -5e-122) {
		tmp = (((fma(a1_m, a1_m, (a2 * a2)) * th) * th) * (-0.5 * 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(Float64(a2 * a2) * t_1) + Float64(t_1 * Float64(a1_m * a1_m))) <= -5e-122)
		tmp = Float64(Float64(Float64(Float64(fma(a1_m, a1_m, Float64(a2 * a2)) * th) * th) * Float64(-0.5 * 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[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$1 * N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-122], N[(N[(N[(N[(N[(a1$95$m * a1$95$m + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision] * th), $MachinePrecision] * N[(-0.5 * N[Sqrt[2.0], $MachinePrecision]), $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))) < -4.9999999999999999e-122

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. frac-add N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

   7. Applied rewrites 43.7%

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

   8. Taylor expanded in th around inf

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

   10. Applied rewrites 43.7%

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

   if -4.9999999999999999e-122 < (+.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-+.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. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

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

      1. lower-sqrt.f64 82.4

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

   7. Applied rewrites 82.4%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} + \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) \leq -5 \cdot 10^{-122}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot th\right) \cdot th\right) \cdot \left(-0.5 \cdot \sqrt{2}\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a2 \cdot a2\right) \cdot t\_1 + t\_1 \cdot \left(a1\_m \cdot a1\_m\right) \leq -5 \cdot 10^{-122}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1\_m \cdot a1\_m\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

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 (<= (+ (* (* a2 a2) t_1) (* t_1 (* a1_m a1_m))) -5e-122)
     (* (* (* (fma -0.5 (* th th) 1.0) (* 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 ((((a2 * a2) * t_1) + (t_1 * (a1_m * a1_m))) <= -5e-122) {
		tmp = ((fma(-0.5, (th * th), 1.0) * (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(Float64(a2 * a2) * t_1) + Float64(t_1 * Float64(a1_m * a1_m))) <= -5e-122)
		tmp = Float64(Float64(Float64(fma(-0.5, Float64(th * th), 1.0) * 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[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$1 * N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-122], N[(N[(N[(N[(-0.5 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * 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))) < -4.9999999999999999e-122

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. frac-add N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

   7. Applied rewrites 43.7%

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

   8. Taylor expanded in a1 around 0

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

   10. Applied rewrites 34.1%

       \[\leadsto \left(\mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{\color{blue}{2}}\right)\right) \cdot 0.5 \]
   11. Step-by-step derivation

   12. Applied rewrites 34.1%

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

   if -4.9999999999999999e-122 < (+.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-+.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. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

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

      1. lower-sqrt.f64 82.4

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

   7. Applied rewrites 82.4%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} + \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) \leq -5 \cdot 10^{-122}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot \left(a2 \cdot a2\right)\right) \cdot \sqrt{2}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a2 \cdot a2\right) \cdot t\_1 + t\_1 \cdot \left(a1\_m \cdot a1\_m\right) \leq -5 \cdot 10^{-122}:\\ \;\;\;\;\left(\left(\left(a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot \left(\left(th \cdot th\right) \cdot -0.5\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1\_m \cdot a1\_m\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

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 (<= (+ (* (* a2 a2) t_1) (* t_1 (* a1_m a1_m))) -5e-122)
     (* (* (* (* a2 a2) (sqrt 2.0)) (* (* th th) -0.5)) 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 ((((a2 * a2) * t_1) + (t_1 * (a1_m * a1_m))) <= -5e-122) {
		tmp = (((a2 * a2) * sqrt(2.0)) * ((th * th) * -0.5)) * 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(Float64(a2 * a2) * t_1) + Float64(t_1 * Float64(a1_m * a1_m))) <= -5e-122)
		tmp = Float64(Float64(Float64(Float64(a2 * a2) * sqrt(2.0)) * Float64(Float64(th * th) * -0.5)) * 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[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$1 * N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-122], N[(N[(N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(th * th), $MachinePrecision] * -0.5), $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))) < -4.9999999999999999e-122

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. frac-add N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

   7. Applied rewrites 43.7%

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

   8. Taylor expanded in a1 around 0

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

   10. Applied rewrites 34.1%

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

   11. Taylor expanded in th around inf

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

   13. Applied rewrites 34.1%

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

   if -4.9999999999999999e-122 < (+.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-+.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. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

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

      1. lower-sqrt.f64 82.4

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

   7. Applied rewrites 82.4%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} + \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) \leq -5 \cdot 10^{-122}:\\ \;\;\;\;\left(\left(\left(a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot \left(\left(th \cdot th\right) \cdot -0.5\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.9× speedup

Math

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

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 a2 a2 (* a1_m a1_m)) (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(a2, a2, (a1_m * a1_m)) / 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(Float64(fma(a2, a2, Float64(a1_m * a1_m)) / 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[(N[(a2 * a2 + N[(a1$95$m * a1$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[th], $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-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 6: 99.6% accurate, 1.9× speedup

Math

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

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 (* (* (cos th) (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) {
	return 0.5 * ((cos(th) * fma(a2, a2, (a1_m * a1_m))) * 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(Float64(cos(th) * fma(a2, a2, Float64(a1_m * a1_m))) * 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[(N[(N[Cos[th], $MachinePrecision] * N[(a2 * a2 + N[(a1$95$m * a1$95$m), $MachinePrecision]), $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-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a2 \cdot \cos th, a2 \cdot \sqrt{2}, \left(a1 \cdot \cos th\right) \cdot \left(a1 \cdot \sqrt{2}\right)\right) \cdot 0.5} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. lift-*.f64 N/A

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

   15. distribute-lft-in N/A

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

   16. lift-fma.f64 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 N/A

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

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

Alternative 7: 99.6% accurate, 1.9× speedup

Math

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

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))) 0.5))

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))) * 0.5;
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(fma(a1_m, a1_m, Float64(a2 * a2)) * Float64(sqrt(2.0) * cos(th))) * 0.5)
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[(N[(a1$95$m * a1$95$m + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $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-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a1 around 0

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

   1. distribute-rgt-out N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 8: 77.7% accurate, 2.0× speedup

Math

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

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 (* (* 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 * a2) * Float64(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 * a2), $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-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 58.3

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

7. Applied rewrites 58.3%

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

Alternative 9: 77.8% accurate, 2.0× speedup

Math

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

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 (* (/ a2 (sqrt 2.0)) (* (cos th) a2)))

C

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

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 / sqrt(2.0d0)) * (cos(th) * a2)
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 / Math.sqrt(2.0)) * (Math.cos(th) * a2);
}

Python

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

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(a2 / sqrt(2.0)) * Float64(cos(th) * a2))
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 / sqrt(2.0)) * (cos(th) * a2);
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[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * a2), $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. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 58.2

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

5. Applied rewrites 58.2%

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

6. Final simplification 58.2%

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

Alternative 10: 77.8% accurate, 2.0× speedup

Math

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

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
 (* (* (* (* (sqrt 2.0) a2) a2) (cos th)) 0.5))

C

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

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 = (((sqrt(2.0d0) * a2) * a2) * cos(th)) * 0.5d0
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 (((Math.sqrt(2.0) * a2) * a2) * Math.cos(th)) * 0.5;
}

Python

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

Julia

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

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = (((sqrt(2.0) * a2) * a2) * cos(th)) * 0.5;
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[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * a2), $MachinePrecision] * a2), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * 0.5), $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-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in th around 0

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

   1. associate-+r+ N/A

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

   2. distribute-lft-out N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt1-in N/A

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

   6. +-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

7. Applied rewrites 59.7%

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

8. Taylor expanded in a1 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-cos.f64 58.2

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

10. Applied rewrites 58.2%

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

Alternative 11: 77.8% accurate, 2.0× speedup

Math

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

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) (* (sqrt 2.0) (cos th))) 0.5))

C

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

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) * (sqrt(2.0d0) * cos(th))) * 0.5d0
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.sqrt(2.0) * Math.cos(th))) * 0.5;
}

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.sqrt(2.0) * math.cos(th))) * 0.5

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(Float64(a2 * a2) * Float64(sqrt(2.0) * cos(th))) * 0.5)
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) * (sqrt(2.0) * cos(th))) * 0.5;
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[(N[(a2 * a2), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $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-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 58.2

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

7. Applied rewrites 58.2%

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

8. Final simplification 58.2%

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

Alternative 12: 65.7% accurate, 8.1× speedup

Math

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

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
 (/ (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) {
	return fma(a2, a2, (a1_m * a1_m)) / 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(a2, a2, Float64(a1_m * a1_m)) / 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 * a2 + N[(a1$95$m * a1$95$m), $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-+.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. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in th around 0

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

   1. lower-sqrt.f64 68.0

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

7. Applied rewrites 68.0%

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

Alternative 13: 65.7% accurate, 8.1× speedup

Math

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

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)) (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)) / 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)) / 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[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-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 68.1

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

7. Applied rewrites 68.1%

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

Alternative 14: 65.8% accurate, 8.3× speedup

Math

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

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)) 0.5))

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)) * 0.5;
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(fma(a1_m, a1_m, Float64(a2 * a2)) * sqrt(2.0)) * 0.5)
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[(N[(a1$95$m * a1$95$m + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $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-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. frac-add N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in th around 0

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

   1. distribute-rgt-out N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 68.0

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

7. Applied rewrites 68.0%

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

Alternative 15: 52.5% accurate, 9.9× speedup

Math

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

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 (sqrt 2.0)) a2))

C

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

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 / sqrt(2.0d0)) * a2
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 / Math.sqrt(2.0)) * a2;
}

Python

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

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(a2 / sqrt(2.0)) * a2)
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 / sqrt(2.0)) * a2;
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[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a2), $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

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*l/ N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sqrt.f64 68.0

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

5. Applied rewrites 68.0%

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

6. Taylor expanded in a1 around 0

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

8. Applied rewrites 40.6%

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

9. Final simplification 40.6%

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

Alternative 16: 26.5% accurate, 9.9× speedup

Math

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

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 (* (/ a1_m (sqrt 2.0)) a1_m))

C

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

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 = (a1_m / sqrt(2.0d0)) * a1_m
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 (a1_m / Math.sqrt(2.0)) * a1_m;
}

Python

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

Julia

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

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = (a1_m / sqrt(2.0)) * a1_m;
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 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a1$95$m), $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

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*l/ N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sqrt.f64 68.0

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

5. Applied rewrites 68.0%

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

6. Taylor expanded in a1 around inf

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

8. Applied rewrites 41.7%

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

9. Final simplification 41.7%

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

Reproduce

herbie shell --seed 2024237 
(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))))