Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 10.8s

Alternatives: 10

Speedup: 1.9×

• 43.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.8× speedup

Math

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

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (/ (fma 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(fma(a2, a2, Float64(a1_m * a1_m)) / Float64(sqrt(2.0) / cos(th)))
end

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. *-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. clear-num N/A

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

   13. un-div-inv N/A

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

   14. lower-/.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-/.f64 99.6

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

4. Applied egg-rr 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 99.6

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

6. Applied egg-rr 99.6%

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

Alternative 2: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1\_m \cdot a1\_m\right) \cdot t\_1 + \left(a2 \cdot a2\right) \cdot t\_1 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \mathsf{fma}\left(a1\_m, \frac{a1\_m}{a2 \cdot \left(a2 \cdot \sqrt{2}\right)}, \frac{-1}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a1\_m, a1\_m, a2 \cdot a2\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

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 (<= (+ (* (* a1_m a1_m) t_1) (* (* a2 a2) t_1)) -1e-306)
     (*
      (* a2 a2)
      (fma a1_m (/ a1_m (* a2 (* a2 (sqrt 2.0)))) (/ -1.0 (sqrt 2.0))))
     (/ (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) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1_m * a1_m) * t_1) + ((a2 * a2) * t_1)) <= -1e-306) {
		tmp = (a2 * a2) * fma(a1_m, (a1_m / (a2 * (a2 * sqrt(2.0)))), (-1.0 / sqrt(2.0)));
	} else {
		tmp = fma(a1_m, a1_m, (a2 * a2)) / sqrt(2.0);
	}
	return tmp;
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1_m * a1_m) * t_1) + Float64(Float64(a2 * a2) * t_1)) <= -1e-306)
		tmp = Float64(Float64(a2 * a2) * fma(a1_m, Float64(a1_m / Float64(a2 * Float64(a2 * sqrt(2.0)))), Float64(-1.0 / sqrt(2.0))));
	else
		tmp = Float64(fma(a1_m, a1_m, Float64(a2 * a2)) / sqrt(2.0));
	end
	return tmp
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -1e-306], N[(N[(a2 * a2), $MachinePrecision] * N[(a1$95$m * N[(a1$95$m / N[(a2 * N[(a2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a1$95$m * a1$95$m + N[(a2 * a2), $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))) < -1.00000000000000003e-306

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\frac{{a1}^{3}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}}} \]
   4. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot \left(a1 \cdot a1\right)}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a1 \cdot \color{blue}{{a1}^{2}}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{a1 \cdot \frac{{a1}^{2}}{\sqrt{2}}} + \frac{{a2}^{3}}{\sqrt{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a1, \frac{{a1}^{2}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, \color{blue}{\frac{{a1}^{2}}{\sqrt{2}}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a1, \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, \frac{a1 \cdot a1}{\color{blue}{\sqrt{2}}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, \frac{a1 \cdot a1}{\sqrt{2}}, \color{blue}{\frac{{a2}^{3}}{\sqrt{2}}}\right) \]

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 28.7

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

   5. Simplified 28.7%

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

   6. Taylor expanded in a2 around -inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-sqrt.f64 27.2

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

   8. Simplified 27.2%

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

   if -1.00000000000000003e-306 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in th around 0

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

      1. lower-sqrt.f64 84.5

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

   7. Simplified 84.5%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -1 \cdot 10^{-306}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \mathsf{fma}\left(a1, \frac{a1}{a2 \cdot \left(a2 \cdot \sqrt{2}\right)}, \frac{-1}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1\_m \cdot a1\_m\right) \cdot t\_1 + \left(a2 \cdot a2\right) \cdot t\_1 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;-\frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a1\_m, a1\_m, a2 \cdot a2\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

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 (<= (+ (* (* a1_m a1_m) t_1) (* (* a2 a2) t_1)) -1e-306)
     (- (/ (* a2 a2) (sqrt 2.0)))
     (/ (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) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1_m * a1_m) * t_1) + ((a2 * a2) * t_1)) <= -1e-306) {
		tmp = -((a2 * a2) / sqrt(2.0));
	} else {
		tmp = fma(a1_m, a1_m, (a2 * a2)) / sqrt(2.0);
	}
	return tmp;
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1_m * a1_m) * t_1) + Float64(Float64(a2 * a2) * t_1)) <= -1e-306)
		tmp = Float64(-Float64(Float64(a2 * a2) / sqrt(2.0)));
	else
		tmp = Float64(fma(a1_m, a1_m, Float64(a2 * a2)) / sqrt(2.0));
	end
	return tmp
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -1e-306], (-N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), N[(N[(a1$95$m * a1$95$m + N[(a2 * a2), $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))) < -1.00000000000000003e-306

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\frac{{a1}^{3}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}}} \]
   4. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot \left(a1 \cdot a1\right)}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a1 \cdot \color{blue}{{a1}^{2}}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{a1 \cdot \frac{{a1}^{2}}{\sqrt{2}}} + \frac{{a2}^{3}}{\sqrt{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a1, \frac{{a1}^{2}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, \color{blue}{\frac{{a1}^{2}}{\sqrt{2}}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a1, \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, \frac{a1 \cdot a1}{\color{blue}{\sqrt{2}}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, \frac{a1 \cdot a1}{\sqrt{2}}, \color{blue}{\frac{{a2}^{3}}{\sqrt{2}}}\right) \]

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 28.7

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

   5. Simplified 28.7%

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

   6. Taylor expanded in a2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{{a2}^{2}}{\sqrt{2}}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

         \[\leadsto \frac{\color{blue}{a2 \cdot a2}}{-1 \cdot \sqrt{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a2 \cdot a2}}{-1 \cdot \sqrt{2}} \]

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sqrt.f64 28.7

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

   8. Simplified 28.7%

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

   if -1.00000000000000003e-306 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in th around 0

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

      1. lower-sqrt.f64 84.5

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

   7. Simplified 84.5%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -1 \cdot 10^{-306}:\\ \;\;\;\;-\frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \begin{array}{l} t_1 := \frac{a2 \cdot a2}{\sqrt{2}}\\ t_2 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1\_m \cdot a1\_m\right) \cdot t\_2 + \left(a2 \cdot a2\right) \cdot t\_2 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;-t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (/ (* a2 a2) (sqrt 2.0))) (t_2 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1_m a1_m) t_2) (* (* a2 a2) t_2)) -1e-306) (- t_1) t_1)))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	double t_1 = (a2 * a2) / sqrt(2.0);
	double t_2 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1_m * a1_m) * t_2) + ((a2 * a2) * t_2)) <= -1e-306) {
		tmp = -t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a2 * a2) / sqrt(2.0d0)
    t_2 = cos(th) / sqrt(2.0d0)
    if ((((a1_m * a1_m) * t_2) + ((a2 * a2) * t_2)) <= (-1d-306)) then
        tmp = -t_1
    else
        tmp = t_1
    end if
    code = tmp
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) {
	double t_1 = (a2 * a2) / Math.sqrt(2.0);
	double t_2 = Math.cos(th) / Math.sqrt(2.0);
	double tmp;
	if ((((a1_m * a1_m) * t_2) + ((a2 * a2) * t_2)) <= -1e-306) {
		tmp = -t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	t_1 = (a2 * a2) / math.sqrt(2.0)
	t_2 = math.cos(th) / math.sqrt(2.0)
	tmp = 0
	if (((a1_m * a1_m) * t_2) + ((a2 * a2) * t_2)) <= -1e-306:
		tmp = -t_1
	else:
		tmp = t_1
	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(Float64(a2 * a2) / sqrt(2.0))
	t_2 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1_m * a1_m) * t_2) + Float64(Float64(a2 * a2) * t_2)) <= -1e-306)
		tmp = Float64(-t_1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp_2 = code(a1_m, a2, th)
	t_1 = (a2 * a2) / sqrt(2.0);
	t_2 = cos(th) / sqrt(2.0);
	tmp = 0.0;
	if ((((a1_m * a1_m) * t_2) + ((a2 * a2) * t_2)) <= -1e-306)
		tmp = -t_1;
	else
		tmp = t_1;
	end
	tmp_2 = 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[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1$95$m * a1$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], -1e-306], (-t$95$1), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -1.00000000000000003e-306

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\frac{{a1}^{3}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}}} \]
   4. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot \left(a1 \cdot a1\right)}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a1 \cdot \color{blue}{{a1}^{2}}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{a1 \cdot \frac{{a1}^{2}}{\sqrt{2}}} + \frac{{a2}^{3}}{\sqrt{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a1, \frac{{a1}^{2}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, \color{blue}{\frac{{a1}^{2}}{\sqrt{2}}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a1, \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, \frac{a1 \cdot a1}{\color{blue}{\sqrt{2}}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, \frac{a1 \cdot a1}{\sqrt{2}}, \color{blue}{\frac{{a2}^{3}}{\sqrt{2}}}\right) \]

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 28.7

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

   5. Simplified 28.7%

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

   6. Taylor expanded in a2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{{a2}^{2}}{\sqrt{2}}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

         \[\leadsto \frac{\color{blue}{a2 \cdot a2}}{-1 \cdot \sqrt{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a2 \cdot a2}}{-1 \cdot \sqrt{2}} \]

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sqrt.f64 28.7

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

   8. Simplified 28.7%

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

   if -1.00000000000000003e-306 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.6%

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

   3. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\frac{{a1}^{3}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}}} \]
   4. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot \left(a1 \cdot a1\right)}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a1 \cdot \color{blue}{{a1}^{2}}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{a1 \cdot \frac{{a1}^{2}}{\sqrt{2}}} + \frac{{a2}^{3}}{\sqrt{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a1, \frac{{a1}^{2}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, \color{blue}{\frac{{a1}^{2}}{\sqrt{2}}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a1, \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, \frac{a1 \cdot a1}{\color{blue}{\sqrt{2}}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, \frac{a1 \cdot a1}{\sqrt{2}}, \color{blue}{\frac{{a2}^{3}}{\sqrt{2}}}\right) \]

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 32.6

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

   5. Simplified 32.6%

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

   6. Taylor expanded in a1 around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 52.0

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

   8. Simplified 52.0%

      \[\leadsto \color{blue}{\frac{a2 \cdot a2}{\sqrt{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -1 \cdot 10^{-306}:\\ \;\;\;\;-\frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. *-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. clear-num N/A

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

   13. un-div-inv N/A

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

   14. lower-/.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-/.f64 99.6

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

4. Applied egg-rr 99.6%

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

Alternative 6: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. lift-/.f64 N/A

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

   11. div-inv N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 7: 77.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. lift-/.f64 N/A

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

   11. div-inv N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in a1 around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 56.9

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

7. Simplified 56.9%

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

8. Final simplification 56.9%

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

Alternative 8: 51.8% accurate, 9.9× speedup

Math

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

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) (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) / 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) / 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.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.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) / 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) / 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[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\frac{{a1}^{3}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}}} \]
4. Step-by-step derivation

   1. cube-mult N/A

      \[\leadsto \frac{\color{blue}{a1 \cdot \left(a1 \cdot a1\right)}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}} \]

   2. unpow2 N/A

      \[\leadsto \frac{a1 \cdot \color{blue}{{a1}^{2}}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{a1 \cdot \frac{{a1}^{2}}{\sqrt{2}}} + \frac{{a2}^{3}}{\sqrt{2}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a1, \frac{{a1}^{2}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right)} \]

   5. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \color{blue}{\frac{{a1}^{2}}{\sqrt{2}}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1 \cdot a1}{\color{blue}{\sqrt{2}}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1 \cdot a1}{\sqrt{2}}, \color{blue}{\frac{{a2}^{3}}{\sqrt{2}}}\right) \]

   10. cube-mult N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-sqrt.f64 31.4

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

5. Simplified 31.4%

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

6. Taylor expanded in a1 around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 36.9

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

8. Simplified 36.9%

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

Alternative 9: 51.8% accurate, 9.9× speedup

Math

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

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 (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 / 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 / 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.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.sqrt(2.0))

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\frac{{a1}^{3}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}}} \]
4. Step-by-step derivation

   1. cube-mult N/A

      \[\leadsto \frac{\color{blue}{a1 \cdot \left(a1 \cdot a1\right)}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}} \]

   2. unpow2 N/A

      \[\leadsto \frac{a1 \cdot \color{blue}{{a1}^{2}}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{a1 \cdot \frac{{a1}^{2}}{\sqrt{2}}} + \frac{{a2}^{3}}{\sqrt{2}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a1, \frac{{a1}^{2}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right)} \]

   5. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \color{blue}{\frac{{a1}^{2}}{\sqrt{2}}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1 \cdot a1}{\color{blue}{\sqrt{2}}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1 \cdot a1}{\sqrt{2}}, \color{blue}{\frac{{a2}^{3}}{\sqrt{2}}}\right) \]

   10. cube-mult N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-sqrt.f64 31.4

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

5. Simplified 31.4%

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

6. Taylor expanded in a1 around -inf

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. sub-neg N/A

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

   5. unpow2 N/A

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

   6. associate-/l* N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-sqrt.f64 23.5

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

8. Simplified 23.5%

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

9. Taylor expanded in a1 around 0

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

    1. unpow2 N/A

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

    2. associate-/l* N/A

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

    3. lower-*.f64 N/A

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

    4. lower-/.f64 N/A

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

    5. lower-sqrt.f64 37.0

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

11. Simplified 37.0%

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

Alternative 10: 3.2% accurate, 12.1× speedup

Math

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

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)))

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);
}

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)
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);
}

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)

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(a1_m / 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 = 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[(a1$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\frac{{a1}^{3}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}}} \]
4. Step-by-step derivation

   1. cube-mult N/A

      \[\leadsto \frac{\color{blue}{a1 \cdot \left(a1 \cdot a1\right)}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}} \]

   2. unpow2 N/A

      \[\leadsto \frac{a1 \cdot \color{blue}{{a1}^{2}}}{\sqrt{2}} + \frac{{a2}^{3}}{\sqrt{2}} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{a1 \cdot \frac{{a1}^{2}}{\sqrt{2}}} + \frac{{a2}^{3}}{\sqrt{2}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a1, \frac{{a1}^{2}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right)} \]

   5. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \color{blue}{\frac{{a1}^{2}}{\sqrt{2}}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1 \cdot a1}{\color{blue}{\sqrt{2}}}, \frac{{a2}^{3}}{\sqrt{2}}\right) \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1 \cdot a1}{\sqrt{2}}, \color{blue}{\frac{{a2}^{3}}{\sqrt{2}}}\right) \]

   10. cube-mult N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-sqrt.f64 31.4

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

5. Simplified 31.4%

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

6. Taylor expanded in a2 around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 36.6

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

8. Simplified 36.6%

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

9. Taylor expanded in a1 around 0

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

    1. lower-/.f64 N/A

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

    2. lower-sqrt.f64 3.6

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

11. Simplified 3.6%

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

Reproduce

herbie shell --seed 2024214 
(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))))