Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 12.3s

Alternatives: 9

Speedup: 1.9×

• 43.0% 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: 99.6% 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, 1.7× speedup

Math

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

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (/ (/ (fma a1 a1 (* a2 a2)) (sqrt 2.0)) (/ 1.0 (cos th))))

C

double code(double a1, double a2, double th) {
	return (fma(a1, a1, (a2 * a2)) / sqrt(2.0)) / (1.0 / cos(th));
}

Julia

function code(a1, a2, th)
	return Float64(Float64(fma(a1, a1, Float64(a2 * a2)) / sqrt(2.0)) / Float64(1.0 / cos(th)))
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 / 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. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. div-inv N/A

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

   6. associate-/r* N/A

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

   7. *-lft-identity N/A

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

   8. associate-*l/ N/A

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

   9. /-lowering-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. /-lowering-/.f64 N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. sqrt-lowering-sqrt.f64 N/A

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

   16. /-lowering-/.f64 N/A

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

   17. cos-lowering-cos.f64 99.6

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

4. Applied egg-rr 99.6%

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

Alternative 2: 47.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1 \cdot a1\right) \cdot t\_1 + \left(a2 \cdot a2\right) \cdot t\_1 \leq -2 \cdot 10^{-196}:\\ \;\;\;\;\left(\left(a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1 a1) t_1) (* (* a2 a2) t_1)) -2e-196)
     (* (* (* a2 a2) (sqrt 2.0)) (fma -0.25 (* th th) 0.5))
     (/ a2 (/ (sqrt 2.0) a2)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -2e-196) {
		tmp = ((a2 * a2) * sqrt(2.0)) * fma(-0.25, (th * th), 0.5);
	} else {
		tmp = a2 / (sqrt(2.0) / a2);
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1 * a1) * t_1) + Float64(Float64(a2 * a2) * t_1)) <= -2e-196)
		tmp = Float64(Float64(Float64(a2 * a2) * sqrt(2.0)) * fma(-0.25, Float64(th * th), 0.5));
	else
		tmp = Float64(a2 / Float64(sqrt(2.0) / a2));
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1 * a1), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -2e-196], N[(N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(th * th), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $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))) < -2.0000000000000001e-196

   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. associate-*l/ N/A

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

      2. associate-*l/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lowering-*.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 egg-rr 99.7%

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

   5. Taylor expanded in a2 around inf

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 44.2

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

   7. Simplified 44.2%

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

   8. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left({a2}^{2} \cdot \left({th}^{2} \cdot \sqrt{2}\right)\right) + \frac{1}{2} \cdot \left({a2}^{2} \cdot \sqrt{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left({a2}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot {th}^{2}\right)}\right) + \frac{1}{2} \cdot \left({a2}^{2} \cdot \sqrt{2}\right) \]

      2. associate-*r* N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(\left({a2}^{2} \cdot \sqrt{2}\right) \cdot {th}^{2}\right)} + \frac{1}{2} \cdot \left({a2}^{2} \cdot \sqrt{2}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \left({a2}^{2} \cdot \sqrt{2}\right)\right) \cdot {th}^{2}} + \frac{1}{2} \cdot \left({a2}^{2} \cdot \sqrt{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({a2}^{2} \cdot \sqrt{2}\right) \cdot \frac{-1}{4}\right)} \cdot {th}^{2} + \frac{1}{2} \cdot \left({a2}^{2} \cdot \sqrt{2}\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({a2}^{2} \cdot \sqrt{2}\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2}\right)} + \frac{1}{2} \cdot \left({a2}^{2} \cdot \sqrt{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \left({a2}^{2} \cdot \sqrt{2}\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2}\right) + \color{blue}{\left({a2}^{2} \cdot \sqrt{2}\right) \cdot \frac{1}{2}} \]

      7. distribute-lft-out N/A

         \[\leadsto \color{blue}{\left({a2}^{2} \cdot \sqrt{2}\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left({a2}^{2} \cdot \sqrt{2}\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot {a2}^{2}\right)} \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{2} \cdot {a2}^{2}\right)} \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{2}} \cdot {a2}^{2}\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(\sqrt{2} \cdot \color{blue}{\left(a2 \cdot a2\right)}\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \color{blue}{\left(a2 \cdot a2\right)}\right) \cdot \left(\frac{-1}{4} \cdot {th}^{2} + \frac{1}{2}\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \left(a2 \cdot a2\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, {th}^{2}, \frac{1}{2}\right)} \]

      15. unpow2 N/A

          \[\leadsto \left(\sqrt{2} \cdot \left(a2 \cdot a2\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{th \cdot th}, \frac{1}{2}\right) \]

      16. *-lowering-*.f64 42.8

          \[\leadsto \left(\sqrt{2} \cdot \left(a2 \cdot a2\right)\right) \cdot \mathsf{fma}\left(-0.25, \color{blue}{th \cdot th}, 0.5\right) \]

   10. Simplified 42.8%

       \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(a2 \cdot a2\right)\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)} \]

   if -2.0000000000000001e-196 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 84.2

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

   5. Simplified 84.2%

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

   6. Taylor expanded in a1 around 0

      \[\leadsto \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
   7. 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. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 47.2

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

   8. Simplified 47.2%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 47.3

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

   10. Applied egg-rr 47.3%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -2 \cdot 10^{-196}:\\ \;\;\;\;\left(\left(a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.25, th \cdot th, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1 \cdot a1\right) \cdot t\_1 + \left(a2 \cdot a2\right) \cdot t\_1 \leq -2 \cdot 10^{-196}:\\ \;\;\;\;th \cdot \left(th \cdot \left(-0.25 \cdot \left(a2 \cdot \left(a2 \cdot \sqrt{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1 a1) t_1) (* (* a2 a2) t_1)) -2e-196)
     (* th (* th (* -0.25 (* a2 (* a2 (sqrt 2.0))))))
     (/ a2 (/ (sqrt 2.0) a2)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -2e-196) {
		tmp = th * (th * (-0.25 * (a2 * (a2 * sqrt(2.0)))));
	} else {
		tmp = a2 / (sqrt(2.0) / a2);
	}
	return tmp;
}

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
    real(8) :: tmp
    t_1 = cos(th) / sqrt(2.0d0)
    if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= (-2d-196)) then
        tmp = th * (th * ((-0.25d0) * (a2 * (a2 * sqrt(2.0d0)))))
    else
        tmp = a2 / (sqrt(2.0d0) / a2)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -2e-196) {
		tmp = th * (th * (-0.25 * (a2 * (a2 * Math.sqrt(2.0)))));
	} else {
		tmp = a2 / (Math.sqrt(2.0) / a2);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	tmp = 0
	if (((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -2e-196:
		tmp = th * (th * (-0.25 * (a2 * (a2 * math.sqrt(2.0)))))
	else:
		tmp = a2 / (math.sqrt(2.0) / a2)
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1 * a1) * t_1) + Float64(Float64(a2 * a2) * t_1)) <= -2e-196)
		tmp = Float64(th * Float64(th * Float64(-0.25 * Float64(a2 * Float64(a2 * sqrt(2.0))))));
	else
		tmp = Float64(a2 / Float64(sqrt(2.0) / a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = 0.0;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -2e-196)
		tmp = th * (th * (-0.25 * (a2 * (a2 * sqrt(2.0)))));
	else
		tmp = a2 / (sqrt(2.0) / a2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1 * a1), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -2e-196], N[(th * N[(th * N[(-0.25 * N[(a2 * N[(a2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $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))) < -2.0000000000000001e-196

   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. associate-*l/ N/A

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

      2. associate-*l/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lowering-*.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 egg-rr 99.7%

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

   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. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. +-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

   7. Simplified 56.5%

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

   8. Taylor expanded in a2 around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 42.8

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

   10. Simplified 42.8%

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

   11. Taylor expanded in th around inf

       \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left({a2}^{2} \cdot \left({th}^{2} \cdot \sqrt{2}\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{-1}{4} \cdot \left({a2}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot {th}^{2}\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(\left({a2}^{2} \cdot \sqrt{2}\right) \cdot {th}^{2}\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \left({a2}^{2} \cdot \sqrt{2}\right)\right) \cdot {th}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \left(\frac{-1}{4} \cdot \left({a2}^{2} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(th \cdot th\right)} \]

       5. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({a2}^{2} \cdot \sqrt{2}\right)\right) \cdot th\right) \cdot th} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({a2}^{2} \cdot \sqrt{2}\right)\right) \cdot th\right) \cdot th} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({a2}^{2} \cdot \sqrt{2}\right)\right) \cdot th\right)} \cdot th \]

       8. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\left({a2}^{2} \cdot \sqrt{2}\right) \cdot \frac{-1}{4}\right)} \cdot th\right) \cdot th \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\left({a2}^{2} \cdot \sqrt{2}\right) \cdot \frac{-1}{4}\right)} \cdot th\right) \cdot th \]

       10. unpow2 N/A

           \[\leadsto \left(\left(\left(\color{blue}{\left(a2 \cdot a2\right)} \cdot \sqrt{2}\right) \cdot \frac{-1}{4}\right) \cdot th\right) \cdot th \]

       11. associate-*l* N/A

           \[\leadsto \left(\left(\color{blue}{\left(a2 \cdot \left(a2 \cdot \sqrt{2}\right)\right)} \cdot \frac{-1}{4}\right) \cdot th\right) \cdot th \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \left(\left(\color{blue}{\left(a2 \cdot \left(a2 \cdot \sqrt{2}\right)\right)} \cdot \frac{-1}{4}\right) \cdot th\right) \cdot th \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \left(\left(\left(a2 \cdot \color{blue}{\left(a2 \cdot \sqrt{2}\right)}\right) \cdot \frac{-1}{4}\right) \cdot th\right) \cdot th \]

       14. sqrt-lowering-sqrt.f64 43.1

           \[\leadsto \left(\left(\left(a2 \cdot \left(a2 \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot -0.25\right) \cdot th\right) \cdot th \]

   13. Simplified 43.1%

       \[\leadsto \color{blue}{\left(\left(\left(a2 \cdot \left(a2 \cdot \sqrt{2}\right)\right) \cdot -0.25\right) \cdot th\right) \cdot th} \]

   if -2.0000000000000001e-196 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 84.2

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

   5. Simplified 84.2%

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

   6. Taylor expanded in a1 around 0

      \[\leadsto \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
   7. 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. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 47.2

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

   8. Simplified 47.2%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 47.3

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

   10. Applied egg-rr 47.3%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -2 \cdot 10^{-196}:\\ \;\;\;\;th \cdot \left(th \cdot \left(-0.25 \cdot \left(a2 \cdot \left(a2 \cdot \sqrt{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (* (cos th) (* (sqrt 2.0) (fma a2 a2 (* a1 a1)))) 0.5))

C

double code(double a1, double a2, double th) {
	return (cos(th) * (sqrt(2.0) * fma(a2, a2, (a1 * a1)))) * 0.5;
}

Julia

function code(a1, a2, th)
	return Float64(Float64(cos(th) * Float64(sqrt(2.0) * fma(a2, a2, Float64(a1 * a1)))) * 0.5)
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a2 * a2 + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $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. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. *-lowering-*.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 egg-rr 99.6%

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

5. Taylor expanded in th around inf

   \[\leadsto \color{blue}{\left({a1}^{2} \cdot \left(\cos th \cdot \sqrt{2}\right) + {a2}^{2} \cdot \left(\cos th \cdot \sqrt{2}\right)\right)} \cdot \frac{1}{2} \]
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. associate-*l* N/A

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

   3. distribute-rgt-out N/A

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

   4. *-lowering-*.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. distribute-rgt-out N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 N/A

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

   9. +-commutative N/A

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

   10. unpow2 N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 99.6

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

7. Simplified 99.6%

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

Alternative 5: 56.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(a2 \cdot a2\right) \cdot \cos th}{\sqrt{2}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a2 * a2), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in a1 around 0

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

   1. /-lowering-/.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 53.0

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

5. Simplified 53.0%

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

Alternative 6: 56.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{a2 \cdot \left(a2 \cdot \cos th\right)}{\sqrt{2}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(a2 * N[(a2 * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. div-inv N/A

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

   6. associate-/r* N/A

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

   7. *-lft-identity N/A

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

   8. associate-*l/ N/A

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

   9. /-lowering-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. /-lowering-/.f64 N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. sqrt-lowering-sqrt.f64 N/A

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

   16. /-lowering-/.f64 N/A

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

   17. cos-lowering-cos.f64 99.6

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in a1 around 0

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

   1. /-lowering-/.f64 N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. cos-lowering-cos.f64 N/A

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

   7. sqrt-lowering-sqrt.f64 53.0

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

7. Simplified 53.0%

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

Alternative 7: 56.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos th \cdot \left(\left(a2 \cdot 0.5\right) \cdot \left(a2 \cdot \sqrt{2}\right)\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (cos th) (* (* a2 0.5) (* a2 (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	return cos(th) * ((a2 * 0.5) * (a2 * sqrt(2.0)));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = cos(th) * ((a2 * 0.5d0) * (a2 * sqrt(2.0d0)))
end function

Java

public static double code(double a1, double a2, double th) {
	return Math.cos(th) * ((a2 * 0.5) * (a2 * Math.sqrt(2.0)));
}

Python

def code(a1, a2, th):
	return math.cos(th) * ((a2 * 0.5) * (a2 * math.sqrt(2.0)))

Julia

function code(a1, a2, th)
	return Float64(cos(th) * Float64(Float64(a2 * 0.5) * Float64(a2 * sqrt(2.0))))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = cos(th) * ((a2 * 0.5) * (a2 * sqrt(2.0)));
end

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 * 0.5), $MachinePrecision] * N[(a2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. *-lowering-*.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 egg-rr 99.6%

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

5. Taylor expanded in a2 around inf

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

   1. *-lowering-*.f64 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 53.1

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

7. Simplified 53.1%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. remove-double-neg N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. *-commutative N/A

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

   9. distribute-lft-neg-out N/A

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

   10. remove-double-neg N/A

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

   11. associate-*r* N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. sqrt-lowering-sqrt.f64 N/A

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

   17. cos-lowering-cos.f64 53.1

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

9. Applied egg-rr 53.1%

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

10. Final simplification 53.1%

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

Alternative 8: 39.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \frac{a2}{\frac{\sqrt{2}}{a2}} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (/ a2 (/ (sqrt 2.0) a2)))

C

double code(double a1, double a2, double th) {
	return a2 / (sqrt(2.0) / a2);
}

Fortran

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

Java

public static double code(double a1, double a2, double th) {
	return a2 / (Math.sqrt(2.0) / a2);
}

Python

def code(a1, a2, th):
	return a2 / (math.sqrt(2.0) / a2)

Julia

function code(a1, a2, th)
	return Float64(a2 / Float64(sqrt(2.0) / a2))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = a2 / (sqrt(2.0) / a2);
end

Wolfram

code[a1_, a2_, th_] := N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. sqrt-lowering-sqrt.f64 66.5

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

5. Simplified 66.5%

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

6. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
7. 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. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 37.5

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

8. Simplified 37.5%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 37.6

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

10. Applied egg-rr 37.6%

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

Alternative 9: 39.5% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{2}\right) \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* 0.5 (* (* a2 a2) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	return 0.5 * ((a2 * a2) * sqrt(2.0));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = 0.5d0 * ((a2 * a2) * sqrt(2.0d0))
end function

Java

public static double code(double a1, double a2, double th) {
	return 0.5 * ((a2 * a2) * Math.sqrt(2.0));
}

Python

def code(a1, a2, th):
	return 0.5 * ((a2 * a2) * math.sqrt(2.0))

Julia

function code(a1, a2, th)
	return Float64(0.5 * Float64(Float64(a2 * a2) * sqrt(2.0)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = 0.5 * ((a2 * a2) * sqrt(2.0));
end

Wolfram

code[a1_, a2_, th_] := N[(0.5 * 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. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. *-lowering-*.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 egg-rr 99.6%

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

5. Taylor expanded in a2 around inf

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

   1. *-lowering-*.f64 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 53.1

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

7. Simplified 53.1%

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

8. Taylor expanded in th around 0

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

   1. sqrt-lowering-sqrt.f64 37.6

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

10. Simplified 37.6%

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

11. Final simplification 37.6%

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

Reproduce

herbie shell --seed 2024203 
(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))))