Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 11.5s

Alternatives: 12

Speedup: 1.9×

• 44.1% 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, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a1_, a2_, th_] := N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

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

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

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

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

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

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

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

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

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

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

   8. sqrt-lowering-sqrt.f64 99.7

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

4. Applied egg-rr 99.7%

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

Alternative 2: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot t\_1 \leq -4 \cdot 10^{-314}:\\ \;\;\;\;\sqrt{2} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* (* a2 a2) t_1)) -4e-314)
     (* (sqrt 2.0) (* (fma a1 a1 (* a2 a2)) -0.5))
     (/ (fma a2 a2 (* a1 a1)) (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + ((a2 * a2) * t_1)) <= -4e-314) {
		tmp = sqrt(2.0) * (fma(a1, a1, (a2 * a2)) * -0.5);
	} else {
		tmp = fma(a2, a2, (a1 * a1)) / sqrt(2.0);
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(Float64(a2 * a2) * t_1)) <= -4e-314)
		tmp = Float64(sqrt(2.0) * Float64(fma(a1, a1, Float64(a2 * a2)) * -0.5));
	else
		tmp = Float64(fma(a2, a2, Float64(a1 * a1)) / sqrt(2.0));
	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[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -4e-314], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * a2 + N[(a1 * a1), $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))) < -3.9999999999e-314

   1. Initial program 99.7%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. 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 1.3

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

   5. Simplified 1.3%

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 1.3

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

   7. Applied egg-rr 1.3%

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

   9. Applied egg-rr 70.8%

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{1}{\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}} \]
   10. Step-by-step derivation

       1. associate-/r/ N/A

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

       2. metadata-eval N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

       9. sqrt-lowering-sqrt.f64 70.9

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

   11. Applied egg-rr 70.9%

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

   if -3.9999999999e-314 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.7%

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

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

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

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

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in th around 0

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

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

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sqrt-lowering-sqrt.f64 86.1

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

   7. Simplified 86.1%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -4 \cdot 10^{-314}:\\ \;\;\;\;\sqrt{2} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\\ t_2 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_2 \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot t\_2 \leq -4 \cdot 10^{-314}:\\ \;\;\;\;\sqrt{2} \cdot \left(t\_1 \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{2}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (fma a1 a1 (* a2 a2))) (t_2 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_2 (* a1 a1)) (* (* a2 a2) t_2)) -4e-314)
     (* (sqrt 2.0) (* t_1 -0.5))
     (* (* t_1 (sqrt 2.0)) 0.5))))

C

double code(double a1, double a2, double th) {
	double t_1 = fma(a1, a1, (a2 * a2));
	double t_2 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_2 * (a1 * a1)) + ((a2 * a2) * t_2)) <= -4e-314) {
		tmp = sqrt(2.0) * (t_1 * -0.5);
	} else {
		tmp = (t_1 * sqrt(2.0)) * 0.5;
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = fma(a1, a1, Float64(a2 * a2))
	t_2 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_2 * Float64(a1 * a1)) + Float64(Float64(a2 * a2) * t_2)) <= -4e-314)
		tmp = Float64(sqrt(2.0) * Float64(t_1 * -0.5));
	else
		tmp = Float64(Float64(t_1 * sqrt(2.0)) * 0.5);
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], -4e-314], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -3.9999999999e-314

   1. Initial program 99.7%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. 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 1.3

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

   5. Simplified 1.3%

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 1.3

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

   7. Applied egg-rr 1.3%

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

   9. Applied egg-rr 70.8%

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{1}{\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}} \]
   10. Step-by-step derivation

       1. associate-/r/ N/A

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

       2. metadata-eval N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

       9. sqrt-lowering-sqrt.f64 70.9

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

   11. Applied egg-rr 70.9%

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

   if -3.9999999999e-314 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.7%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. 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 86.1

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

   5. Simplified 86.1%

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 86.0

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

   7. Applied egg-rr 86.0%

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

      1. div-inv N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. *-lowering-*.f64 86.0

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

   9. Applied egg-rr 86.0%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -4 \cdot 10^{-314}:\\ \;\;\;\;\sqrt{2} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{2}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot t\_1 \leq -4 \cdot 10^{-314}:\\ \;\;\;\;-0.5 \cdot \left(a2 \cdot \left(a2 \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* (* a2 a2) t_1)) -4e-314)
     (* -0.5 (* a2 (* a2 (sqrt 2.0))))
     (/ (* a2 a2) (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + ((a2 * a2) * t_1)) <= -4e-314) {
		tmp = -0.5 * (a2 * (a2 * sqrt(2.0)));
	} else {
		tmp = (a2 * a2) / sqrt(2.0);
	}
	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 (((t_1 * (a1 * a1)) + ((a2 * a2) * t_1)) <= (-4d-314)) then
        tmp = (-0.5d0) * (a2 * (a2 * sqrt(2.0d0)))
    else
        tmp = (a2 * a2) / sqrt(2.0d0)
    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 (((t_1 * (a1 * a1)) + ((a2 * a2) * t_1)) <= -4e-314) {
		tmp = -0.5 * (a2 * (a2 * Math.sqrt(2.0)));
	} else {
		tmp = (a2 * a2) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	tmp = 0
	if ((t_1 * (a1 * a1)) + ((a2 * a2) * t_1)) <= -4e-314:
		tmp = -0.5 * (a2 * (a2 * math.sqrt(2.0)))
	else:
		tmp = (a2 * a2) / math.sqrt(2.0)
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(Float64(a2 * a2) * t_1)) <= -4e-314)
		tmp = Float64(-0.5 * Float64(a2 * Float64(a2 * sqrt(2.0))));
	else
		tmp = Float64(Float64(a2 * a2) / sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = 0.0;
	if (((t_1 * (a1 * a1)) + ((a2 * a2) * t_1)) <= -4e-314)
		tmp = -0.5 * (a2 * (a2 * sqrt(2.0)));
	else
		tmp = (a2 * a2) / sqrt(2.0);
	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[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -4e-314], N[(-0.5 * N[(a2 * N[(a2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * a2), $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))) < -3.9999999999e-314

   1. Initial program 99.7%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. 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 1.3

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

   5. Simplified 1.3%

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 1.3

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

   7. Applied egg-rr 1.3%

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

   9. Applied egg-rr 70.8%

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

   10. Taylor expanded in a1 around 0

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

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

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

       7. sqrt-lowering-sqrt.f64 37.8

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

   12. Simplified 37.8%

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

   if -3.9999999999e-314 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.7%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. 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 86.1

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

   5. Simplified 86.1%

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

   6. Taylor expanded in a1 around 0

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

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

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

      4. sqrt-lowering-sqrt.f64 56.4

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

   8. Simplified 56.4%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -4 \cdot 10^{-314}:\\ \;\;\;\;-0.5 \cdot \left(a2 \cdot \left(a2 \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot t\_1 \leq -4 \cdot 10^{-314}:\\ \;\;\;\;-0.5 \cdot \left(a2 \cdot \left(a2 \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* (* a2 a2) t_1)) -4e-314)
     (* -0.5 (* a2 (* a2 (sqrt 2.0))))
     (* a2 (/ a2 (sqrt 2.0))))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + ((a2 * a2) * t_1)) <= -4e-314) {
		tmp = -0.5 * (a2 * (a2 * sqrt(2.0)));
	} else {
		tmp = a2 * (a2 / sqrt(2.0));
	}
	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 (((t_1 * (a1 * a1)) + ((a2 * a2) * t_1)) <= (-4d-314)) then
        tmp = (-0.5d0) * (a2 * (a2 * sqrt(2.0d0)))
    else
        tmp = a2 * (a2 / sqrt(2.0d0))
    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 (((t_1 * (a1 * a1)) + ((a2 * a2) * t_1)) <= -4e-314) {
		tmp = -0.5 * (a2 * (a2 * Math.sqrt(2.0)));
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	tmp = 0
	if ((t_1 * (a1 * a1)) + ((a2 * a2) * t_1)) <= -4e-314:
		tmp = -0.5 * (a2 * (a2 * math.sqrt(2.0)))
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(Float64(a2 * a2) * t_1)) <= -4e-314)
		tmp = Float64(-0.5 * Float64(a2 * Float64(a2 * sqrt(2.0))));
	else
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = 0.0;
	if (((t_1 * (a1 * a1)) + ((a2 * a2) * t_1)) <= -4e-314)
		tmp = -0.5 * (a2 * (a2 * sqrt(2.0)));
	else
		tmp = a2 * (a2 / sqrt(2.0));
	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[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -4e-314], N[(-0.5 * N[(a2 * N[(a2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -3.9999999999e-314

   1. Initial program 99.7%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. 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 1.3

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

   5. Simplified 1.3%

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 1.3

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

   7. Applied egg-rr 1.3%

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

   9. Applied egg-rr 70.8%

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

   10. Taylor expanded in a1 around 0

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

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

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

       7. sqrt-lowering-sqrt.f64 37.8

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

   12. Simplified 37.8%

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

   if -3.9999999999e-314 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.7%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. 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 86.1

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

   5. Simplified 86.1%

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

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. associate-*r/ N/A

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

      4. frac-add N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      18. sqrt-lowering-sqrt.f64 86.0

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

   7. Applied egg-rr 86.0%

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

   8. Taylor expanded in a2 around inf

      \[\leadsto \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
   9. 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 56.4

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

   10. Simplified 56.4%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -4 \cdot 10^{-314}:\\ \;\;\;\;-0.5 \cdot \left(a2 \cdot \left(a2 \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a2 \cdot \left(a2 \cdot \sqrt{2}\right)\\ t_2 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_2 \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot t\_2 \leq -4 \cdot 10^{-314}:\\ \;\;\;\;-0.5 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (* a2 (* a2 (sqrt 2.0)))) (t_2 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_2 (* a1 a1)) (* (* a2 a2) t_2)) -4e-314)
     (* -0.5 t_1)
     (* 0.5 t_1))))

C

double code(double a1, double a2, double th) {
	double t_1 = a2 * (a2 * sqrt(2.0));
	double t_2 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_2 * (a1 * a1)) + ((a2 * a2) * t_2)) <= -4e-314) {
		tmp = -0.5 * t_1;
	} else {
		tmp = 0.5 * t_1;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = a2 * (a2 * sqrt(2.0d0))
    t_2 = cos(th) / sqrt(2.0d0)
    if (((t_2 * (a1 * a1)) + ((a2 * a2) * t_2)) <= (-4d-314)) then
        tmp = (-0.5d0) * t_1
    else
        tmp = 0.5d0 * t_1
    end if
    code = tmp
end function

Java

public static double code(double a1, 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 (((t_2 * (a1 * a1)) + ((a2 * a2) * t_2)) <= -4e-314) {
		tmp = -0.5 * t_1;
	} else {
		tmp = 0.5 * t_1;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = a2 * (a2 * math.sqrt(2.0))
	t_2 = math.cos(th) / math.sqrt(2.0)
	tmp = 0
	if ((t_2 * (a1 * a1)) + ((a2 * a2) * t_2)) <= -4e-314:
		tmp = -0.5 * t_1
	else:
		tmp = 0.5 * t_1
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(a2 * Float64(a2 * sqrt(2.0)))
	t_2 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_2 * Float64(a1 * a1)) + Float64(Float64(a2 * a2) * t_2)) <= -4e-314)
		tmp = Float64(-0.5 * t_1);
	else
		tmp = Float64(0.5 * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = a2 * (a2 * sqrt(2.0));
	t_2 = cos(th) / sqrt(2.0);
	tmp = 0.0;
	if (((t_2 * (a1 * a1)) + ((a2 * a2) * t_2)) <= -4e-314)
		tmp = -0.5 * t_1;
	else
		tmp = 0.5 * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(a2 * N[(a2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], -4e-314], N[(-0.5 * t$95$1), $MachinePrecision], N[(0.5 * t$95$1), $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))) < -3.9999999999e-314

   1. Initial program 99.7%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. 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 1.3

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

   5. Simplified 1.3%

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 1.3

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

   7. Applied egg-rr 1.3%

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

   9. Applied egg-rr 70.8%

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

   10. Taylor expanded in a1 around 0

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

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

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

       7. sqrt-lowering-sqrt.f64 37.8

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

   12. Simplified 37.8%

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

   if -3.9999999999e-314 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.7%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. 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 86.1

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

   5. Simplified 86.1%

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 86.0

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

   7. Applied egg-rr 86.0%

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

   8. Taylor expanded in a2 around inf

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 56.4

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

   10. Simplified 56.4%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -4 \cdot 10^{-314}:\\ \;\;\;\;-0.5 \cdot \left(a2 \cdot \left(a2 \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(a2 \cdot \left(a2 \cdot \sqrt{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq -0.01:\\ \;\;\;\;\sqrt{2} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) -0.01)
   (* (sqrt 2.0) (* (fma a1 a1 (* a2 a2)) -0.5))
   (/ (* a2 a2) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= -0.01) {
		tmp = sqrt(2.0) * (fma(a1, a1, (a2 * a2)) * -0.5);
	} else {
		tmp = (a2 * a2) / sqrt(2.0);
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= -0.01)
		tmp = Float64(sqrt(2.0) * Float64(fma(a1, a1, Float64(a2 * a2)) * -0.5));
	else
		tmp = Float64(Float64(a2 * a2) / sqrt(2.0));
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], -0.01], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -0.0100000000000000002

   1. Initial program 99.7%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. 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 5.9

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

   5. Simplified 5.9%

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. frac-add N/A

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

      4. rem-square-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 5.9

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

   7. Applied egg-rr 5.9%

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

   9. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{1}{\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}} \]
   10. Step-by-step derivation

       1. associate-/r/ N/A

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

       2. metadata-eval N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

       9. sqrt-lowering-sqrt.f64 72.3

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

   11. Applied egg-rr 72.3%

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

   if -0.0100000000000000002 < (cos.f64 th)

   1. Initial program 99.7%

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

   3. 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 85.9

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

   5. Simplified 85.9%

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

   6. Taylor expanded in a1 around 0

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

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

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

      4. sqrt-lowering-sqrt.f64 55.7

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

   8. Simplified 55.7%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -0.01:\\ \;\;\;\;\sqrt{2} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-out N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

   12. cos-lowering-cos.f64 99.7

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 9: 47.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq -0.01:\\ \;\;\;\;\frac{a2 \cdot a2}{-\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) -0.01)
   (/ (* a2 a2) (- (sqrt 2.0)))
   (/ (* a2 a2) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= -0.01) {
		tmp = (a2 * a2) / -sqrt(2.0);
	} else {
		tmp = (a2 * a2) / sqrt(2.0);
	}
	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) :: tmp
    if (cos(th) <= (-0.01d0)) then
        tmp = (a2 * a2) / -sqrt(2.0d0)
    else
        tmp = (a2 * a2) / sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= -0.01) {
		tmp = (a2 * a2) / -Math.sqrt(2.0);
	} else {
		tmp = (a2 * a2) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= -0.01:
		tmp = (a2 * a2) / -math.sqrt(2.0)
	else:
		tmp = (a2 * a2) / math.sqrt(2.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= -0.01)
		tmp = Float64(Float64(a2 * a2) / Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(a2 * a2) / sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= -0.01)
		tmp = (a2 * a2) / -sqrt(2.0);
	else
		tmp = (a2 * a2) / sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], -0.01], N[(N[(a2 * a2), $MachinePrecision] / (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -0.0100000000000000002

   1. Initial program 99.7%

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

   3. Taylor expanded in th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. 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 5.9

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

   5. Simplified 5.9%

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

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. associate-*r/ N/A

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

      4. frac-add N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      18. sqrt-lowering-sqrt.f64 5.9

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

   7. Applied egg-rr 5.9%

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

   9. Applied egg-rr 72.3%

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

   10. Taylor expanded in a2 around inf

       \[\leadsto \color{blue}{-1 \cdot \frac{{a2}^{2}}{\sqrt{2}}} \]
   11. 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. /-lowering-/.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. *-lowering-*.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. neg-lowering-neg.f64 N/A

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

       9. sqrt-lowering-sqrt.f64 40.7

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

   12. Simplified 40.7%

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

   if -0.0100000000000000002 < (cos.f64 th)

   1. Initial program 99.7%

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

   3. 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 85.9

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

   5. Simplified 85.9%

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

   6. Taylor expanded in a1 around 0

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

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

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

      4. sqrt-lowering-sqrt.f64 55.7

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

   8. Simplified 55.7%

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

Alternative 10: 57.4% accurate, 2.0× speedup

Math

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

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(a2 * a2) * Float64(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 * a2), $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

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

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

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

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

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

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

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

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

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

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

   8. sqrt-lowering-sqrt.f64 99.7

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 60.7

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

7. Simplified 60.7%

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

Alternative 11: 40.2% accurate, 10.2× speedup

Math

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

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(a2 * Float64(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[(a2 * N[(a2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. 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 65.9

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

5. Simplified 65.9%

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f64 65.8

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

7. Applied egg-rr 65.8%

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

8. Taylor expanded in a2 around inf

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

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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. sqrt-lowering-sqrt.f64 43.4

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

10. Simplified 43.4%

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

Alternative 12: 39.0% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(a1 \cdot \left(a1 \cdot \sqrt{2}\right)\right) \end{array} \]

FPCore

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

C

double code(double a1, double a2, double th) {
	return 0.5 * (a1 * (a1 * 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 * (a1 * (a1 * sqrt(2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(0.5 * N[(a1 * N[(a1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. 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 65.9

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

5. Simplified 65.9%

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. frac-add N/A

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

   4. rem-square-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f64 65.8

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

7. Applied egg-rr 65.8%

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

8. Taylor expanded in a2 around 0

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

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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. sqrt-lowering-sqrt.f64 38.5

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

10. Simplified 38.5%

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

Reproduce

herbie shell --seed 2024198 
(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))))