Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 11.6s

Alternatives: 16

Speedup: 1.9×

• 43.7% 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, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 2: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot t\_1 \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a2}{\sqrt{2}}, a2, \frac{a1 \cdot a1}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* (* a2 a2) t_1)) -1e-27)
     (* (fma -0.5 (* th th) 1.0) (/ (* a2 a2) (sqrt 2.0)))
     (fma (/ a2 (sqrt 2.0)) 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)) <= -1e-27) {
		tmp = fma(-0.5, (th * th), 1.0) * ((a2 * a2) / sqrt(2.0));
	} else {
		tmp = fma((a2 / sqrt(2.0)), 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)) <= -1e-27)
		tmp = Float64(fma(-0.5, Float64(th * th), 1.0) * Float64(Float64(a2 * a2) / sqrt(2.0)));
	else
		tmp = fma(Float64(a2 / sqrt(2.0)), a2, Float64(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], -1e-27], N[(N[(-0.5 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a2 + N[(N[(a1 * a1), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -1e-27

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in th around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

   7. Applied rewrites 55.1%

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

   8. Taylor expanded in a2 around inf

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

   10. Applied rewrites 47.9%

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

   if -1e-27 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.6%

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

   3. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\frac{{a1}^{2}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}}} \]
   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. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

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

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

      9. lower-sqrt.f64 87.2

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

   5. Applied rewrites 87.2%

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

   7. Applied rewrites 87.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a2}{\sqrt{2}}, a2, \frac{a1 \cdot a1}{\sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t_2 := \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot t\_1 \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))) (t_2 (/ (* a2 a2) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* (* a2 a2) t_1)) -1e-27)
     (* (fma -0.5 (* th th) 1.0) t_2)
     (fma a1 (/ a1 (sqrt 2.0)) t_2))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double t_2 = (a2 * a2) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + ((a2 * a2) * t_1)) <= -1e-27) {
		tmp = fma(-0.5, (th * th), 1.0) * t_2;
	} else {
		tmp = fma(a1, (a1 / sqrt(2.0)), t_2);
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	t_2 = Float64(Float64(a2 * a2) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(Float64(a2 * a2) * t_1)) <= -1e-27)
		tmp = Float64(fma(-0.5, Float64(th * th), 1.0) * t_2);
	else
		tmp = fma(a1, Float64(a1 / sqrt(2.0)), t_2);
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a2 * a2), $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], -1e-27], N[(N[(-0.5 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision], N[(a1 * N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + t$95$2), $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))) < -1e-27

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in th around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

   7. Applied rewrites 55.1%

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

   8. Taylor expanded in a2 around inf

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

   10. Applied rewrites 47.9%

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

   if -1e-27 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.6%

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

   3. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\frac{{a1}^{2}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}}} \]
   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. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

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

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

      9. lower-sqrt.f64 87.2

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

   5. Applied rewrites 87.2%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{a2 \cdot a2}{\sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot t\_1 \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{2}}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* (* a2 a2) t_1)) -1e-27)
     (* (fma -0.5 (* th th) 1.0) (/ (* a2 a2) (sqrt 2.0)))
     (/ 1.0 (/ (sqrt 2.0) (fma a2 a2 (* a1 a1)))))))

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)) <= -1e-27) {
		tmp = fma(-0.5, (th * th), 1.0) * ((a2 * a2) / sqrt(2.0));
	} else {
		tmp = 1.0 / (sqrt(2.0) / fma(a2, a2, (a1 * a1)));
	}
	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)) <= -1e-27)
		tmp = Float64(fma(-0.5, Float64(th * th), 1.0) * Float64(Float64(a2 * a2) / sqrt(2.0)));
	else
		tmp = Float64(1.0 / Float64(sqrt(2.0) / fma(a2, a2, Float64(a1 * a1))));
	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], -1e-27], N[(N[(-0.5 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a2 * a2 + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -1e-27

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in th around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

   7. Applied rewrites 55.1%

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

   8. Taylor expanded in a2 around inf

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

   10. Applied rewrites 47.9%

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

   if -1e-27 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{2}}{\color{blue}{{a2}^{2} + {a1}^{2}}}} \]

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.2

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

   7. Applied rewrites 87.2%

      \[\leadsto \frac{1}{\frac{\sqrt{2}}{\color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{2}}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot t\_1 \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* (* a2 a2) t_1)) -1e-27)
     (* (fma -0.5 (* th th) 1.0) (/ (* a2 a2) (sqrt 2.0)))
     (/ (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)) <= -1e-27) {
		tmp = fma(-0.5, (th * th), 1.0) * ((a2 * a2) / sqrt(2.0));
	} 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)) <= -1e-27)
		tmp = Float64(fma(-0.5, Float64(th * th), 1.0) * Float64(Float64(a2 * a2) / sqrt(2.0)));
	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], -1e-27], N[(N[(-0.5 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $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))) < -1e-27

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in th around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

   7. Applied rewrites 55.1%

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

   8. Taylor expanded in a2 around inf

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

   10. Applied rewrites 47.9%

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

   if -1e-27 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in th around 0

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

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

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

      7. lower-sqrt.f64 87.2

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

   7. Applied rewrites 87.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot \frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot t\_1 \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* (* a2 a2) t_1)) -1e-27)
     (* (fma -0.5 (* th th) 1.0) (/ (* a1 a1) (sqrt 2.0)))
     (/ (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)) <= -1e-27) {
		tmp = fma(-0.5, (th * th), 1.0) * ((a1 * a1) / sqrt(2.0));
	} 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)) <= -1e-27)
		tmp = Float64(fma(-0.5, Float64(th * th), 1.0) * Float64(Float64(a1 * a1) / sqrt(2.0)));
	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], -1e-27], N[(N[(-0.5 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $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))) < -1e-27

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in th around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

   7. Applied rewrites 55.1%

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

   8. Taylor expanded in a2 around 0

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

   10. Applied rewrites 40.9%

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

   if -1e-27 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in th around 0

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

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

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

      7. lower-sqrt.f64 87.2

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

   7. Applied rewrites 87.2%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 99.7

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

4. Applied rewrites 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 8: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lower-/.f64 99.7

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l/ N/A

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

   5. div-inv N/A

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

   6. associate-*l* N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-rgt-out N/A

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

   10. div-inv N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r/ N/A

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

   13. lift-sqrt.f64 N/A

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

   14. div-inv N/A

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

   15. lift-*.f64 N/A

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

   16. lift-sqrt.f64 N/A

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

   17. lift-sqrt.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. lift-sqrt.f64 N/A

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 9: 57.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-sqrt.f64 50.1

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

7. Applied rewrites 50.1%

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

Alternative 10: 57.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lower-/.f64 99.7

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l/ N/A

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

   5. div-inv N/A

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

   6. associate-*l* N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-rgt-out N/A

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

   10. div-inv N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r/ N/A

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

   13. lift-sqrt.f64 N/A

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

   14. div-inv N/A

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

   15. lift-*.f64 N/A

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

   16. lift-sqrt.f64 N/A

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

   17. lift-sqrt.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. lift-sqrt.f64 N/A

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

8. Applied rewrites 99.7%

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

9. Taylor expanded in a1 around 0

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

    1. unpow2 N/A

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

    2. lower-*.f64 50.0

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

11. Applied rewrites 50.0%

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

12. Final simplification 50.0%

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

Alternative 11: 66.6% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in th around 0

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

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

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

   7. lower-sqrt.f64 70.9

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

7. Applied rewrites 70.9%

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

Alternative 12: 66.6% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. 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. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

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

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

   9. lower-sqrt.f64 70.9

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

5. Applied rewrites 70.9%

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

7. Applied rewrites 70.9%

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

9. Applied rewrites 70.9%

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

10. Final simplification 70.9%

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

Alternative 13: 40.1% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\frac{{a1}^{2}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}}} \]
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. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

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

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

   9. lower-sqrt.f64 70.9

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

5. Applied rewrites 70.9%

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

6. Taylor expanded in a1 around 0

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

8. Applied rewrites 36.7%

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

Alternative 14: 40.1% accurate, 10.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in th around 0

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

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

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

   9. lower-sqrt.f64 70.9

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

5. Applied rewrites 70.9%

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

7. Applied rewrites 70.9%

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

8. Taylor expanded in a1 around inf

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

10. Applied rewrites 47.9%

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

11. Taylor expanded in a1 around 0

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

13. Applied rewrites 36.6%

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

14. Final simplification 36.6%

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

Alternative 15: 40.1% accurate, 10.2× speedup

Math

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

FPCore

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

C

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

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 * (sqrt(2.0d0) * 0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\frac{{a1}^{2}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}}} \]
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. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

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

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

   9. lower-sqrt.f64 70.9

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

5. Applied rewrites 70.9%

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

7. Applied rewrites 70.9%

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

8. Taylor expanded in a1 around 0

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

10. Applied rewrites 36.6%

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

Alternative 16: 39.9% accurate, 10.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\frac{{a1}^{2}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}}} \]
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. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

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

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

   9. lower-sqrt.f64 70.9

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

5. Applied rewrites 70.9%

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

7. Applied rewrites 70.9%

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

8. Taylor expanded in a1 around inf

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

10. Applied rewrites 47.9%

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

Reproduce

herbie shell --seed 2024231 
(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))))