Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.0% → 99.7%

Time: 12.8s

Alternatives: 28

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Initial Program: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (hypot (sin ky) (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / hypot(sin(ky), sin(kx))) * sin(th);
}

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

   3. +-commutative N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

   5. unpow2 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

   7. unpow2 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

   8. lower-hypot.f64 99.7

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

4. Applied rewrites 99.7%

   \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
5. Add Preprocessing

Alternative 2: 67.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \cos \left(kx \cdot -2\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_3}}\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(0.5, t\_1, t\_3\right)}} \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\right)\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_1} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;t\_2 \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;t\_2 \leq 0.996:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- 1.0 (cos (* kx -2.0))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (fma -0.5 (cos (* ky -2.0)) 0.5)))
   (if (<= t_2 -1.0)
     (* (sin th) (/ (sin ky) (sqrt t_3)))
     (if (<= t_2 -0.2)
       (*
        th
        (*
         (sqrt (/ 1.0 (fma 0.5 t_1 t_3)))
         (* (sin ky) (fma -0.16666666666666666 (* th th) 1.0))))
       (if (<= t_2 4e-144)
         (* (sin th) (/ (sin ky) (* (sqrt t_1) (sqrt 0.5))))
         (if (<= t_2 0.1)
           (* (sin th) (/ (sin ky) (sin kx)))
           (if (<= t_2 0.996)
             (/
              1.0
              (/
               (sqrt
                (fma
                 (- 1.0 (cos (+ kx kx)))
                 0.5
                 (+ 0.5 (* -0.5 (cos (+ ky ky))))))
               (*
                (sin ky)
                (fma
                 th
                 (*
                  (* th th)
                  (fma 0.008333333333333333 (* th th) -0.16666666666666666))
                 th))))
             (*
              (sin th)
              (/ (sin ky) (fma kx (* kx (/ 0.5 ky)) (sin ky)))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = 1.0 - cos((kx * -2.0));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = fma(-0.5, cos((ky * -2.0)), 0.5);
	double tmp;
	if (t_2 <= -1.0) {
		tmp = sin(th) * (sin(ky) / sqrt(t_3));
	} else if (t_2 <= -0.2) {
		tmp = th * (sqrt((1.0 / fma(0.5, t_1, t_3))) * (sin(ky) * fma(-0.16666666666666666, (th * th), 1.0)));
	} else if (t_2 <= 4e-144) {
		tmp = sin(th) * (sin(ky) / (sqrt(t_1) * sqrt(0.5)));
	} else if (t_2 <= 0.1) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else if (t_2 <= 0.996) {
		tmp = 1.0 / (sqrt(fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky)))))) / (sin(ky) * fma(th, ((th * th) * fma(0.008333333333333333, (th * th), -0.16666666666666666)), th)));
	} else {
		tmp = sin(th) * (sin(ky) / fma(kx, (kx * (0.5 / ky)), sin(ky)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(1.0 - cos(Float64(kx * -2.0)))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = fma(-0.5, cos(Float64(ky * -2.0)), 0.5)
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(t_3)));
	elseif (t_2 <= -0.2)
		tmp = Float64(th * Float64(sqrt(Float64(1.0 / fma(0.5, t_1, t_3))) * Float64(sin(ky) * fma(-0.16666666666666666, Float64(th * th), 1.0))));
	elseif (t_2 <= 4e-144)
		tmp = Float64(sin(th) * Float64(sin(ky) / Float64(sqrt(t_1) * sqrt(0.5))));
	elseif (t_2 <= 0.1)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	elseif (t_2 <= 0.996)
		tmp = Float64(1.0 / Float64(sqrt(fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))) / Float64(sin(ky) * fma(th, Float64(Float64(th * th) * fma(0.008333333333333333, Float64(th * th), -0.16666666666666666)), th))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / fma(kx, Float64(kx * Float64(0.5 / ky)), sin(ky))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], N[(th * N[(N[Sqrt[N[(1.0 / N[(0.5 * t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e-144], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.1], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.996], N[(1.0 / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * N[(th * N[(N[(th * th), $MachinePrecision] * N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(kx * N[(kx * N[(0.5 / ky), $MachinePrecision]), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1

   1. Initial program 90.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 71.2

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

   7. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 71.4%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 99.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(\frac{-1}{6} \cdot \left(\left({th}^{2} \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}\right) + \sin ky \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}\right)} \]

   7. Applied rewrites 58.9%

      \[\leadsto \color{blue}{th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}} \cdot \left(\mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right) \cdot \sin ky\right)\right)} \]

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-144

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. count-2 N/A

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

      10. cos-diff N/A

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

      11. cos-sin-sum N/A

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

      12. lower--.f64 N/A

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

      13. count-2 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-+.f64 80.7

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{\frac{1}{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{\frac{1}{2}}}} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      11. lower-sqrt.f64 78.0

          \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \color{blue}{\sqrt{0.5}}} \cdot \sin th \]

   7. Applied rewrites 78.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}} \cdot \sin th \]

   if 3.9999999999999998e-144 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 55.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 55.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.996

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\left(th \cdot \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \left(th \cdot \color{blue}{\left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right) + 1\right)}\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\left(th \cdot \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) + th \cdot 1\right)}}} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \left(th \cdot \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) + \color{blue}{th}\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\mathsf{fma}\left(th, {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right)}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \color{blue}{{th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)}, th\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right)} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right)} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right)}} \]

      8. sub-neg N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \color{blue}{\left(\frac{1}{120} \cdot {th}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, th\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \left(\frac{1}{120} \cdot {th}^{2} + \color{blue}{\frac{-1}{6}}\right), th\right)}} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {th}^{2}, \frac{-1}{6}\right)}, th\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{th \cdot th}, \frac{-1}{6}\right), th\right)}} \]

      12. lower-*.f64 58.0

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, \color{blue}{th \cdot th}, -0.16666666666666666\right), th\right)}} \]

   7. Applied rewrites 58.0%

      \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)}}} \]

   if 0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 84.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + \frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky} + \sin ky}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2}}{\sin ky} \cdot \frac{1}{2}} + \sin ky} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2} \cdot \frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{kx}^{2} \cdot \frac{\frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{\sin ky} + \sin ky} \cdot \sin th \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)} + \sin ky} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left(kx \cdot kx\right)} \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right) + \sin ky} \cdot \sin th \]

      8. associate-*l* N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{kx \cdot \left(kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)\right)} + \sin ky} \cdot \sin th \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right), \sin ky\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, \color{blue}{kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)}, \sin ky\right)} \cdot \sin th \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2} \cdot 1}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\color{blue}{\frac{1}{2}}}{\sin ky}, \sin ky\right)} \cdot \sin th \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2}}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      14. lower-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      15. lower-sin.f64 92.7

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   5. Applied rewrites 92.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 92.0%

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
3. Recombined 6 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}} \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.996:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \cos \left(kx \cdot -2\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_3}}\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(0.5, t\_1, t\_3\right)}} \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\right)\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_1} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;t\_2 \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;t\_2 \leq 0.996:\\ \;\;\;\;\frac{\sin ky \cdot \left(th \cdot \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), 1\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- 1.0 (cos (* kx -2.0))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (fma -0.5 (cos (* ky -2.0)) 0.5)))
   (if (<= t_2 -1.0)
     (* (sin th) (/ (sin ky) (sqrt t_3)))
     (if (<= t_2 -0.2)
       (*
        th
        (*
         (sqrt (/ 1.0 (fma 0.5 t_1 t_3)))
         (* (sin ky) (fma -0.16666666666666666 (* th th) 1.0))))
       (if (<= t_2 4e-144)
         (* (sin th) (/ (sin ky) (* (sqrt t_1) (sqrt 0.5))))
         (if (<= t_2 0.1)
           (* (sin th) (/ (sin ky) (sin kx)))
           (if (<= t_2 0.996)
             (/
              (*
               (sin ky)
               (*
                th
                (fma
                 (* th th)
                 (fma 0.008333333333333333 (* th th) -0.16666666666666666)
                 1.0)))
              (sqrt
               (fma
                (- 1.0 (cos (+ kx kx)))
                0.5
                (+ 0.5 (* -0.5 (cos (+ ky ky)))))))
             (*
              (sin th)
              (/ (sin ky) (fma kx (* kx (/ 0.5 ky)) (sin ky)))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = 1.0 - cos((kx * -2.0));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = fma(-0.5, cos((ky * -2.0)), 0.5);
	double tmp;
	if (t_2 <= -1.0) {
		tmp = sin(th) * (sin(ky) / sqrt(t_3));
	} else if (t_2 <= -0.2) {
		tmp = th * (sqrt((1.0 / fma(0.5, t_1, t_3))) * (sin(ky) * fma(-0.16666666666666666, (th * th), 1.0)));
	} else if (t_2 <= 4e-144) {
		tmp = sin(th) * (sin(ky) / (sqrt(t_1) * sqrt(0.5)));
	} else if (t_2 <= 0.1) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else if (t_2 <= 0.996) {
		tmp = (sin(ky) * (th * fma((th * th), fma(0.008333333333333333, (th * th), -0.16666666666666666), 1.0))) / sqrt(fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky))))));
	} else {
		tmp = sin(th) * (sin(ky) / fma(kx, (kx * (0.5 / ky)), sin(ky)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(1.0 - cos(Float64(kx * -2.0)))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = fma(-0.5, cos(Float64(ky * -2.0)), 0.5)
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(t_3)));
	elseif (t_2 <= -0.2)
		tmp = Float64(th * Float64(sqrt(Float64(1.0 / fma(0.5, t_1, t_3))) * Float64(sin(ky) * fma(-0.16666666666666666, Float64(th * th), 1.0))));
	elseif (t_2 <= 4e-144)
		tmp = Float64(sin(th) * Float64(sin(ky) / Float64(sqrt(t_1) * sqrt(0.5))));
	elseif (t_2 <= 0.1)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	elseif (t_2 <= 0.996)
		tmp = Float64(Float64(sin(ky) * Float64(th * fma(Float64(th * th), fma(0.008333333333333333, Float64(th * th), -0.16666666666666666), 1.0))) / sqrt(fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / fma(kx, Float64(kx * Float64(0.5 / ky)), sin(ky))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], N[(th * N[(N[Sqrt[N[(1.0 / N[(0.5 * t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e-144], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.1], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.996], N[(N[(N[Sin[ky], $MachinePrecision] * N[(th * N[(N[(th * th), $MachinePrecision] * N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(kx * N[(kx * N[(0.5 / ky), $MachinePrecision]), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1

   1. Initial program 90.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 71.2

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

   7. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 71.4%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 99.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(\frac{-1}{6} \cdot \left(\left({th}^{2} \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}\right) + \sin ky \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}\right)} \]

   7. Applied rewrites 58.9%

      \[\leadsto \color{blue}{th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}} \cdot \left(\mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right) \cdot \sin ky\right)\right)} \]

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-144

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. count-2 N/A

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

      10. cos-diff N/A

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

      11. cos-sin-sum N/A

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

      12. lower--.f64 N/A

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

      13. count-2 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-+.f64 80.7

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{\frac{1}{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{\frac{1}{2}}}} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      11. lower-sqrt.f64 78.0

          \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \color{blue}{\sqrt{0.5}}} \cdot \sin th \]

   7. Applied rewrites 78.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}} \cdot \sin th \]

   if 3.9999999999999998e-144 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 55.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 55.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.996

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. lower-*.f64 99.3

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \]

      11. sin-mult N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \]

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky \cdot \color{blue}{\left(th \cdot \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \color{blue}{\left(th \cdot \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \left(th \cdot \color{blue}{\left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right) + 1\right)}\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \left(th \cdot \color{blue}{\mathsf{fma}\left({th}^{2}, \frac{1}{120} \cdot {th}^{2} - \frac{1}{6}, 1\right)}\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin ky \cdot \left(th \cdot \mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{1}{120} \cdot {th}^{2} - \frac{1}{6}, 1\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \left(th \cdot \mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{1}{120} \cdot {th}^{2} - \frac{1}{6}, 1\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      6. sub-neg N/A

         \[\leadsto \frac{\sin ky \cdot \left(th \cdot \mathsf{fma}\left(th \cdot th, \color{blue}{\frac{1}{120} \cdot {th}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \left(th \cdot \mathsf{fma}\left(th \cdot th, \frac{1}{120} \cdot {th}^{2} + \color{blue}{\frac{-1}{6}}, 1\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \left(th \cdot \mathsf{fma}\left(th \cdot th, \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {th}^{2}, \frac{-1}{6}\right)}, 1\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky \cdot \left(th \cdot \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(\frac{1}{120}, \color{blue}{th \cdot th}, \frac{-1}{6}\right), 1\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      10. lower-*.f64 58.0

          \[\leadsto \frac{\sin ky \cdot \left(th \cdot \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(0.008333333333333333, \color{blue}{th \cdot th}, -0.16666666666666666\right), 1\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \]

   7. Applied rewrites 58.0%

      \[\leadsto \frac{\sin ky \cdot \color{blue}{\left(th \cdot \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), 1\right)\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \]

   if 0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 84.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + \frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky} + \sin ky}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2}}{\sin ky} \cdot \frac{1}{2}} + \sin ky} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2} \cdot \frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{kx}^{2} \cdot \frac{\frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{\sin ky} + \sin ky} \cdot \sin th \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)} + \sin ky} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left(kx \cdot kx\right)} \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right) + \sin ky} \cdot \sin th \]

      8. associate-*l* N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{kx \cdot \left(kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)\right)} + \sin ky} \cdot \sin th \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right), \sin ky\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, \color{blue}{kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)}, \sin ky\right)} \cdot \sin th \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2} \cdot 1}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\color{blue}{\frac{1}{2}}}{\sin ky}, \sin ky\right)} \cdot \sin th \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2}}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      14. lower-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      15. lower-sin.f64 92.7

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   5. Applied rewrites 92.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 92.0%

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
3. Recombined 6 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}} \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.996:\\ \;\;\;\;\frac{\sin ky \cdot \left(th \cdot \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), 1\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \cos \left(kx \cdot -2\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\\ t_4 := th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(0.5, t\_1, t\_3\right)}} \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\right)\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_3}}\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_1} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;t\_2 \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;t\_2 \leq 0.996:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- 1.0 (cos (* kx -2.0))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (fma -0.5 (cos (* ky -2.0)) 0.5))
        (t_4
         (*
          th
          (*
           (sqrt (/ 1.0 (fma 0.5 t_1 t_3)))
           (* (sin ky) (fma -0.16666666666666666 (* th th) 1.0))))))
   (if (<= t_2 -1.0)
     (* (sin th) (/ (sin ky) (sqrt t_3)))
     (if (<= t_2 -0.2)
       t_4
       (if (<= t_2 4e-144)
         (* (sin th) (/ (sin ky) (* (sqrt t_1) (sqrt 0.5))))
         (if (<= t_2 0.1)
           (* (sin th) (/ (sin ky) (sin kx)))
           (if (<= t_2 0.996)
             t_4
             (*
              (sin th)
              (/ (sin ky) (fma kx (* kx (/ 0.5 ky)) (sin ky)))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = 1.0 - cos((kx * -2.0));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = fma(-0.5, cos((ky * -2.0)), 0.5);
	double t_4 = th * (sqrt((1.0 / fma(0.5, t_1, t_3))) * (sin(ky) * fma(-0.16666666666666666, (th * th), 1.0)));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = sin(th) * (sin(ky) / sqrt(t_3));
	} else if (t_2 <= -0.2) {
		tmp = t_4;
	} else if (t_2 <= 4e-144) {
		tmp = sin(th) * (sin(ky) / (sqrt(t_1) * sqrt(0.5)));
	} else if (t_2 <= 0.1) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else if (t_2 <= 0.996) {
		tmp = t_4;
	} else {
		tmp = sin(th) * (sin(ky) / fma(kx, (kx * (0.5 / ky)), sin(ky)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(1.0 - cos(Float64(kx * -2.0)))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = fma(-0.5, cos(Float64(ky * -2.0)), 0.5)
	t_4 = Float64(th * Float64(sqrt(Float64(1.0 / fma(0.5, t_1, t_3))) * Float64(sin(ky) * fma(-0.16666666666666666, Float64(th * th), 1.0))))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(t_3)));
	elseif (t_2 <= -0.2)
		tmp = t_4;
	elseif (t_2 <= 4e-144)
		tmp = Float64(sin(th) * Float64(sin(ky) / Float64(sqrt(t_1) * sqrt(0.5))));
	elseif (t_2 <= 0.1)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	elseif (t_2 <= 0.996)
		tmp = t_4;
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / fma(kx, Float64(kx * Float64(0.5 / ky)), sin(ky))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$4 = N[(th * N[(N[Sqrt[N[(1.0 / N[(0.5 * t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], t$95$4, If[LessEqual[t$95$2, 4e-144], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.1], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.996], t$95$4, N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(kx * N[(kx * N[(0.5 / ky), $MachinePrecision]), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1

   1. Initial program 90.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 71.2

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

   7. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 71.4%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.996

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(\frac{-1}{6} \cdot \left(\left({th}^{2} \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}\right) + \sin ky \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}\right)} \]

   7. Applied rewrites 58.2%

      \[\leadsto \color{blue}{th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}} \cdot \left(\mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right) \cdot \sin ky\right)\right)} \]

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-144

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. count-2 N/A

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

      10. cos-diff N/A

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

      11. cos-sin-sum N/A

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

      12. lower--.f64 N/A

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

      13. count-2 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-+.f64 80.7

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{\frac{1}{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{\frac{1}{2}}}} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      11. lower-sqrt.f64 78.0

          \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \color{blue}{\sqrt{0.5}}} \cdot \sin th \]

   7. Applied rewrites 78.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}} \cdot \sin th \]

   if 3.9999999999999998e-144 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 55.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 55.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 84.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + \frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky} + \sin ky}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2}}{\sin ky} \cdot \frac{1}{2}} + \sin ky} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2} \cdot \frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{kx}^{2} \cdot \frac{\frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{\sin ky} + \sin ky} \cdot \sin th \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)} + \sin ky} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left(kx \cdot kx\right)} \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right) + \sin ky} \cdot \sin th \]

      8. associate-*l* N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{kx \cdot \left(kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)\right)} + \sin ky} \cdot \sin th \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right), \sin ky\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, \color{blue}{kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)}, \sin ky\right)} \cdot \sin th \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2} \cdot 1}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\color{blue}{\frac{1}{2}}}{\sin ky}, \sin ky\right)} \cdot \sin th \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2}}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      14. lower-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      15. lower-sin.f64 92.7

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   5. Applied rewrites 92.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 92.0%

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}} \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.996:\\ \;\;\;\;th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}} \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;\frac{th \cdot \left(\sin ky \cdot t\_1\right)}{t\_3}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;t\_2 \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;t\_2 \leq 0.996:\\ \;\;\;\;\frac{t\_1 \cdot \left(\sin ky \cdot th\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fma -0.16666666666666666 (* th th) 1.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3
         (sqrt
          (fma (- 1.0 (cos (+ kx kx))) 0.5 (+ 0.5 (* -0.5 (cos (+ ky ky))))))))
   (if (<= t_2 -1.0)
     (* (sin th) (/ (sin ky) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_2 -0.2)
       (/ (* th (* (sin ky) t_1)) t_3)
       (if (<= t_2 4e-144)
         (*
          (sin th)
          (/ (sin ky) (* (sqrt (- 1.0 (cos (* kx -2.0)))) (sqrt 0.5))))
         (if (<= t_2 0.1)
           (* (sin th) (/ (sin ky) (sin kx)))
           (if (<= t_2 0.996)
             (/ (* t_1 (* (sin ky) th)) t_3)
             (*
              (sin th)
              (/ (sin ky) (fma kx (* kx (/ 0.5 ky)) (sin ky)))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma(-0.16666666666666666, (th * th), 1.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = sqrt(fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky))))));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_2 <= -0.2) {
		tmp = (th * (sin(ky) * t_1)) / t_3;
	} else if (t_2 <= 4e-144) {
		tmp = sin(th) * (sin(ky) / (sqrt((1.0 - cos((kx * -2.0)))) * sqrt(0.5)));
	} else if (t_2 <= 0.1) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else if (t_2 <= 0.996) {
		tmp = (t_1 * (sin(ky) * th)) / t_3;
	} else {
		tmp = sin(th) * (sin(ky) / fma(kx, (kx * (0.5 / ky)), sin(ky)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = fma(-0.16666666666666666, Float64(th * th), 1.0)
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = sqrt(fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky))))))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_2 <= -0.2)
		tmp = Float64(Float64(th * Float64(sin(ky) * t_1)) / t_3);
	elseif (t_2 <= 4e-144)
		tmp = Float64(sin(th) * Float64(sin(ky) / Float64(sqrt(Float64(1.0 - cos(Float64(kx * -2.0)))) * sqrt(0.5))));
	elseif (t_2 <= 0.1)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	elseif (t_2 <= 0.996)
		tmp = Float64(Float64(t_1 * Float64(sin(ky) * th)) / t_3);
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / fma(kx, Float64(kx * Float64(0.5 / ky)), sin(ky))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(-0.16666666666666666 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], N[(N[(th * N[(N[Sin[ky], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$2, 4e-144], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.1], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.996], N[(N[(t$95$1 * N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(kx * N[(kx * N[(0.5 / ky), $MachinePrecision]), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1

   1. Initial program 90.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 71.2

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

   7. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 71.4%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 99.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. lower-*.f64 99.4

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \]

      11. sin-mult N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \]

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 94.8%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \left(\sin ky + \frac{-1}{6} \cdot \left({th}^{2} \cdot \sin ky\right)\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{th \cdot \left(\sin ky + \frac{-1}{6} \cdot \left({th}^{2} \cdot \sin ky\right)\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{th \cdot \left(\color{blue}{1 \cdot \sin ky} + \frac{-1}{6} \cdot \left({th}^{2} \cdot \sin ky\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{th \cdot \left(1 \cdot \sin ky + \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot \sin ky}\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{th \cdot \color{blue}{\left(\sin ky \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{th \cdot \color{blue}{\left(\sin ky \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{th \cdot \left(\color{blue}{\sin ky} \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      7. +-commutative N/A

         \[\leadsto \frac{th \cdot \left(\sin ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{th \cdot \left(\sin ky \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {th}^{2}, 1\right)}\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      9. unpow2 N/A

         \[\leadsto \frac{th \cdot \left(\sin ky \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{th \cdot th}, 1\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      10. lower-*.f64 58.7

          \[\leadsto \frac{th \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{th \cdot th}, 1\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \]

   7. Applied rewrites 58.7%

      \[\leadsto \frac{\color{blue}{th \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \]

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-144

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. count-2 N/A

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

      10. cos-diff N/A

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

      11. cos-sin-sum N/A

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

      12. lower--.f64 N/A

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

      13. count-2 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-+.f64 80.7

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{\frac{1}{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{\frac{1}{2}}}} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      11. lower-sqrt.f64 78.0

          \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \color{blue}{\sqrt{0.5}}} \cdot \sin th \]

   7. Applied rewrites 78.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}} \cdot \sin th \]

   if 3.9999999999999998e-144 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 55.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 55.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.996

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. lower-*.f64 99.3

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \]

      11. sin-mult N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \]

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky \cdot \color{blue}{\left(th \cdot \left(1 + {th}^{2} \cdot \left({th}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {th}^{2}\right) - \frac{1}{6}\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \left(th \cdot \color{blue}{\left({th}^{2} \cdot \left({th}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {th}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky \cdot \color{blue}{\left(th \cdot \left({th}^{2} \cdot \left({th}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {th}^{2}\right) - \frac{1}{6}\right)\right) + th \cdot 1\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky \cdot \left(th \cdot \left({th}^{2} \cdot \left({th}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {th}^{2}\right) - \frac{1}{6}\right)\right) + \color{blue}{th}\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \color{blue}{\mathsf{fma}\left(th, {th}^{2} \cdot \left({th}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {th}^{2}\right) - \frac{1}{6}\right), th\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \mathsf{fma}\left(th, \color{blue}{{th}^{2} \cdot \left({th}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {th}^{2}\right) - \frac{1}{6}\right)}, th\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky \cdot \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right)} \cdot \left({th}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {th}^{2}\right) - \frac{1}{6}\right), th\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right)} \cdot \left({th}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {th}^{2}\right) - \frac{1}{6}\right), th\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      8. sub-neg N/A

         \[\leadsto \frac{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \color{blue}{\left({th}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {th}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, th\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \left({th}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {th}^{2}\right) + \color{blue}{\frac{-1}{6}}\right), th\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \color{blue}{\mathsf{fma}\left({th}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {th}^{2}, \frac{-1}{6}\right)}, th\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{1}{120} + \frac{-1}{5040} \cdot {th}^{2}, \frac{-1}{6}\right), th\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{1}{120} + \frac{-1}{5040} \cdot {th}^{2}, \frac{-1}{6}\right), th\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(th \cdot th, \color{blue}{\frac{-1}{5040} \cdot {th}^{2} + \frac{1}{120}}, \frac{-1}{6}\right), th\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(th \cdot th, \color{blue}{\mathsf{fma}\left(\frac{-1}{5040}, {th}^{2}, \frac{1}{120}\right)}, \frac{-1}{6}\right), th\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(\frac{-1}{5040}, \color{blue}{th \cdot th}, \frac{1}{120}\right), \frac{-1}{6}\right), th\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      16. lower-*.f64 58.6

          \[\leadsto \frac{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(-0.0001984126984126984, \color{blue}{th \cdot th}, 0.008333333333333333\right), -0.16666666666666666\right), th\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \]

   7. Applied rewrites 58.6%

      \[\leadsto \frac{\sin ky \cdot \color{blue}{\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(-0.0001984126984126984, th \cdot th, 0.008333333333333333\right), -0.16666666666666666\right), th\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \left(\sin ky + \frac{-1}{6} \cdot \left({th}^{2} \cdot \sin ky\right)\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{th \cdot \left(\color{blue}{1 \cdot \sin ky} + \frac{-1}{6} \cdot \left({th}^{2} \cdot \sin ky\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{th \cdot \left(1 \cdot \sin ky + \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot \sin ky}\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{th \cdot \color{blue}{\left(\sin ky \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(th \cdot \sin ky\right) \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(th \cdot \sin ky\right) \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(th \cdot \sin ky\right)} \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\left(th \cdot \color{blue}{\sin ky}\right) \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\left(th \cdot \sin ky\right) \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(th \cdot \sin ky\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {th}^{2}, 1\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      10. unpow2 N/A

          \[\leadsto \frac{\left(th \cdot \sin ky\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{th \cdot th}, 1\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      11. lower-*.f64 57.7

          \[\leadsto \frac{\left(th \cdot \sin ky\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{th \cdot th}, 1\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \]

   10. Applied rewrites 57.7%

       \[\leadsto \frac{\color{blue}{\left(th \cdot \sin ky\right) \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \]

   if 0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 84.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + \frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky} + \sin ky}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2}}{\sin ky} \cdot \frac{1}{2}} + \sin ky} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2} \cdot \frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{kx}^{2} \cdot \frac{\frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{\sin ky} + \sin ky} \cdot \sin th \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)} + \sin ky} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left(kx \cdot kx\right)} \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right) + \sin ky} \cdot \sin th \]

      8. associate-*l* N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{kx \cdot \left(kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)\right)} + \sin ky} \cdot \sin th \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right), \sin ky\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, \color{blue}{kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)}, \sin ky\right)} \cdot \sin th \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2} \cdot 1}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\color{blue}{\frac{1}{2}}}{\sin ky}, \sin ky\right)} \cdot \sin th \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2}}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      14. lower-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      15. lower-sin.f64 92.7

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   5. Applied rewrites 92.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 92.0%

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
3. Recombined 6 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\frac{th \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.996:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right) \cdot \left(\sin ky \cdot th\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \frac{th \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;t\_1 \leq 0.996:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2
         (/
          (* th (* (sin ky) (fma -0.16666666666666666 (* th th) 1.0)))
          (sqrt
           (fma
            (- 1.0 (cos (+ kx kx)))
            0.5
            (+ 0.5 (* -0.5 (cos (+ ky ky)))))))))
   (if (<= t_1 -1.0)
     (* (sin th) (/ (sin ky) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_1 -0.2)
       t_2
       (if (<= t_1 4e-144)
         (*
          (sin th)
          (/ (sin ky) (* (sqrt (- 1.0 (cos (* kx -2.0)))) (sqrt 0.5))))
         (if (<= t_1 0.1)
           (* (sin th) (/ (sin ky) (sin kx)))
           (if (<= t_1 0.996)
             t_2
             (*
              (sin th)
              (/ (sin ky) (fma kx (* kx (/ 0.5 ky)) (sin ky)))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = (th * (sin(ky) * fma(-0.16666666666666666, (th * th), 1.0))) / sqrt(fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky))))));
	double tmp;
	if (t_1 <= -1.0) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_1 <= -0.2) {
		tmp = t_2;
	} else if (t_1 <= 4e-144) {
		tmp = sin(th) * (sin(ky) / (sqrt((1.0 - cos((kx * -2.0)))) * sqrt(0.5)));
	} else if (t_1 <= 0.1) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else if (t_1 <= 0.996) {
		tmp = t_2;
	} else {
		tmp = sin(th) * (sin(ky) / fma(kx, (kx * (0.5 / ky)), sin(ky)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(Float64(th * Float64(sin(ky) * fma(-0.16666666666666666, Float64(th * th), 1.0))) / sqrt(fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_1 <= -0.2)
		tmp = t_2;
	elseif (t_1 <= 4e-144)
		tmp = Float64(sin(th) * Float64(sin(ky) / Float64(sqrt(Float64(1.0 - cos(Float64(kx * -2.0)))) * sqrt(0.5))));
	elseif (t_1 <= 0.1)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	elseif (t_1 <= 0.996)
		tmp = t_2;
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / fma(kx, Float64(kx * Float64(0.5 / ky)), sin(ky))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(th * N[(N[Sin[ky], $MachinePrecision] * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], t$95$2, If[LessEqual[t$95$1, 4e-144], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.996], t$95$2, N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(kx * N[(kx * N[(0.5 / ky), $MachinePrecision]), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1

   1. Initial program 90.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 71.2

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

   7. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 71.4%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.996

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. lower-*.f64 99.3

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \]

      11. sin-mult N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \]

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 95.5%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \left(\sin ky + \frac{-1}{6} \cdot \left({th}^{2} \cdot \sin ky\right)\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{th \cdot \left(\sin ky + \frac{-1}{6} \cdot \left({th}^{2} \cdot \sin ky\right)\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{th \cdot \left(\color{blue}{1 \cdot \sin ky} + \frac{-1}{6} \cdot \left({th}^{2} \cdot \sin ky\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{th \cdot \left(1 \cdot \sin ky + \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot \sin ky}\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{th \cdot \color{blue}{\left(\sin ky \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{th \cdot \color{blue}{\left(\sin ky \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{th \cdot \left(\color{blue}{\sin ky} \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      7. +-commutative N/A

         \[\leadsto \frac{th \cdot \left(\sin ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{th \cdot \left(\sin ky \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {th}^{2}, 1\right)}\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      9. unpow2 N/A

         \[\leadsto \frac{th \cdot \left(\sin ky \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{th \cdot th}, 1\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      10. lower-*.f64 58.2

          \[\leadsto \frac{th \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{th \cdot th}, 1\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \]

   7. Applied rewrites 58.2%

      \[\leadsto \frac{\color{blue}{th \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \]

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-144

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. count-2 N/A

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

      10. cos-diff N/A

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

      11. cos-sin-sum N/A

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

      12. lower--.f64 N/A

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

      13. count-2 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-+.f64 80.7

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{\frac{1}{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{\frac{1}{2}}}} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      11. lower-sqrt.f64 78.0

          \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \color{blue}{\sqrt{0.5}}} \cdot \sin th \]

   7. Applied rewrites 78.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}} \cdot \sin th \]

   if 3.9999999999999998e-144 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 55.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 55.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 84.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + \frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky} + \sin ky}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2}}{\sin ky} \cdot \frac{1}{2}} + \sin ky} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2} \cdot \frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{kx}^{2} \cdot \frac{\frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{\sin ky} + \sin ky} \cdot \sin th \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)} + \sin ky} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left(kx \cdot kx\right)} \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right) + \sin ky} \cdot \sin th \]

      8. associate-*l* N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{kx \cdot \left(kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)\right)} + \sin ky} \cdot \sin th \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right), \sin ky\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, \color{blue}{kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)}, \sin ky\right)} \cdot \sin th \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2} \cdot 1}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\color{blue}{\frac{1}{2}}}{\sin ky}, \sin ky\right)} \cdot \sin th \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2}}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      14. lower-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      15. lower-sin.f64 92.7

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   5. Applied rewrites 92.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 92.0%

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\frac{th \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.996:\\ \;\;\;\;\frac{th \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \cos \left(kx \cdot -2\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\\ t_4 := \sqrt{\frac{1}{\mathsf{fma}\left(0.5, t\_1, t\_3\right)}} \cdot \left(\sin ky \cdot th\right)\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_3}}\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_1} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;t\_2 \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;t\_2 \leq 0.996:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- 1.0 (cos (* kx -2.0))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (fma -0.5 (cos (* ky -2.0)) 0.5))
        (t_4 (* (sqrt (/ 1.0 (fma 0.5 t_1 t_3))) (* (sin ky) th))))
   (if (<= t_2 -1.0)
     (* (sin th) (/ (sin ky) (sqrt t_3)))
     (if (<= t_2 -0.2)
       t_4
       (if (<= t_2 4e-144)
         (* (sin th) (/ (sin ky) (* (sqrt t_1) (sqrt 0.5))))
         (if (<= t_2 0.1)
           (* (sin th) (/ (sin ky) (sin kx)))
           (if (<= t_2 0.996)
             t_4
             (*
              (sin th)
              (/ (sin ky) (fma kx (* kx (/ 0.5 ky)) (sin ky)))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = 1.0 - cos((kx * -2.0));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = fma(-0.5, cos((ky * -2.0)), 0.5);
	double t_4 = sqrt((1.0 / fma(0.5, t_1, t_3))) * (sin(ky) * th);
	double tmp;
	if (t_2 <= -1.0) {
		tmp = sin(th) * (sin(ky) / sqrt(t_3));
	} else if (t_2 <= -0.2) {
		tmp = t_4;
	} else if (t_2 <= 4e-144) {
		tmp = sin(th) * (sin(ky) / (sqrt(t_1) * sqrt(0.5)));
	} else if (t_2 <= 0.1) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else if (t_2 <= 0.996) {
		tmp = t_4;
	} else {
		tmp = sin(th) * (sin(ky) / fma(kx, (kx * (0.5 / ky)), sin(ky)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(1.0 - cos(Float64(kx * -2.0)))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = fma(-0.5, cos(Float64(ky * -2.0)), 0.5)
	t_4 = Float64(sqrt(Float64(1.0 / fma(0.5, t_1, t_3))) * Float64(sin(ky) * th))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(t_3)));
	elseif (t_2 <= -0.2)
		tmp = t_4;
	elseif (t_2 <= 4e-144)
		tmp = Float64(sin(th) * Float64(sin(ky) / Float64(sqrt(t_1) * sqrt(0.5))));
	elseif (t_2 <= 0.1)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	elseif (t_2 <= 0.996)
		tmp = t_4;
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / fma(kx, Float64(kx * Float64(0.5 / ky)), sin(ky))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(1.0 / N[(0.5 * t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], t$95$4, If[LessEqual[t$95$2, 4e-144], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.1], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.996], t$95$4, N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(kx * N[(kx * N[(0.5 / ky), $MachinePrecision]), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1

   1. Initial program 90.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 71.2

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

   7. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 71.4%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.996

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, 1 - \cos \left(2 \cdot kx\right), \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \]

      9. metadata-eval N/A

         \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{1}{2}, 1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right), \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{1}{2}, 1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \]

      11. lower--.f64 N/A

          \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \]

      12. cos-neg N/A

          \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{1}{2}, 1 - \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \]

      13. lower-cos.f64 N/A

          \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{1}{2}, 1 - \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \]

      14. *-commutative N/A

          \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{1}{2}, 1 - \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{1}{2}, 1 - \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \]

      16. +-commutative N/A

          \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{1}{2}, 1 - \cos \left(kx \cdot -2\right), \color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}}\right)}} \]

   7. Applied rewrites 57.5%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-144

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. count-2 N/A

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

      10. cos-diff N/A

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

      11. cos-sin-sum N/A

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

      12. lower--.f64 N/A

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

      13. count-2 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-+.f64 80.7

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{\frac{1}{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{\frac{1}{2}}}} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      11. lower-sqrt.f64 78.0

          \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \color{blue}{\sqrt{0.5}}} \cdot \sin th \]

   7. Applied rewrites 78.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}} \cdot \sin th \]

   if 3.9999999999999998e-144 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 55.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 55.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 84.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + \frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky} + \sin ky}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2}}{\sin ky} \cdot \frac{1}{2}} + \sin ky} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2} \cdot \frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{kx}^{2} \cdot \frac{\frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{\sin ky} + \sin ky} \cdot \sin th \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)} + \sin ky} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left(kx \cdot kx\right)} \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right) + \sin ky} \cdot \sin th \]

      8. associate-*l* N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{kx \cdot \left(kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)\right)} + \sin ky} \cdot \sin th \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right), \sin ky\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, \color{blue}{kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)}, \sin ky\right)} \cdot \sin th \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2} \cdot 1}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\color{blue}{\frac{1}{2}}}{\sin ky}, \sin ky\right)} \cdot \sin th \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2}}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      14. lower-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      15. lower-sin.f64 92.7

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   5. Applied rewrites 92.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 92.0%

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}} \cdot \left(\sin ky \cdot th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.996:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}} \cdot \left(\sin ky \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;t\_1 \leq 0.996:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2
         (/
          (* (sin ky) th)
          (sqrt
           (fma
            (- 1.0 (cos (+ kx kx)))
            0.5
            (+ 0.5 (* -0.5 (cos (+ ky ky)))))))))
   (if (<= t_1 -1.0)
     (* (sin th) (/ (sin ky) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_1 -0.2)
       t_2
       (if (<= t_1 4e-144)
         (*
          (sin th)
          (/ (sin ky) (* (sqrt (- 1.0 (cos (* kx -2.0)))) (sqrt 0.5))))
         (if (<= t_1 0.1)
           (* (sin th) (/ (sin ky) (sin kx)))
           (if (<= t_1 0.996)
             t_2
             (*
              (sin th)
              (/ (sin ky) (fma kx (* kx (/ 0.5 ky)) (sin ky)))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = (sin(ky) * th) / sqrt(fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky))))));
	double tmp;
	if (t_1 <= -1.0) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_1 <= -0.2) {
		tmp = t_2;
	} else if (t_1 <= 4e-144) {
		tmp = sin(th) * (sin(ky) / (sqrt((1.0 - cos((kx * -2.0)))) * sqrt(0.5)));
	} else if (t_1 <= 0.1) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else if (t_1 <= 0.996) {
		tmp = t_2;
	} else {
		tmp = sin(th) * (sin(ky) / fma(kx, (kx * (0.5 / ky)), sin(ky)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(Float64(sin(ky) * th) / sqrt(fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_1 <= -0.2)
		tmp = t_2;
	elseif (t_1 <= 4e-144)
		tmp = Float64(sin(th) * Float64(sin(ky) / Float64(sqrt(Float64(1.0 - cos(Float64(kx * -2.0)))) * sqrt(0.5))));
	elseif (t_1 <= 0.1)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	elseif (t_1 <= 0.996)
		tmp = t_2;
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / fma(kx, Float64(kx * Float64(0.5 / ky)), sin(ky))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], t$95$2, If[LessEqual[t$95$1, 4e-144], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.996], t$95$2, N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(kx * N[(kx * N[(0.5 / ky), $MachinePrecision]), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1

   1. Initial program 90.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 71.2

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

   7. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 71.4%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.996

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. lower-*.f64 99.3

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \]

      11. sin-mult N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \]

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 95.5%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \]

      2. lower-sin.f64 57.3

         \[\leadsto \frac{th \cdot \color{blue}{\sin ky}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \]

   7. Applied rewrites 57.3%

      \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \]

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-144

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. count-2 N/A

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

      10. cos-diff N/A

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

      11. cos-sin-sum N/A

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

      12. lower--.f64 N/A

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

      13. count-2 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-+.f64 80.7

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{\frac{1}{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{\frac{1}{2}}}} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      11. lower-sqrt.f64 78.0

          \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \color{blue}{\sqrt{0.5}}} \cdot \sin th \]

   7. Applied rewrites 78.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}} \cdot \sin th \]

   if 3.9999999999999998e-144 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 55.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 55.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 84.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + \frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky} + \sin ky}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2}}{\sin ky} \cdot \frac{1}{2}} + \sin ky} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2} \cdot \frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{kx}^{2} \cdot \frac{\frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{\sin ky} + \sin ky} \cdot \sin th \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)} + \sin ky} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left(kx \cdot kx\right)} \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right) + \sin ky} \cdot \sin th \]

      8. associate-*l* N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{kx \cdot \left(kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)\right)} + \sin ky} \cdot \sin th \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right), \sin ky\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, \color{blue}{kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)}, \sin ky\right)} \cdot \sin th \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2} \cdot 1}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\color{blue}{\frac{1}{2}}}{\sin ky}, \sin ky\right)} \cdot \sin th \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2}}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      14. lower-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      15. lower-sin.f64 92.7

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   5. Applied rewrites 92.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 92.0%

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.996:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_1 + kx \cdot kx}}\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;t\_2 \leq 0.005:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{elif}\;t\_2 \leq 0.996:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \sin ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -1.0)
     (* (sin th) (/ (sin ky) (sqrt (+ t_1 (* kx kx)))))
     (if (<= t_2 -0.2)
       (*
        (/ (sin ky) (hypot (sin ky) (sin kx)))
        (fma th (* -0.16666666666666666 (* th th)) th))
       (if (<= t_2 0.005)
         (*
          (sin th)
          (/
           (sin ky)
           (hypot (fma ky (* (* ky ky) -0.16666666666666666) ky) (sin kx))))
         (if (<= t_2 0.996)
           (/
            1.0
            (/
             (sqrt
              (fma
               (- 1.0 (cos (+ kx kx)))
               0.5
               (+ 0.5 (* -0.5 (cos (+ ky ky))))))
             (*
              (sin ky)
              (fma
               th
               (*
                (* th th)
                (fma 0.008333333333333333 (* th th) -0.16666666666666666))
               th))))
           (*
            (sin th)
            (/ (sin ky) (fma kx (* kx (/ 0.5 (sin ky))) (sin ky))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = sin(th) * (sin(ky) / sqrt((t_1 + (kx * kx))));
	} else if (t_2 <= -0.2) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * fma(th, (-0.16666666666666666 * (th * th)), th);
	} else if (t_2 <= 0.005) {
		tmp = sin(th) * (sin(ky) / hypot(fma(ky, ((ky * ky) * -0.16666666666666666), ky), sin(kx)));
	} else if (t_2 <= 0.996) {
		tmp = 1.0 / (sqrt(fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky)))))) / (sin(ky) * fma(th, ((th * th) * fma(0.008333333333333333, (th * th), -0.16666666666666666)), th)));
	} else {
		tmp = sin(th) * (sin(ky) / fma(kx, (kx * (0.5 / sin(ky))), sin(ky)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(Float64(t_1 + Float64(kx * kx)))));
	elseif (t_2 <= -0.2)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th));
	elseif (t_2 <= 0.005)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(fma(ky, Float64(Float64(ky * ky) * -0.16666666666666666), ky), sin(kx))));
	elseif (t_2 <= 0.996)
		tmp = Float64(1.0 / Float64(sqrt(fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))) / Float64(sin(ky) * fma(th, Float64(Float64(th * th) * fma(0.008333333333333333, Float64(th * th), -0.16666666666666666)), th))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / fma(kx, Float64(kx * Float64(0.5 / sin(ky))), sin(ky))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.005], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.996], N[(1.0 / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * N[(th * N[(N[(th * th), $MachinePrecision] * N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(kx * N[(kx * N[(0.5 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1

   1. Initial program 90.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 90.9

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 90.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 99.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.2

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 63.3

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   7. Applied rewrites 63.3%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0050000000000000001

   1. Initial program 98.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.5

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}, \sin kx\right)} \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1}, \sin kx\right)} \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}, \sin kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}, \sin kx\right)} \cdot \sin th \]

      5. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right), \sin kx\right)} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right), \sin kx\right)} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \frac{-1}{6}, ky\right), \sin kx\right)} \cdot \sin th \]

      8. lower-*.f64 97.5

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot -0.16666666666666666, ky\right), \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 97.5%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}, \sin kx\right)} \cdot \sin th \]

   if 0.0050000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.996

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 96.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\left(th \cdot \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \left(th \cdot \color{blue}{\left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right) + 1\right)}\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\left(th \cdot \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) + th \cdot 1\right)}}} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \left(th \cdot \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) + \color{blue}{th}\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\mathsf{fma}\left(th, {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right)}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \color{blue}{{th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)}, th\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right)} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right)} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right)}} \]

      8. sub-neg N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \color{blue}{\left(\frac{1}{120} \cdot {th}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, th\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \left(\frac{1}{120} \cdot {th}^{2} + \color{blue}{\frac{-1}{6}}\right), th\right)}} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {th}^{2}, \frac{-1}{6}\right)}, th\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{th \cdot th}, \frac{-1}{6}\right), th\right)}} \]

      12. lower-*.f64 55.8

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, \color{blue}{th \cdot th}, -0.16666666666666666\right), th\right)}} \]

   7. Applied rewrites 55.8%

      \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)}}} \]

   if 0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 84.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + \frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky} + \sin ky}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2}}{\sin ky} \cdot \frac{1}{2}} + \sin ky} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2} \cdot \frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{kx}^{2} \cdot \frac{\frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{\sin ky} + \sin ky} \cdot \sin th \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)} + \sin ky} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left(kx \cdot kx\right)} \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right) + \sin ky} \cdot \sin th \]

      8. associate-*l* N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{kx \cdot \left(kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)\right)} + \sin ky} \cdot \sin th \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right), \sin ky\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, \color{blue}{kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)}, \sin ky\right)} \cdot \sin th \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2} \cdot 1}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\color{blue}{\frac{1}{2}}}{\sin ky}, \sin ky\right)} \cdot \sin th \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2}}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      14. lower-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      15. lower-sin.f64 92.7

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   5. Applied rewrites 92.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \sin ky\right)}} \cdot \sin th \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin ky}^{2} + kx \cdot kx}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.005:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.996:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_1 + kx \cdot kx}}\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;t\_2 \leq 0.005:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{elif}\;t\_2 \leq 0.996:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -1.0)
     (* (sin th) (/ (sin ky) (sqrt (+ t_1 (* kx kx)))))
     (if (<= t_2 -0.2)
       (*
        (/ (sin ky) (hypot (sin ky) (sin kx)))
        (fma th (* -0.16666666666666666 (* th th)) th))
       (if (<= t_2 0.005)
         (*
          (sin th)
          (/
           (sin ky)
           (hypot (fma ky (* (* ky ky) -0.16666666666666666) ky) (sin kx))))
         (if (<= t_2 0.996)
           (/
            1.0
            (/
             (sqrt
              (fma
               (- 1.0 (cos (+ kx kx)))
               0.5
               (+ 0.5 (* -0.5 (cos (+ ky ky))))))
             (*
              (sin ky)
              (fma
               th
               (*
                (* th th)
                (fma 0.008333333333333333 (* th th) -0.16666666666666666))
               th))))
           (* (sin th) (/ (sin ky) (fma kx (* kx (/ 0.5 ky)) (sin ky))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = sin(th) * (sin(ky) / sqrt((t_1 + (kx * kx))));
	} else if (t_2 <= -0.2) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * fma(th, (-0.16666666666666666 * (th * th)), th);
	} else if (t_2 <= 0.005) {
		tmp = sin(th) * (sin(ky) / hypot(fma(ky, ((ky * ky) * -0.16666666666666666), ky), sin(kx)));
	} else if (t_2 <= 0.996) {
		tmp = 1.0 / (sqrt(fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky)))))) / (sin(ky) * fma(th, ((th * th) * fma(0.008333333333333333, (th * th), -0.16666666666666666)), th)));
	} else {
		tmp = sin(th) * (sin(ky) / fma(kx, (kx * (0.5 / ky)), sin(ky)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(Float64(t_1 + Float64(kx * kx)))));
	elseif (t_2 <= -0.2)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th));
	elseif (t_2 <= 0.005)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(fma(ky, Float64(Float64(ky * ky) * -0.16666666666666666), ky), sin(kx))));
	elseif (t_2 <= 0.996)
		tmp = Float64(1.0 / Float64(sqrt(fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))) / Float64(sin(ky) * fma(th, Float64(Float64(th * th) * fma(0.008333333333333333, Float64(th * th), -0.16666666666666666)), th))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / fma(kx, Float64(kx * Float64(0.5 / ky)), sin(ky))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.005], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.996], N[(1.0 / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * N[(th * N[(N[(th * th), $MachinePrecision] * N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(kx * N[(kx * N[(0.5 / ky), $MachinePrecision]), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1

   1. Initial program 90.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 90.9

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 90.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 99.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.2

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 63.3

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   7. Applied rewrites 63.3%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0050000000000000001

   1. Initial program 98.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.5

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}, \sin kx\right)} \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1}, \sin kx\right)} \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}, \sin kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}, \sin kx\right)} \cdot \sin th \]

      5. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right), \sin kx\right)} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right), \sin kx\right)} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \frac{-1}{6}, ky\right), \sin kx\right)} \cdot \sin th \]

      8. lower-*.f64 97.5

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot -0.16666666666666666, ky\right), \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 97.5%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}, \sin kx\right)} \cdot \sin th \]

   if 0.0050000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.996

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 96.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\left(th \cdot \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \left(th \cdot \color{blue}{\left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right) + 1\right)}\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\left(th \cdot \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) + th \cdot 1\right)}}} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \left(th \cdot \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) + \color{blue}{th}\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\mathsf{fma}\left(th, {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right)}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \color{blue}{{th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)}, th\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right)} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right)} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right)}} \]

      8. sub-neg N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \color{blue}{\left(\frac{1}{120} \cdot {th}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, th\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \left(\frac{1}{120} \cdot {th}^{2} + \color{blue}{\frac{-1}{6}}\right), th\right)}} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {th}^{2}, \frac{-1}{6}\right)}, th\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{th \cdot th}, \frac{-1}{6}\right), th\right)}} \]

      12. lower-*.f64 55.8

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, \color{blue}{th \cdot th}, -0.16666666666666666\right), th\right)}} \]

   7. Applied rewrites 55.8%

      \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)}}} \]

   if 0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 84.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + \frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky} + \sin ky}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2}}{\sin ky} \cdot \frac{1}{2}} + \sin ky} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2} \cdot \frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{kx}^{2} \cdot \frac{\frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{\sin ky} + \sin ky} \cdot \sin th \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)} + \sin ky} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left(kx \cdot kx\right)} \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right) + \sin ky} \cdot \sin th \]

      8. associate-*l* N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{kx \cdot \left(kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)\right)} + \sin ky} \cdot \sin th \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right), \sin ky\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, \color{blue}{kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)}, \sin ky\right)} \cdot \sin th \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2} \cdot 1}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\color{blue}{\frac{1}{2}}}{\sin ky}, \sin ky\right)} \cdot \sin th \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2}}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      14. lower-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      15. lower-sin.f64 92.7

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   5. Applied rewrites 92.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 92.0%

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin ky}^{2} + kx \cdot kx}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.005:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.996:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;t\_1 \leq 0.005:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{elif}\;t\_1 \leq 0.996:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -1.0)
     (* (sin th) (/ (sin ky) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_1 -0.2)
       (*
        (/ (sin ky) (hypot (sin ky) (sin kx)))
        (fma th (* -0.16666666666666666 (* th th)) th))
       (if (<= t_1 0.005)
         (*
          (sin th)
          (/
           (sin ky)
           (hypot (fma ky (* (* ky ky) -0.16666666666666666) ky) (sin kx))))
         (if (<= t_1 0.996)
           (/
            1.0
            (/
             (sqrt
              (fma
               (- 1.0 (cos (+ kx kx)))
               0.5
               (+ 0.5 (* -0.5 (cos (+ ky ky))))))
             (*
              (sin ky)
              (fma
               th
               (*
                (* th th)
                (fma 0.008333333333333333 (* th th) -0.16666666666666666))
               th))))
           (* (sin th) (/ (sin ky) (fma kx (* kx (/ 0.5 ky)) (sin ky))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -1.0) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_1 <= -0.2) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * fma(th, (-0.16666666666666666 * (th * th)), th);
	} else if (t_1 <= 0.005) {
		tmp = sin(th) * (sin(ky) / hypot(fma(ky, ((ky * ky) * -0.16666666666666666), ky), sin(kx)));
	} else if (t_1 <= 0.996) {
		tmp = 1.0 / (sqrt(fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky)))))) / (sin(ky) * fma(th, ((th * th) * fma(0.008333333333333333, (th * th), -0.16666666666666666)), th)));
	} else {
		tmp = sin(th) * (sin(ky) / fma(kx, (kx * (0.5 / ky)), sin(ky)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_1 <= -0.2)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th));
	elseif (t_1 <= 0.005)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(fma(ky, Float64(Float64(ky * ky) * -0.16666666666666666), ky), sin(kx))));
	elseif (t_1 <= 0.996)
		tmp = Float64(1.0 / Float64(sqrt(fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))) / Float64(sin(ky) * fma(th, Float64(Float64(th * th) * fma(0.008333333333333333, Float64(th * th), -0.16666666666666666)), th))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / fma(kx, Float64(kx * Float64(0.5 / ky)), sin(ky))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.005], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.996], N[(1.0 / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * N[(th * N[(N[(th * th), $MachinePrecision] * N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(kx * N[(kx * N[(0.5 / ky), $MachinePrecision]), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1

   1. Initial program 90.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 71.2

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

   7. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 71.4%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 99.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.2

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 63.3

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   7. Applied rewrites 63.3%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0050000000000000001

   1. Initial program 98.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.5

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}, \sin kx\right)} \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1}, \sin kx\right)} \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}, \sin kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}, \sin kx\right)} \cdot \sin th \]

      5. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right), \sin kx\right)} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right), \sin kx\right)} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \frac{-1}{6}, ky\right), \sin kx\right)} \cdot \sin th \]

      8. lower-*.f64 97.5

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot -0.16666666666666666, ky\right), \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 97.5%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}, \sin kx\right)} \cdot \sin th \]

   if 0.0050000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.996

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 96.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\left(th \cdot \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \left(th \cdot \color{blue}{\left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right) + 1\right)}\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\left(th \cdot \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) + th \cdot 1\right)}}} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \left(th \cdot \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) + \color{blue}{th}\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\mathsf{fma}\left(th, {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right)}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \color{blue}{{th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)}, th\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right)} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right)} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right)}} \]

      8. sub-neg N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \color{blue}{\left(\frac{1}{120} \cdot {th}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, th\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \left(\frac{1}{120} \cdot {th}^{2} + \color{blue}{\frac{-1}{6}}\right), th\right)}} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {th}^{2}, \frac{-1}{6}\right)}, th\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{th \cdot th}, \frac{-1}{6}\right), th\right)}} \]

      12. lower-*.f64 55.8

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, \color{blue}{th \cdot th}, -0.16666666666666666\right), th\right)}} \]

   7. Applied rewrites 55.8%

      \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)}}} \]

   if 0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 84.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + \frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky} + \sin ky}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2}}{\sin ky} \cdot \frac{1}{2}} + \sin ky} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2} \cdot \frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{kx}^{2} \cdot \frac{\frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{\sin ky} + \sin ky} \cdot \sin th \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)} + \sin ky} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left(kx \cdot kx\right)} \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right) + \sin ky} \cdot \sin th \]

      8. associate-*l* N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{kx \cdot \left(kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)\right)} + \sin ky} \cdot \sin th \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right), \sin ky\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, \color{blue}{kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)}, \sin ky\right)} \cdot \sin th \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2} \cdot 1}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\color{blue}{\frac{1}{2}}}{\sin ky}, \sin ky\right)} \cdot \sin th \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2}}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      14. lower-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      15. lower-sin.f64 92.7

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   5. Applied rewrites 92.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 92.0%

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.005:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.996:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_2}}\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), t\_2\right)}} \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.005:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{elif}\;t\_1 \leq 0.996:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (fma -0.5 (cos (* ky -2.0)) 0.5)))
   (if (<= t_1 -1.0)
     (* (sin th) (/ (sin ky) (sqrt t_2)))
     (if (<= t_1 -0.2)
       (*
        th
        (*
         (sqrt (/ 1.0 (fma 0.5 (- 1.0 (cos (* kx -2.0))) t_2)))
         (* (sin ky) (fma -0.16666666666666666 (* th th) 1.0))))
       (if (<= t_1 0.005)
         (*
          (sin th)
          (/
           (sin ky)
           (hypot (fma ky (* (* ky ky) -0.16666666666666666) ky) (sin kx))))
         (if (<= t_1 0.996)
           (/
            1.0
            (/
             (sqrt
              (fma
               (- 1.0 (cos (+ kx kx)))
               0.5
               (+ 0.5 (* -0.5 (cos (+ ky ky))))))
             (*
              (sin ky)
              (fma
               th
               (*
                (* th th)
                (fma 0.008333333333333333 (* th th) -0.16666666666666666))
               th))))
           (* (sin th) (/ (sin ky) (fma kx (* kx (/ 0.5 ky)) (sin ky))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = fma(-0.5, cos((ky * -2.0)), 0.5);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = sin(th) * (sin(ky) / sqrt(t_2));
	} else if (t_1 <= -0.2) {
		tmp = th * (sqrt((1.0 / fma(0.5, (1.0 - cos((kx * -2.0))), t_2))) * (sin(ky) * fma(-0.16666666666666666, (th * th), 1.0)));
	} else if (t_1 <= 0.005) {
		tmp = sin(th) * (sin(ky) / hypot(fma(ky, ((ky * ky) * -0.16666666666666666), ky), sin(kx)));
	} else if (t_1 <= 0.996) {
		tmp = 1.0 / (sqrt(fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky)))))) / (sin(ky) * fma(th, ((th * th) * fma(0.008333333333333333, (th * th), -0.16666666666666666)), th)));
	} else {
		tmp = sin(th) * (sin(ky) / fma(kx, (kx * (0.5 / ky)), sin(ky)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = fma(-0.5, cos(Float64(ky * -2.0)), 0.5)
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(t_2)));
	elseif (t_1 <= -0.2)
		tmp = Float64(th * Float64(sqrt(Float64(1.0 / fma(0.5, Float64(1.0 - cos(Float64(kx * -2.0))), t_2))) * Float64(sin(ky) * fma(-0.16666666666666666, Float64(th * th), 1.0))));
	elseif (t_1 <= 0.005)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(fma(ky, Float64(Float64(ky * ky) * -0.16666666666666666), ky), sin(kx))));
	elseif (t_1 <= 0.996)
		tmp = Float64(1.0 / Float64(sqrt(fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))) / Float64(sin(ky) * fma(th, Float64(Float64(th * th) * fma(0.008333333333333333, Float64(th * th), -0.16666666666666666)), th))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / fma(kx, Float64(kx * Float64(0.5 / ky)), sin(ky))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], N[(th * N[(N[Sqrt[N[(1.0 / N[(0.5 * N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.005], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.996], N[(1.0 / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * N[(th * N[(N[(th * th), $MachinePrecision] * N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(kx * N[(kx * N[(0.5 / ky), $MachinePrecision]), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1

   1. Initial program 90.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 71.2

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

   7. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 71.4%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 99.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(\frac{-1}{6} \cdot \left(\left({th}^{2} \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}\right) + \sin ky \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}\right)} \]

   7. Applied rewrites 58.9%

      \[\leadsto \color{blue}{th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}} \cdot \left(\mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right) \cdot \sin ky\right)\right)} \]

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0050000000000000001

   1. Initial program 98.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.5

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}, \sin kx\right)} \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1}, \sin kx\right)} \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}, \sin kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}, \sin kx\right)} \cdot \sin th \]

      5. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right), \sin kx\right)} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right), \sin kx\right)} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \frac{-1}{6}, ky\right), \sin kx\right)} \cdot \sin th \]

      8. lower-*.f64 97.5

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot -0.16666666666666666, ky\right), \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 97.5%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}, \sin kx\right)} \cdot \sin th \]

   if 0.0050000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.996

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 96.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\left(th \cdot \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \left(th \cdot \color{blue}{\left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right) + 1\right)}\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\left(th \cdot \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) + th \cdot 1\right)}}} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \left(th \cdot \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) + \color{blue}{th}\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\mathsf{fma}\left(th, {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right)}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \color{blue}{{th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)}, th\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right)} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right)} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right)}} \]

      8. sub-neg N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \color{blue}{\left(\frac{1}{120} \cdot {th}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, th\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \left(\frac{1}{120} \cdot {th}^{2} + \color{blue}{\frac{-1}{6}}\right), th\right)}} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {th}^{2}, \frac{-1}{6}\right)}, th\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{th \cdot th}, \frac{-1}{6}\right), th\right)}} \]

      12. lower-*.f64 55.8

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, \color{blue}{th \cdot th}, -0.16666666666666666\right), th\right)}} \]

   7. Applied rewrites 55.8%

      \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \color{blue}{\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)}}} \]

   if 0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 84.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + \frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky} + \sin ky}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2}}{\sin ky} \cdot \frac{1}{2}} + \sin ky} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2} \cdot \frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{kx}^{2} \cdot \frac{\frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{\sin ky} + \sin ky} \cdot \sin th \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{{kx}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)} + \sin ky} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left(kx \cdot kx\right)} \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right) + \sin ky} \cdot \sin th \]

      8. associate-*l* N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{kx \cdot \left(kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)\right)} + \sin ky} \cdot \sin th \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right), \sin ky\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, \color{blue}{kx \cdot \left(\frac{1}{2} \cdot \frac{1}{\sin ky}\right)}, \sin ky\right)} \cdot \sin th \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2} \cdot 1}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\color{blue}{\frac{1}{2}}}{\sin ky}, \sin ky\right)} \cdot \sin th \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \color{blue}{\frac{\frac{1}{2}}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      14. lower-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      15. lower-sin.f64 92.7

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   5. Applied rewrites 92.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{\frac{1}{2}}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 92.0%

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}} \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.005:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.996:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{fma}\left(kx, kx \cdot \frac{0.5}{ky}, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.715:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.715)
     (* (sin th) (/ (sin ky) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_1 4e-144)
       (*
        (sin th)
        (/ (sin ky) (* (sqrt (- 1.0 (cos (* kx -2.0)))) (sqrt 0.5))))
       (if (<= t_1 0.1) (* (sin th) (/ (sin ky) (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.715) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_1 <= 4e-144) {
		tmp = sin(th) * (sin(ky) / (sqrt((1.0 - cos((kx * -2.0)))) * sqrt(0.5)));
	} else if (t_1 <= 0.1) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.715)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_1 <= 4e-144)
		tmp = Float64(sin(th) * Float64(sin(ky) / Float64(sqrt(Float64(1.0 - cos(Float64(kx * -2.0)))) * sqrt(0.5))));
	elseif (t_1 <= 0.1)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.715], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-144], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.714999999999999969

   1. Initial program 92.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 62.3

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

   7. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 62.4%

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

   if -0.714999999999999969 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-144

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. count-2 N/A

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

      10. cos-diff N/A

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

      11. cos-sin-sum N/A

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

      12. lower--.f64 N/A

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

      13. count-2 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-+.f64 82.7

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

   4. Applied rewrites 82.7%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{\frac{1}{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{\frac{1}{2}}}} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \sqrt{\frac{1}{2}}} \cdot \sin th \]

      11. lower-sqrt.f64 71.7

          \[\leadsto \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \color{blue}{\sqrt{0.5}}} \cdot \sin th \]

   7. Applied rewrites 71.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}} \cdot \sin th \]

   if 3.9999999999999998e-144 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 55.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 55.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.715:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.715:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{0.5 \cdot \left(1 - \cos \left(kx \cdot -2\right)\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.715)
     (* (sin th) (/ (sin ky) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_1 4e-144)
       (/ (* (sin ky) (sin th)) (sqrt (* 0.5 (- 1.0 (cos (* kx -2.0))))))
       (if (<= t_1 0.1) (* (sin th) (/ (sin ky) (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.715) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_1 <= 4e-144) {
		tmp = (sin(ky) * sin(th)) / sqrt((0.5 * (1.0 - cos((kx * -2.0)))));
	} else if (t_1 <= 0.1) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.715)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_1 <= 4e-144)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(kx * -2.0))))));
	elseif (t_1 <= 0.1)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.715], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-144], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.714999999999999969

   1. Initial program 92.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 62.3

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

   7. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 62.4%

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

   if -0.714999999999999969 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-144

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. lower-*.f64 99.2

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \]

      11. sin-mult N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \]

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 80.8%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \]

   5. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)}}} \]

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower--.f64 N/A

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

      5. cos-neg N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(1 - \color{blue}{\cos \left(-2 \cdot kx\right)}\right)}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(1 - \color{blue}{\cos \left(-2 \cdot kx\right)}\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \color{blue}{\left(kx \cdot -2\right)}\right)}} \]

      8. lower-*.f64 71.6

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{0.5 \cdot \left(1 - \cos \color{blue}{\left(kx \cdot -2\right)}\right)}} \]

   7. Applied rewrites 71.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{0.5 \cdot \left(1 - \cos \left(kx \cdot -2\right)\right)}}} \]

   if 3.9999999999999998e-144 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 55.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 55.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.715:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{0.5 \cdot \left(1 - \cos \left(kx \cdot -2\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.02)
     (* (sin th) (/ (sin ky) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_1 4e-144)
       (*
        (sin th)
        (* (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0))))) (* ky (sqrt 2.0))))
       (if (<= t_1 0.1) (* (sin th) (/ (sin ky) (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.02) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_1 <= 4e-144) {
		tmp = sin(th) * (sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * sqrt(2.0)));
	} else if (t_1 <= 0.1) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.02)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_1 <= 4e-144)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0))))) * Float64(ky * sqrt(2.0))));
	elseif (t_1 <= 0.1)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.02], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-144], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0200000000000000004

   1. Initial program 93.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 78.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 56.4

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

   7. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.5%

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

   if -0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-144

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. count-2 N/A

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

      10. cos-diff N/A

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

      11. cos-sin-sum N/A

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

      12. lower--.f64 N/A

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

      13. count-2 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-+.f64 80.4

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

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      5. metadata-eval N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower--.f64 N/A

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

      8. cos-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      10. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \color{blue}{\left(ky \cdot \sqrt{2}\right)}\right) \cdot \sin th \]

      13. lower-sqrt.f64 78.3

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \sin th \]

   7. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

   if 3.9999999999999998e-144 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 55.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 55.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.02:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.02)
     (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_1 4e-144)
       (*
        (sin th)
        (* (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0))))) (* ky (sqrt 2.0))))
       (if (<= t_1 0.1) (* (sin th) (/ (sin ky) (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.02) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_1 <= 4e-144) {
		tmp = sin(th) * (sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * sqrt(2.0)));
	} else if (t_1 <= 0.1) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.02)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_1 <= 4e-144)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0))))) * Float64(ky * sqrt(2.0))));
	elseif (t_1 <= 0.1)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.02], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-144], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0200000000000000004

   1. Initial program 93.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 78.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 56.4

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

   7. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.4%

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

   11. Applied rewrites 56.5%

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

   if -0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-144

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. count-2 N/A

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

      10. cos-diff N/A

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

      11. cos-sin-sum N/A

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

      12. lower--.f64 N/A

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

      13. count-2 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-+.f64 80.4

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

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      5. metadata-eval N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower--.f64 N/A

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

      8. cos-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      10. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \color{blue}{\left(ky \cdot \sqrt{2}\right)}\right) \cdot \sin th \]

      13. lower-sqrt.f64 78.3

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \sin th \]

   7. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

   if 3.9999999999999998e-144 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 55.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 55.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.02:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 53.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.2)
     (* th (* (sin ky) (sqrt (/ 1.0 (fma -0.5 (cos (* ky -2.0)) 0.5)))))
     (if (<= t_1 4e-144)
       (*
        (sin th)
        (* (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0))))) (* ky (sqrt 2.0))))
       (if (<= t_1 0.1) (* (sin th) (/ (sin ky) (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = th * (sin(ky) * sqrt((1.0 / fma(-0.5, cos((ky * -2.0)), 0.5))));
	} else if (t_1 <= 4e-144) {
		tmp = sin(th) * (sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * sqrt(2.0)));
	} else if (t_1 <= 0.1) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.2)
		tmp = Float64(th * Float64(sin(ky) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(ky * -2.0)), 0.5)))));
	elseif (t_1 <= 4e-144)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0))))) * Float64(ky * sqrt(2.0))));
	elseif (t_1 <= 0.1)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], N[(th * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-144], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 93.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 57.0

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

   7. Applied rewrites 57.0%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 30.3%

       \[\leadsto th \cdot \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}} \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\right)} \]

   11. Taylor expanded in th around 0

       \[\leadsto th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-2 \cdot ky\right)}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 30.9%

       \[\leadsto th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(-2 \cdot ky\right), 0.5\right)}}\right) \]

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-144

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. count-2 N/A

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

      10. cos-diff N/A

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

      11. cos-sin-sum N/A

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

      12. lower--.f64 N/A

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

      13. count-2 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-+.f64 80.7

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      5. metadata-eval N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower--.f64 N/A

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

      8. cos-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      10. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \color{blue}{\left(ky \cdot \sqrt{2}\right)}\right) \cdot \sin th \]

      13. lower-sqrt.f64 77.2

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \sin th \]

   7. Applied rewrites 77.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

   if 3.9999999999999998e-144 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 55.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 55.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 53.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.2)
     (* th (* (sin ky) (sqrt (/ 1.0 (fma -0.5 (cos (* ky -2.0)) 0.5)))))
     (if (<= t_1 4e-144)
       (*
        (sin th)
        (* (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0))))) (* ky (sqrt 2.0))))
       (if (<= t_1 0.01) (* (sin th) (/ ky (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = th * (sin(ky) * sqrt((1.0 / fma(-0.5, cos((ky * -2.0)), 0.5))));
	} else if (t_1 <= 4e-144) {
		tmp = sin(th) * (sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * sqrt(2.0)));
	} else if (t_1 <= 0.01) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.2)
		tmp = Float64(th * Float64(sin(ky) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(ky * -2.0)), 0.5)))));
	elseif (t_1 <= 4e-144)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0))))) * Float64(ky * sqrt(2.0))));
	elseif (t_1 <= 0.01)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], N[(th * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-144], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.01], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 93.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 57.0

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

   7. Applied rewrites 57.0%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 30.3%

       \[\leadsto th \cdot \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}} \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\right)} \]

   11. Taylor expanded in th around 0

       \[\leadsto th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-2 \cdot ky\right)}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 30.9%

       \[\leadsto th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(-2 \cdot ky\right), 0.5\right)}}\right) \]

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-144

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. count-2 N/A

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

      10. cos-diff N/A

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

      11. cos-sin-sum N/A

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

      12. lower--.f64 N/A

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

      13. count-2 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-+.f64 80.7

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      5. metadata-eval N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower--.f64 N/A

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

      8. cos-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      10. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \color{blue}{\left(ky \cdot \sqrt{2}\right)}\right) \cdot \sin th \]

      13. lower-sqrt.f64 77.2

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \sin th \]

   7. Applied rewrites 77.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

   if 3.9999999999999998e-144 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 95.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 58.7

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 62.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 52.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(\sqrt{2} \cdot \left(ky \cdot \sin th\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.2)
     (* th (* (sin ky) (sqrt (/ 1.0 (fma -0.5 (cos (* ky -2.0)) 0.5)))))
     (if (<= t_1 4e-144)
       (*
        (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0)))))
        (* (sqrt 2.0) (* ky (sin th))))
       (if (<= t_1 0.01) (* (sin th) (/ ky (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = th * (sin(ky) * sqrt((1.0 / fma(-0.5, cos((ky * -2.0)), 0.5))));
	} else if (t_1 <= 4e-144) {
		tmp = sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (sqrt(2.0) * (ky * sin(th)));
	} else if (t_1 <= 0.01) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.2)
		tmp = Float64(th * Float64(sin(ky) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(ky * -2.0)), 0.5)))));
	elseif (t_1 <= 4e-144)
		tmp = Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0))))) * Float64(sqrt(2.0) * Float64(ky * sin(th))));
	elseif (t_1 <= 0.01)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], N[(th * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-144], N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.01], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 93.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 57.0

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

   7. Applied rewrites 57.0%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 30.3%

       \[\leadsto th \cdot \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}} \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\right)} \]

   11. Taylor expanded in th around 0

       \[\leadsto th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-2 \cdot ky\right)}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 30.9%

       \[\leadsto th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(-2 \cdot ky\right), 0.5\right)}}\right) \]

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-144

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 77.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(ky \cdot \sin th\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \]

      5. lower-sin.f64 N/A

         \[\leadsto \left(\left(ky \cdot \color{blue}{\sin th}\right) \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(ky \cdot \sin th\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \]

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower--.f64 N/A

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

      12. cos-neg N/A

          \[\leadsto \left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \]

      13. lower-cos.f64 N/A

          \[\leadsto \left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \]

      14. *-commutative N/A

          \[\leadsto \left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \]

      15. lower-*.f64 76.9

          \[\leadsto \left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \]

   7. Applied rewrites 76.9%

      \[\leadsto \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}}} \]

   if 3.9999999999999998e-144 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 95.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 58.7

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 62.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(\sqrt{2} \cdot \left(ky \cdot \sin th\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 50.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\right)\\ \mathbf{elif}\;t\_1 \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.2)
     (* th (* (sin ky) (sqrt (/ 1.0 (fma -0.5 (cos (* ky -2.0)) 0.5)))))
     (if (<= t_1 0.01) (* (sin th) (/ ky (sin kx))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = th * (sin(ky) * sqrt((1.0 / fma(-0.5, cos((ky * -2.0)), 0.5))));
	} else if (t_1 <= 0.01) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.2)
		tmp = Float64(th * Float64(sin(ky) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(ky * -2.0)), 0.5)))));
	elseif (t_1 <= 0.01)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], N[(th * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.01], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 93.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   4. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 57.0

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

   7. Applied rewrites 57.0%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 30.3%

       \[\leadsto th \cdot \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}} \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\right)} \]

   11. Taylor expanded in th around 0

       \[\leadsto th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-2 \cdot ky\right)}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 30.9%

       \[\leadsto th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(-2 \cdot ky\right), 0.5\right)}}\right) \]

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 98.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 63.6

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 62.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 43.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.01)
   (* (sin th) (/ ky (sin kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.01) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.01d0) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.01) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.01:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.01)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.01)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.01], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 96.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 37.3

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 37.3%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 62.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 43.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.01)
   (/ (* ky (sin th)) (sin kx))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.01) {
		tmp = (ky * sin(th)) / sin(kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.01d0) then
        tmp = (ky * sin(th)) / sin(kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.01) {
		tmp = (ky * Math.sin(th)) / Math.sin(kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.01:
		tmp = (ky * math.sin(th)) / math.sin(kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.01)
		tmp = Float64(Float64(ky * sin(th)) / sin(kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.01)
		tmp = (ky * sin(th)) / sin(kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.01], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 96.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{ky \cdot \sin th}}{\sin kx} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \color{blue}{\sin th}}{\sin kx} \]

      4. lower-sin.f64 36.8

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sin kx}} \]

   5. Applied rewrites 36.8%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 62.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 15.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-295}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;th \cdot \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (sin th)
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
      5e-295)
   (* -0.16666666666666666 (* th (* th th)))
   (*
    th
    (fma
     (* th th)
     (fma 0.008333333333333333 (* th th) -0.16666666666666666)
     1.0))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(th) * (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0))))) <= 5e-295) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = th * fma((th * th), fma(0.008333333333333333, (th * th), -0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(th) * Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 5e-295)
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	else
		tmp = Float64(th * fma(Float64(th * th), fma(0.008333333333333333, Float64(th * th), -0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-295], N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(th * N[(N[(th * th), $MachinePrecision] * N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 5.00000000000000008e-295

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 22.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 22.9%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 13.1%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{-0.16666666666666666 \cdot \left(th \cdot th\right)}, th\right) \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 16.1%

       \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   if 5.00000000000000008e-295 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 94.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 23.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 23.0%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 11.4%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{-0.16666666666666666 \cdot \left(th \cdot th\right)}, th\right) \]

   9. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 11.3%

       \[\leadsto th \cdot \color{blue}{\mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-295}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;th \cdot \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 15.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-295}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (sin th)
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
      5e-295)
   (* -0.16666666666666666 (* th (* th th)))
   (fma th (* -0.16666666666666666 (* th th)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(th) * (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0))))) <= 5e-295) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(th) * Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 5e-295)
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	else
		tmp = fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-295], N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 5.00000000000000008e-295

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 22.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 22.9%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 13.1%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{-0.16666666666666666 \cdot \left(th \cdot th\right)}, th\right) \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 16.1%

       \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   if 5.00000000000000008e-295 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 94.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 23.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 23.0%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 11.4%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{-0.16666666666666666 \cdot \left(th \cdot th\right)}, th\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-295}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 30.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-90}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 2e-90)
   (* -0.16666666666666666 (* th (* th th)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 2e-90) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 2d-90) then
        tmp = (-0.16666666666666666d0) * (th * (th * th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 2e-90) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 2e-90:
		tmp = -0.16666666666666666 * (th * (th * th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-90)
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-90)
		tmp = -0.16666666666666666 * (th * (th * th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-90], N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.99999999999999999e-90

   1. Initial program 96.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 3.5%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 3.4%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{-0.16666666666666666 \cdot \left(th \cdot th\right)}, th\right) \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 14.7%

       \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   if 1.99999999999999999e-90 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 53.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 75.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.003:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.003)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot (fma ky (* (* ky ky) -0.16666666666666666) ky) (sin kx))))
   (*
    (* (sin ky) (sin th))
    (sqrt
     (/
      1.0
      (fma (- 1.0 (cos (+ kx kx))) 0.5 (+ 0.5 (* -0.5 (cos (+ ky ky))))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.003) {
		tmp = sin(th) * (sin(ky) / hypot(fma(ky, ((ky * ky) * -0.16666666666666666), ky), sin(kx)));
	} else {
		tmp = (sin(ky) * sin(th)) * sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky)))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.003)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(fma(ky, Float64(Float64(ky * ky) * -0.16666666666666666), ky), sin(kx))));
	else
		tmp = Float64(Float64(sin(ky) * sin(th)) * sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky))))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.003], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.0030000000000000001

   1. Initial program 92.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.6

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}, \sin kx\right)} \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1}, \sin kx\right)} \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}, \sin kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}, \sin kx\right)} \cdot \sin th \]

      5. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right), \sin kx\right)} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right), \sin kx\right)} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \frac{-1}{6}, ky\right), \sin kx\right)} \cdot \sin th \]

      8. lower-*.f64 67.0

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot -0.16666666666666666, ky\right), \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 67.0%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}, \sin kx\right)} \cdot \sin th \]

   if 0.0030000000000000001 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.003:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 75.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.0032:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.0032)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot (fma ky (* (* ky ky) -0.16666666666666666) ky) (sin kx))))
   (/
    (* (sin ky) (sin th))
    (sqrt (fma (- 1.0 (cos (+ kx kx))) 0.5 (fma (cos (+ ky ky)) -0.5 0.5))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0032) {
		tmp = sin(th) * (sin(ky) / hypot(fma(ky, ((ky * ky) * -0.16666666666666666), ky), sin(kx)));
	} else {
		tmp = (sin(ky) * sin(th)) / sqrt(fma((1.0 - cos((kx + kx))), 0.5, fma(cos((ky + ky)), -0.5, 0.5)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.0032)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(fma(ky, Float64(Float64(ky * ky) * -0.16666666666666666), ky), sin(kx))));
	else
		tmp = Float64(Float64(sin(ky) * sin(th)) / sqrt(fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, fma(cos(Float64(ky + ky)), -0.5, 0.5))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.0032], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.00320000000000000015

   1. Initial program 92.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.6

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}, \sin kx\right)} \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1}, \sin kx\right)} \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}, \sin kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}, \sin kx\right)} \cdot \sin th \]

      5. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right), \sin kx\right)} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right), \sin kx\right)} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \frac{-1}{6}, ky\right), \sin kx\right)} \cdot \sin th \]

      8. lower-*.f64 67.0

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot -0.16666666666666666, ky\right), \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 67.0%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}, \sin kx\right)} \cdot \sin th \]

   if 0.00320000000000000015 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. lower-*.f64 99.5

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \]

      11. sin-mult N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \]

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{\frac{-1}{2} \cdot \cos \left(ky + ky\right) + \frac{1}{2}}\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{\frac{-1}{2} \cdot \cos \left(ky + ky\right)} + \frac{1}{2}\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{\cos \left(ky + ky\right) \cdot \frac{-1}{2}} + \frac{1}{2}\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \cos \color{blue}{\left(ky + ky\right)} \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \]

      6. flip-+ N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \cos \color{blue}{\left(\frac{ky \cdot ky - ky \cdot ky}{ky - ky}\right)} \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \]

      7. +-inverses N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \cos \left(\frac{\color{blue}{0}}{ky - ky}\right) \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \]

      8. +-inverses N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \cos \left(\frac{\color{blue}{kx \cdot kx - kx \cdot kx}}{ky - ky}\right) \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \]

      9. +-inverses N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \cos \left(\frac{kx \cdot kx - kx \cdot kx}{\color{blue}{0}}\right) \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \]

      10. +-inverses N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \cos \left(\frac{kx \cdot kx - kx \cdot kx}{\color{blue}{kx - kx}}\right) \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \]

      11. flip-+ N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \]

      13. lower-fma.f64 11.3

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \color{blue}{\mathsf{fma}\left(\cos \left(kx + kx\right), -0.5, 0.5\right)}\right)}} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \mathsf{fma}\left(\cos \color{blue}{\left(kx + kx\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}} \]

      15. flip-+ N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \mathsf{fma}\left(\cos \color{blue}{\left(\frac{kx \cdot kx - kx \cdot kx}{kx - kx}\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}} \]

      16. +-inverses N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{\color{blue}{0}}{kx - kx}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \]

      17. +-inverses N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{\color{blue}{ky \cdot ky - ky \cdot ky}}{kx - kx}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \]

      18. +-inverses N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{ky \cdot ky - ky \cdot ky}{\color{blue}{0}}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \]

      19. +-inverses N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{ky \cdot ky - ky \cdot ky}{\color{blue}{ky - ky}}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \]

      20. flip-+ N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \mathsf{fma}\left(\cos \color{blue}{\left(ky + ky\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}} \]

      21. lift-+.f64 97.1

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \mathsf{fma}\left(\cos \color{blue}{\left(ky + ky\right)}, -0.5, 0.5\right)\right)}} \]

   6. Applied rewrites 97.1%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \color{blue}{\mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.0032:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 11.0% accurate, 39.5× speedup

Math

\[\begin{array}{l} \\ -0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right) \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* -0.16666666666666666 (* th (* th th))))

C

double code(double kx, double ky, double th) {
	return -0.16666666666666666 * (th * (th * th));
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (-0.16666666666666666d0) * (th * (th * th))
end function

Java

public static double code(double kx, double ky, double th) {
	return -0.16666666666666666 * (th * (th * th));
}

Python

def code(kx, ky, th):
	return -0.16666666666666666 * (th * (th * th))

Julia

function code(kx, ky, th)
	return Float64(-0.16666666666666666 * Float64(th * Float64(th * th)))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = -0.16666666666666666 * (th * (th * th));
end

Wolfram

code[kx_, ky_, th_] := N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 23.0

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 23.0%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

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

8. Applied rewrites 12.3%

   \[\leadsto \mathsf{fma}\left(th, \color{blue}{-0.16666666666666666 \cdot \left(th \cdot th\right)}, th\right) \]

9. Taylor expanded in th around inf

   \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
10. Step-by-step derivation

11. Applied rewrites 10.5%

    \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  :precision binary64
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))